Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 658936 | 37 pages | https://doi.org/10.1155/2011/658936

Solutions of Smooth Nonlinear Partial Differential Equations

Academic Editor: Stephen Clark
Received01 Feb 2011
Accepted15 Apr 2011
Published30 Jun 2011

Abstract

The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrรถdinger equation in one spatial dimension.

1. Introduction

In the 1994 monograph [1] Oberguggenberger and Rosinger presented a general and type-independent theory for the existence and basic regularity of the solutions of a large class of systems of nonlinear PDEs, based on the Dedekind order completion of spaces of piecewise smooth functions. In the mentioned monograph, it is shown that the solutions satisfy a blanket regularity property. Namely, the solutions may be assimilated with usual real measurable functions, or even nearly finite Hausdorff continuous functions [2], defined on the Euclidean domain of definition the respective system of equations. The latter result is based on the highly nontrivial characterization of the Dedekind order completion of sets of continuous functions in terms of spaces of Hausdorff continuous interval valued functions [3]. Recently, the regularity of the solutions constructed through the order completion method has been significantly improved upon by introducing suitable uniform convergence structures on appropriate spaces of piecewise smooth functions; see [4โ€“7]. This new approach also gives new insight into the structure of the solutions obtained through the original order completion method [1].

The generality and type independence of the solution method introduced in [1, 4โ€“6] has to date not been obtained in any of the usual theories of generalized solutions of linear and nonlinear PDEs. Indeed, and perhaps as a result of the insufficiency of the spaces of generalized functions that are typical in the study of generalized solutions of PDEs, at least from the point of view of the existence of solutions of PDEs, it is often believed that such a general theory is not possible, see for instance [8, 9]. Within the setting of the linear topological spaces of generalized functions that form the basis for most studies of PDEs, this may perhaps turn out to be the case. As a clarification and motivation of the above remarks, the following general comments may be of interest.

For over 135 years by now, there has been a general and type-independent existence and regularity result for the solutions of systems of analytic nonlinear PDEs. Indeed, in 1875 Kovalevskaia [10], upon the suggestion of Weierstrass, gave a rigorous proof of an earlier result of Cauchy, published in 1821 in his Course d'Analyse. This result, although restricted to the realm of analytic PDEs, is completely general as far as the type of nonlinearities involved are concerned. The analytic solutions of such a systems of PDEs can, however, be guaranteed to exists only on a neighborhood of the noncharacteristic analytic hypersurface on which the analytic initial data is specified. The nonexistence of solutions of a system of analytic PDEs on the whole domain of definition of the respective system of equations is not due to the particular techniques used in the proof of the result, but may rather be attributed to the very nature of nonlinear PDEs. Indeed, rather simple examples, such as the nonlinear conservation law ๐‘ˆ ๐‘ก + ๐‘ˆ ๐‘ฅ ๐‘ˆ = 0 , ๐‘ก > 0 , ๐‘ฅ โˆˆ โ„ ( 1 . 1 ) with the initial condition ๐‘ˆ ( 0 , ๐‘ฅ ) = ๐‘ข ( ๐‘ฅ ) , ๐‘ฅ โˆˆ โ„ ( 1 . 2 ) show that, irrespective of the smoothness of the initial data (1.2), the solution of the initial value problem may fail to exist on the whole domain of definition [ 0 , โˆž ) ร— โ„ of the equation, see for instance [11, 12]. Furthermore, in the case of the nonlinear conservation law (1.1), it is exactly the points where the solution fails to exists that are of interest, since these may represent the formation and propagation of shock waves, as well as other chaotic phenomena such as turbulence.

In view of the above remarks, it is clear that any general and type-independent theory for the existence and regularity of the solutions of nonlinear PDEs must alow for sufficiently singular objects to act as generalized solutions of such equations. In particular, the solutions may fail to be continuous, let alone sufficiently smooth, on the whole domain of definition of respective system of equations. In many cases, it happens that the spaces of generalized functions that are used in the study of PDEs do not admit such sufficiently singular objects. Indeed, we may recall that, due to the well-known Sobolev Embedding theorem, see for instance [13], the Sobolev Space ๐ป ๐‘˜ ( โ„ ๐‘› ) will, for sufficiently large values of ๐‘˜ , contain only continuous functions.

Moreover, even in case a given system of PDEs admits a solution which is classical, indeed even analytic, everywhere except at a single point of its Euclidean domain of definition, it may happen that such a solution does not belong to any of the customary spaces of generalized functions. For example, given a function ๎€ฝ ๐‘ง ๐‘ข โˆถ โ„‚ โงต 0 ๎€พ โŸถ โ„‚ ( 1 . 3 ) which is analytic everywhere except at the single point ๐‘ง 0 โˆˆ โ„‚ , and with an essential singularity at ๐‘ง 0 , Picard's Theorem states that ๐‘ข attains every complex value, with possibly one exception, in every neighborhood of ๐‘ง 0 . Clearly such a function does not satisfy any of the usual growth conditions that are, rather as a rule, imposed on generalized functions. Indeed, we may recall that the elements of a Sobolev space are locally integrable, while the elements of the Colombeau algebras [14], which contain the ๐’Ÿ ๎…ž distributions, must satisfy certain polynomial type growth conditions near singularities. Therefore these concepts of generalized functions cannot accommodate the mentioned singularity of the function in (1.3).

In this paper, we present further developments of the general and type-independent solution method presented in [1], and in particular the uniform convergence spaces of generalized functions introduced in [4โ€“6]. Furthermore, and in contradistinction with the spaces of generalized functions introduced in [6], we construct here a space of generalized functions that admit generalized partial derivatives of arbitrary order. While, following the methods introduced in [6], one may easily construct such a space of generalized functions, the existence of generalized solutions of systems of nonlinear PDEs in this space is nontrivial. Here we present the mentioned construction of the space of generalized functions, and show how generalized solutions of a large class of ๐’ž โˆž -smooth nonlinear PDEs may be obtained in this space.

As an application of the general theory, we discuss also the existence and regularity of generalized solutions of the parametrically driven, damped nonlinear Schrรถdinger equation in one spatial dimension. In this regard, we show that for a large class of ๐’ž โˆž -smooth initial values, the mentioned Schrรถdinger equation admits a generalized solution that satisfies the initial condition in a suitable generalized sense. We also introduce the concept of a strongly generic weak solution of this equation, and show that the solution we construct is such a weak solution.

The paper is organized as follows. In Section 2 we recall some basic facts concerning normal lower semicontinuous functions from the literature. The construction of spaces of generalized functions is given in Section 3, while Section 4 is concerned with the existence of generalized solutions of ๐’ž โˆž -smooth nonlinear PDEs. Lastly, in Section 5, we apply the general method to the parametrically driven, damped nonlinear Schrรถdinger equation in one spatial dimension. For all details on convergence spaces we refer the reader to the excellent book [15] and the paper [16].

2. Normal Lower Semicontinuous Functions

The concept of a normal lower semicontinuous function was first introduced by Dilworth [17] in connection with his attempts at characterizing the Dedekind order completion of spaces of continuous functions, a problem that was solved only recently by Anguelov [3]. Here we recall some facts concerning normal semicontinuous functions. For more details, and the proofs of some of the results, we refer the reader to the more recent presentations in [4, 18].

Denote by โ„ = โ„ โˆช { ยฑ โˆž } the extended real line, ordered as usual. The set of all extended real-valued functions on a topological space ๐‘‹ is denoted ๐’œ ( ๐‘‹ ) . A function ๐‘ข โˆˆ ๐’œ ( ๐‘‹ ) is said to be nearly finite whenever { ๐‘ฅ โˆˆ ๐‘‹ โˆถ ๐‘ข ( ๐‘ฅ ) โˆˆ โ„ } i s o p e n a n d d e n s e . ( 2 . 1 ) Two fundamental operations on the space ๐’œ ( ๐‘‹ ) are the Lower and Upper Baire Operators ๐ผ โˆถ ๐’œ ( ๐‘‹ ) โŸถ ๐’œ ( ๐‘‹ ) , ๐‘† โˆถ ๐’œ ( ๐‘‹ ) โŸถ ๐’œ ( ๐‘‹ ) , ( 2 . 2 ) introduced by Baire [19], see also [3], which are defined by ๐ผ ๎€ฝ ( ๐‘ข ) โˆถ ๐‘‹ โˆ‹ ๐‘ฅ โŸผ s u p i n f { ๐‘ข ( ๐‘ฆ ) โˆถ ๐‘ฆ โˆˆ ๐‘‰ } โˆถ ๐‘‰ โˆˆ ๐’ฑ ๐‘ฅ ๎€พ ๎€ฝ , ( 2 . 3 ) ๐‘† ( ๐‘ข ) โˆถ ๐‘‹ โˆ‹ ๐‘ฅ โŸผ i n f s u p { ๐‘ข ( ๐‘ฆ ) โˆถ ๐‘ฆ โˆˆ ๐‘‰ } โˆถ ๐‘‰ โˆˆ ๐’ฑ ๐‘ฅ ๎€พ , ( 2 . 4 ) respectively, with ๐’ฑ ๐‘ฅ denoting the neighborhood filter at ๐‘ฅ โˆˆ ๐‘‹ . Clearly, the Baire operators ๐ผ and ๐‘† satisfy ๐ผ ( ๐‘ข ) โ‰ค ๐‘ข โ‰ค ๐‘† ( ๐‘ข ) , ๐‘ข โˆˆ ๐’œ ( ๐‘‹ ) , ( 2 . 5 ) when ๐’œ ( ๐‘‹ ) is equipped with the usual pointwise order ๐‘ข โ‰ค ๐‘ฃ โŸบ ( โˆ€ ๐‘ฅ โˆˆ ๐‘‹ โˆถ ๐‘ข ( ๐‘ฅ ) โ‰ค ๐‘ฃ ( ๐‘ฅ ) ) . ( 2 . 6 ) Furthermore, the Baire operators, as well as their compositions, are idempotent and monotone with respect to the pointwise order. That is, โŽ› โŽœ โŽœ โŽœ โŽ ( โŽž โŽŸ โŽŸ โŽŸ โŽ  . โˆ€ ๐‘ข โˆˆ ๐’œ ( ฮฉ ) โˆถ ( 1 ) ๐ผ ( ๐ผ ( ๐‘ข ) ) = ๐ผ ( ๐‘ข ) , ( 2 ) ๐‘† ( ๐‘† ( ๐‘ข ) ) = ๐‘† ( ๐‘ข ) , ( 3 ) ( ๐ผ โˆ˜ ๐‘† ) ( ( ๐ผ โˆ˜ ๐‘† ) ( ๐‘ข ) ) = ( ๐ผ โˆ˜ ๐‘† ) ( ๐‘ข ) , โˆ€ ๐‘ข , ๐‘ฃ โˆˆ ๐’œ ( ฮฉ ) โˆถ ๐‘ข โ‰ค ๐‘ฃ โŸน ( 1 ) ๐ผ ( ๐‘ข ) โ‰ค ๐ผ ( ๐‘ฃ ) ( 2 ) ๐‘† ( ๐‘ข ) โ‰ค ๐‘† ( ๐‘ฃ ) 3 ) ( ๐ผ โˆ˜ ๐‘† ) ( ๐‘ข ) โ‰ค ( ๐ผ โˆ˜ ๐‘† ) ( ๐‘ฃ ) ( 2 . 7 )

The operators ๐ผ and ๐‘† , as well as their compositions ๐ผ โˆ˜ ๐‘† and ๐‘† โˆ˜ ๐ผ , are useful tools for the study of (extended) real-valued functions. In this regard, we may mention that these mappings characterize certain continuity properties of functions in ๐’œ ( ๐‘‹ ) . In particular, we have ๐‘ข โˆˆ ๐’œ ( ๐‘‹ ) i s l o w e r s e m i c o n t i n u o u s โŸบ ๐ผ ( ๐‘ข ) = ๐‘ข , ๐‘ข โˆˆ ๐’œ ( ๐‘‹ ) i s u p p e r s e m i c o n t i n u o u s โŸบ ๐‘† ( ๐‘ข ) = ๐‘ข . ( 2 . 8 ) Furthermore, a function ๐‘ข โˆˆ ๐’œ ( ๐‘‹ ) is normal lower semicontinuous on ๐‘‹ whenever ( ๐ผ โˆ˜ ๐‘† ) ( ๐‘ข ) = ๐‘ข . ( 2 . 9 )

We denote the set of nearly finite normal lower semicontinuous functions on ๐‘‹ by ๐’ฉ โ„’ ( ๐‘‹ ) . The concept of normal lower semicontinuity of extended real-valued functions extends that of continuity of usual real-valued functions. In particular, each continuous function is nearly finite and normal lower semicontinuous so that we have the inclusion ๐’ž ( ๐‘‹ ) โŠ† ๐’ฉ โ„’ ( ๐‘‹ ) . ( 2 . 1 0 ) Conversely, a normal lower semicontinuous function is generically continuous in the sense that โˆ€ ๐‘ข โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) โˆถ โˆƒ ๐ต โŠ† ๐‘‹ o f ๏ฌ r s t B a i r e c a t e g o r y โˆถ ๐‘ฅ โˆˆ ๐‘‹ โงต ๐ต โŸน ๐‘ข c o n t i n u o u s a t ๐‘ฅ . ( 2 . 1 1 ) In particular, in case ๐‘‹ is a Baire space, it follows that each ๐‘ข โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) is continuous on some residual set, which is dense in ๐‘‹ . Furthermore, the following well-known property of continuous functions holds also for normal lower semicontinuous functions. Namely, we have โˆ€ ๐‘ข , ๐‘ฃ โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) , ๐ท โŠ† ๐‘‹ d e n s e โˆถ ( โˆ€ ๐‘ฅ โˆˆ ๐ท โˆถ ๐‘ข ( ๐‘ฅ ) โ‰ค ๐‘ฃ ( ๐‘ฅ ) ) โŸน ๐‘ข โ‰ค ๐‘ฃ . ( 2 . 1 2 )

With respect to the usual pointwise order, the space ๐’ฉ โ„’ ( ๐‘‹ ) is a fully distributive lattice. That is, suprema and infima of finite sets always exists and ๐‘ข โˆ€ ๐ด โŠ‚ ๐’ฉ โ„’ ( ๐‘‹ ) , ๐‘ฃ โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) โˆถ 0 ๎€ฝ = s u p ๐ด โŸน i n f ๐‘ฃ , ๐‘ข 0 ๎€พ = s u p { i n f { ๐‘ข , ๐‘ฃ } โˆถ ๐‘ข โˆˆ ๐ด } . ( 2 . 1 3 ) Furthermore, ๐’ฉ โ„’ ( ๐‘‹ ) is Dedekind order complete. That is, every set which is order bounded from above, respectively, below, has a least upper bound, respectively, greatest lower bound. In particular, the supremum and infimum of a set ๐ด โŠ‚ ๐’ฉ โ„’ ( ๐‘‹ ) , is given by s u p ๐ด = ( ๐ผ โˆ˜ ๐‘† ) ( ๐œ‘ ) , i n f ๐ด = ( ๐ผ โˆ˜ ๐‘† ) ( ๐œ“ ) , ( 2 . 1 4 ) respectively, with ๐œ‘ and ๐œ“ given by ๐œ‘ โˆถ ๐‘‹ โˆ‹ ๐‘ฅ โŸผ s u p { ๐‘ข ( ๐‘ฅ ) โˆถ ๐‘ข โˆˆ ๐ด } โˆˆ โ„ , ( 2 . 1 5 ) ๐œ“ โˆถ ๐‘‹ โˆ‹ ๐‘ฅ โŸผ i n f { ๐‘ข ( ๐‘ฅ ) โˆถ ๐‘ข โˆˆ ๐ด } โˆˆ โ„ . ( 2 . 1 6 ) A useful characterization of order bounded sets in ๐’ฉ โ„’ ( ๐‘‹ ) is the following: If ๐‘‹ is a Baire space, then for any set ๐ด โŠ‚ ๐’ฉ โ„’ ( ๐‘‹ ) we have โˆƒ ๐‘ข 0 โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) โˆถ ๐‘ข โ‰ค ๐‘ข 0 , ๐‘ข โˆˆ ๐ด ( 2 . 1 7 ) if and only if โˆƒ ๐‘… โŠ† ๐‘‹ a r e s i d u a l s e t โˆถ โˆ€ ๐‘ฅ โˆˆ ๐‘… โˆถ s u p { ๐‘ข ( ๐‘ฅ ) โˆถ ๐‘ข โˆˆ ๐ด } < โˆž . ( 2 . 1 8 ) Indeed, suppose that a set ๐ด โŠ‚ ๐’ฉ โ„’ ( ๐‘‹ ) satisfies (2.18). Then the function ๐œ‘ associated with ๐ด through (2.15) satisfies ๐œ‘ ( ๐‘ฅ ) < โˆž , ๐‘ฅ โˆˆ ๐‘… . ( 2 . 1 9 ) The function ๐‘ข 0 = ( ๐ผ โˆ˜ ๐‘† ) ( ๐œ‘ ) is normal lower semicontinuous and satisfies ๐‘ข โ‰ค ๐‘ข 0 , ๐‘ข โˆˆ ๐ด . ( 2 . 2 0 ) It is sufficient to show that ๐‘ข 0 is finite on a dense subset of ๐‘‹ . To see that ๐‘ข 0 satisfies this condition, assume the opposite. That is, we assume ๐‘ข โˆƒ ๐‘‰ โŠ† ๐‘‹ n o n e m p t y a n d o p e n โˆถ โˆ€ ๐‘ฅ โˆˆ ๐‘‰ โˆถ 0 ( ๐‘ฅ ) = โˆž . ( 2 . 2 1 ) It follows by (2.5) that ๐‘† ( ๐œ‘ ) ( ๐‘ฅ ) = โˆž , ๐‘ฅ โˆˆ ๐‘‰ . ( 2 . 2 2 ) Thus (2.4) implies that โˆ€ ๐‘ฅ โˆˆ ๐‘‰ , ๐‘Š โˆˆ ๐’ฑ ๐‘ฅ , ๐‘€ > 0 โˆถ โˆƒ ๐‘ฅ ๐‘€ ๐œ‘ ๎€ท ๐‘ฅ โˆˆ ๐‘‰ โˆฉ ๐‘Š โˆถ ๐‘€ ๎€ธ > ๐‘€ . ( 2 . 2 3 ) Note that, since each ๐‘ข โˆˆ ๐ด is lower semicontinuous, the function ๐œ‘ is also lower semicontinuous. Therefore (2.23) implies โˆ€ ๐‘€ > 0 โˆถ โˆƒ ๐ท ๐‘€ โŠ† ๐‘‰ o p e n a n d d e n s e i n ๐‘‰ โˆถ ๐œ‘ ( ๐‘ฅ ) > ๐‘€ , ๐‘ฅ โˆˆ ๐ท ๐‘€ . ( 2 . 2 4 ) There is therefore a residual set ๐‘… ๎…ž โŠ† ๐‘‰ such that ๐œ‘ ( ๐‘ฅ ) = โˆž , ๐‘ฅ โˆˆ ๐‘… ๎…ž . ( 2 . 2 5 ) Since ๐‘‹ is a Baire space, so is the open set ๐‘‰ in the subspace topology. Furthermore, ๐‘… โˆฉ ๐‘‰ is a residual set in ๐‘‰ . But ๐‘… โˆฉ ๐‘‰ โŠ† ๐‘‰ โงต ๐‘… ๎…ž so that ๐‘… โˆฉ ๐‘‰ must be of first Baire category in ๐‘‰ , which is a contradiction. Therefore ๐‘ข 0 is finite on a dense subset of ๐‘‹ . The dual statement for sets bounded from below also holds.

A subspace of ๐’ฉ โ„’ ( ๐‘‹ ) which is of particular interest to us here is the space โ„ณ โ„’ ( ๐‘‹ ) which consists of all functions in ๐’ฉ โ„’ ( ๐‘‹ ) which are real-valued and continuous on an open and dense subset of ๐‘‹ . That is, โ„ณ โ„’ ( ๐‘‹ ) = { ๐‘ข โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) โˆฃ โˆƒ ฮ“ โŠ‚ ๐‘‹ c l o s e d n o w h e r e d e n s e โˆถ ๐‘ข โˆˆ ๐’ž ( ๐‘‹ โงต ฮ“ ) } . ( 2 . 2 6 ) The space โ„ณ โ„’ ( ๐‘‹ ) is a sublattice of ๐’ฉ โ„’ ( ๐‘‹ ) . As such, it is also fully distributive. Furthermore, whenever ๐‘‹ is a metric space, the following order denseness property is satisfied: โˆ€ ๐‘ข โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) โˆถ s u p { ๐‘ฃ โˆˆ โ„ณ โ„’ ( ๐‘‹ ) โˆถ ๐‘ฃ โ‰ค ๐‘ข } = ๐‘ข = i n f { ๐‘ค โˆˆ โ„ณ โ„’ ( ๐‘‹ ) โˆถ ๐‘ข โ‰ค ๐‘ค } . ( 2 . 2 7 ) Moreover, the following sequential version of (2.27) holds: โˆƒ ๎€ท ๐œ† โˆ€ ๐‘ข โˆˆ ๐’ฉ โ„’ ( ๐‘‹ ) โˆถ ๐‘› ๎€ธ , ๎€ท ๐œ‡ ๐‘› ๎€ธ โŠ‚ โ„ณ โ„’ ( ๐‘‹ ) โˆถ ( 1 ) ๐œ† ๐‘› โ‰ค ๐œ† ๐‘› + 1 โ‰ค ๐‘ข โ‰ค ๐œ‡ ๐‘› + 1 โ‰ค ๐œ‡ ๐‘› ๎€ฝ ๐œ† , ๐‘› โˆˆ โ„• , ( 2 ) s u p ๐‘› ๎€พ ๎€ฝ ๐œ‡ โˆถ ๐‘› โˆˆ โ„• = ๐‘ข = i n f ๐‘› ๎€พ . โˆถ ๐‘› โˆˆ โ„• ( 2 . 2 8 ) We may note that the space โ„ณ โ„’ ( ๐‘‹ ) , and suitable subspaces of it, also play an important role in the theory of rings of continuous functions [20], and the theory of differential algebras of generalized functions [21] where such spaces arise in connection with the so-called closed nowhere dense ideals. In the next section we construct a space of generalized functions as the completion of a suitable uniform convergence space, the elements of which are functions in โ„ณ โ„’ ( ๐‘‹ ) , when ๐‘‹ is an open subset of Euclidean ๐‘› -space โ„ ๐‘› .

3. Spaces of Generalized Functions

We now consider the construction of spaces of generalized functions based on the spaces of normal lower semicontinuous functions discussed in Section 2. This follows closely the method used in [6], with the exception that we consider here the case of infinitely differentiable functions, this being the main topic of the current investigation.

In this regard, let ฮฉ be an open, nonempty and possibly unbounded subset of โ„ ๐‘› . For ๐‘š โˆˆ โ„• โˆช { โˆž } we denote by โ„ณ โ„’ ๐‘š ( ฮฉ ) the set of those functions in ๐’ฉ โ„’ ( ฮฉ ) that are ๐’ž ๐‘š -smooth everywhere except on some closed nowhere dense set ฮ“ โŠ‚ ฮฉ . That is, โ„ณ โ„’ ๐‘š ๎€ฝ ( ฮฉ ) = ๐‘ข โˆˆ ๐’ฉ โ„’ ( ฮฉ ) โˆฃ โˆƒ ฮ“ โŠ‚ ฮฉ c l o s e d n o w h e r e d e n s e โˆถ ๐‘ข โˆˆ ๐’ž ๐‘š ๎€พ ( ฮฉ โงต ฮ“ ) . ( 3 . 1 ) One should note that, while the singularity set ฮ“ associated with a function ๐‘ข โˆˆ โ„ณ โ„’ ๐‘š ( ฮฉ ) through (3.1) is a topologically small set, it may be large in the sense of measure [22]. That is, the set ฮ“ may have arbitrarily large positive Lebesgue measure. Furthermore, a function ๐‘ข โˆˆ โ„ณ โ„’ ๐‘š ( ฮฉ ) typically does not satisfy any of the usual growth conditions that are imposed on generalized functions. In particular, ๐‘ข is, in general, not locally integrable on any neighborhood of any point ๐‘ฅ โˆˆ ฮ“ . Moreover, ๐‘ข will typically not satisfy any of the polynomial type growth conditions that are imposed on elements of the Colombeau algebras of generalized functions [14].

For ๐‘š = 0 , the space (3.1) reduces to โ„ณ โ„’ 0 ( ฮฉ ) = โ„ณ โ„’ ( ฮฉ ) , as defined in (2.26). For ๐‘š = โˆž , the usual partial differential operators ๐ท ๐›ผ โˆถ ๐’ž โˆž ( ฮฉ ) โŸถ ๐’ž โˆž ( ฮฉ ) โŠ‚ ๐’ž 0 ( ฮฉ ) , ๐›ผ โˆˆ โ„• ๐‘› ( 3 . 2 ) extend in a straight forward way to mappings ๐’Ÿ ๐›ผ โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) โŸถ โ„ณ โ„’ โˆž ( ฮฉ ) โŠ† โ„ณ โ„’ 0 ( ฮฉ ) , ๐›ผ โˆˆ โ„• ๐‘› ( 3 . 3 ) which are defined through ๐’Ÿ ๐›ผ โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ( ๐ผ โˆ˜ ๐‘† ) ( ๐ท ๐›ผ ๐‘ข ) โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) , ๐›ผ โˆˆ โ„• ๐‘› . ( 3 . 4 )

The space of generalized functions we consider here are constructed as the completion of the space โ„ณ โ„’ โˆž ( ฮฉ ) equipped with a suitable uniform convergence structure. In this regard, see [4], we consider on โ„ณ โ„’ 0 ( ฮฉ ) the uniform order convergence structure.

Definition 3.1. Let ฮฃ consist of all nonempty order intervals in โ„ณ โ„’ 0 ( ฮฉ ) . The family ๐’ฅ ๐‘œ of filters on โ„ณ โ„’ 0 ( ฮฉ ) ร— โ„ณ โ„’ 0 ( ฮฉ ) consists of all filters that satisfy the following: there exists ๐‘˜ โˆˆ โ„• such that โˆ€ ๐‘– = 1 , โ€ฆ , ๐‘˜ โˆถ โˆƒ ฮฃ ๐‘– = ๎€ท ๐ผ ๐‘– ๐‘› ๎€ธ โŠ† ฮฃ , ๐‘ข ๐‘– โˆˆ ๐’ฉ โ„’ ( ฮฉ ) โˆถ ( 1 ) ๐ผ ๐‘– ๐‘› + 1 โŠ† ๐ผ ๐‘– ๐‘› ๎€ฝ , ๐‘› โˆˆ โ„• , ( 2 ) s u p i n f ๐ผ ๐‘– ๐‘› ๎€พ โˆถ ๐‘› โˆˆ โ„• = ๐‘ข ๐‘– ๎€ฝ = i n f s u p ๐ผ ๐‘– ๐‘› ๎€พ , ฮฃ โˆถ ๐‘› โˆˆ โ„• ( 3 ) ๎€ท ๎€บ 1 ๎€ป ร— ๎€บ ฮฃ 1 ฮฃ ๎€ป ๎€ธ โˆฉ โ‹ฏ โˆฉ ๎€ท ๎€บ ๐‘˜ ๎€ป ร— ๎€บ ฮฃ ๐‘˜ ๎€ป ๎€ธ โŠ† ๐’ฐ . ( 3 . 5 )

The family of filters ๐’ฅ ๐‘œ is a uniformly Hausdorff and first countable uniform convergence structure. Furthermore, the induced convergence structure is the order convergence structure [23]. That is, a filter โ„ฑ on โ„ณ โ„’ 0 ( ฮฉ ) converges to ๐‘ข โˆˆ โ„ณ โ„’ 0 ( ฮฉ ) if and only if โˆƒ ๎€ท ๐œ† ๐‘› ๎€ธ , ๎€ท ๐œ‡ ๐‘› ๎€ธ โŠ‚ โ„ณ โ„’ 0 ( ฮฉ ) โˆถ ( 1 ) ๐œ† ๐‘› โ‰ค ๐œ† ๐‘› + 1 โ‰ค ๐‘ข โ‰ค ๐œ‡ ๐‘› + 1 โ‰ค ๐œ‡ ๐‘› ๎€ฝ ๐œ† , ๐‘› โˆˆ โ„• , ( 2 ) s u p ๐‘› ๎€พ ๎€ฝ ๐œ‡ โˆถ ๐‘› โˆˆ โ„• = ๐‘ข = i n f ๐‘› ๎€พ , ๎€บ ๐œ† โˆถ ๐‘› โˆˆ โ„• ( 3 ) ๐‘› , ๐œ‡ ๐‘› ๎€ป โˆˆ โ„ฑ , ๐‘› โˆˆ โ„• . ( 3 . 6 )

The completion of the uniform convergence space โ„ณ โ„’ 0 ( ฮฉ ) may be characterized in terms of the space ๐’ฉ โ„’ ( ฮฉ ) equipped with a suitable uniform convergence structure ๐’ฅ โ™ฏ ๐‘œ , see [4]. In particular, this means that ๐’ฉ โ„’ ( ฮฉ ) is complete and contains โ„ณ โ„’ 0 ( ฮฉ ) as a dense subspace. Furthermore, given any complete Hausdorff uniform convergence space ๐‘Œ , and any uniformly continuous mapping ฮจ โˆถ โ„ณ โ„’ 0 ( ฮฉ ) โŸถ ๐‘Œ , ( 3 . 7 ) there exists a unique uniformly continuous mapping ฮจ โ™ฏ โˆถ ๐’ฉ โ„’ ( ฮฉ ) โŸถ ๐‘Œ ( 3 . 8 ) which extends ฮจ .

The space โ„ณ โ„’ โˆž ( ฮฉ ) is equipped with the initial uniform convergence structure with respect to the family of mappings (3.3). That is, ๐’ฐ โˆˆ ๐’ฅ ๐ท โŸบ ๎€ท โˆ€ ๐›ผ โˆˆ โ„• ๐‘› โˆถ ๎€ท ๐’Ÿ ๐›ผ ร— ๐’Ÿ ๐›ผ ๎€ธ ( ๐’ฐ ) โˆˆ ๐’ฅ ๐‘œ ๎€ธ . ( 3 . 9 ) Each of the mappings (3.3) is uniformly continuous with respect to the uniform convergence structures ๐’ฅ ๐ท and ๐’ฅ ๐‘œ on โ„ณ โ„’ โˆž ( ฮฉ ) and โ„ณ โ„’ 0 ( ฮฉ ) , respectively. In fact, (3.9) is the finest uniform convergence structure on โ„ณ โ„’ โˆž ( ฮฉ ) making the mappings (3.3) uniformly continuous.

Since the family of mappings (3.3) is countable and separates the points of โ„ณ โ„’ โˆž ( ฮฉ ) , that is, โˆ€ ๐‘ข , ๐‘ฃ โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) โˆถ โˆƒ ๐›ผ โˆˆ โ„• ๐‘› โˆถ ๐’Ÿ ๐›ผ ๐‘ข โ‰  ๐’Ÿ ๐›ผ ๐‘ฃ , ( 3 . 1 0 ) it follows from the corresponding properties of โ„ณ โ„’ 0 ( ฮฉ ) that the uniform convergence structure ๐’ฅ ๐ท is uniformly Hausdorff and first countable. Furthermore, a filter โ„ฑ on โ„ณ โ„’ โˆž ( ฮฉ ) converges to ๐‘ข โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) with respect to the convergence structure ๐œ† ๐ท induced by ๐’ฅ ๐ท if and only if โˆ€ ๐›ผ โˆˆ โ„• ๐‘› โˆถ ๐’Ÿ ๐›ผ ( โ„ฑ ) c o n v e r g e s t o ๐’Ÿ ๐›ผ ๐‘ข i n โ„ณ โ„’ 0 ( ฮฉ ) . ( 3 . 1 1 )

The completion of โ„ณ โ„’ โˆž ( ฮฉ ) , which we denote by ๐’ฉ โ„’ โˆž ( ฮฉ ) , is related to the completion ๐’ฉ โ„’ ( ฮฉ ) of โ„ณ โ„’ 0 ( ฮฉ ) in the following way. Since โ„ณ โ„’ โˆž ( ฮฉ ) carries the initial uniform convergence structure with respect to the family of mappings (3.3), it follows [16] that the mapping ๐ƒ โˆถ โ„ณ โ„’ โˆž ๎€ท ๐’Ÿ ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ๐›ผ ๐‘ข ๎€ธ ๐›ผ โˆˆ โ„• ๐‘› โˆˆ โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› ( 3 . 1 2 ) is a uniformly continuous embedding, with โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› equipped with the product uniform convergence structure with respect to ๐’ฅ ๐‘œ . In particular, the diagram Eq1(3.13) commutes for each ๐›ผ โˆˆ โ„• ๐‘› , with ๐œ‹ ๐›ผ the projection. In view of the uniform continuity of the mappings (3.3) and (3.12), there are unique extensions of these mappings to the completion ๐’ฉ โ„’ โˆž ( ฮฉ ) of โ„ณ โ„’ โˆž ( ฮฉ ) . That is, we have uniformly continuous mappings ๐’Ÿ ๐›ผ โ™ฏ โˆถ ๐’ฉ โ„’ โˆž ( ฮฉ ) โŸถ ๐’ฉ โ„’ ( ฮฉ ) , ๐›ผ โˆˆ โ„• ๐‘› , ๐ƒ ( 3 . 1 4 ) โ™ฏ โˆถ ๐’ฉ โ„’ โˆž ( ๎€ท ฮฉ ) โŸถ โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› ๎€ธ โ™ฏ ( 3 . 1 5 ) which extend the mappings (3.3) and (3.12), respectively. Here ( โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› ) โ™ฏ denotes the completion of โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› . In particular, the mapping (3.15) is injective. Note that there exists a canonical, bijective uniformly continuous mapping ๎€ท ๐œ„ โˆถ โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› ๎€ธ โ™ฏ โŸถ ๐’ฉ โ„’ ( ฮฉ ) โ„• ๐‘› , ( 3 . 1 6 ) see [16]. We may therefore consider (3.15) as an injective uniformly continuous mapping ๐ƒ โ™ฏ โˆถ ๐’ฉ โ„’ โˆž ( ฮฉ ) โŸถ ๐’ฉ โ„’ ( ฮฉ ) โ„• ๐‘› , ( 3 . 1 7 ) where ๐’ฉ โ„’ ( ฮฉ ) โ„• ๐‘› carries the product uniform convergence structure with respect to ๐’ฅ โ™ฏ ๐‘œ . Furthermore, the commutative diagram (3.13) extends to the diagramEq2(3.18)

The interpretation of the existence of the injective, uniformly continuous mapping (3.17) and the commutative diagram (3.18) is as follows. Each generalized function ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) may be represented in a canonical way through its generalized partial derivatives ๐’Ÿ ๐›ผ โ™ฏ ๐‘ข โ™ฏ , which are usual nearly finite normal lower semicontinuous functions. This gives a first clarification of the structure of generalized functions. Furthermore, this also provides a basic blanket regularity for the generalized functions in ๐’ฉ โ„’ โˆž ( ฮฉ ) . Namely, each such generalized function is identified with the vector of normal lower semicontinuous functions ๐ƒ โ™ฏ ๐‘ข โ™ฏ = ๎‚€ ๐’Ÿ ๐›ผ โ™ฏ ๐‘ข โ™ฏ ๎‚ . ( 3 . 1 9 ) Now, in view of (2.11), we have โˆ€ ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) โˆถ โˆƒ ๐‘… โŠ† ฮฉ a r e s i d u a l s e t โˆถ โˆ€ ๐›ผ โˆˆ โ„• ๐‘› โˆถ ๐‘ฅ โˆˆ ๐‘… โŸน ๐’Ÿ ๐›ผ โ™ฏ ๐‘ข โ™ฏ c o n t i n u o u s a t ๐‘ฅ . ( 3 . 2 0 ) Thus the singularity set associated with each generalized function ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) , that is, the set where ๐‘ข โ™ฏ or any of its generalized partial derivatives are discontinuous, is of first Baire category. This set, while small in a topological sense, may be dense in ฮฉ . Furthermore, it may have arbitrarily large positive Lebesgue measure [22]. We note that such highly singular objects may be of interest in connection with turbulence in fluids and other types chaotical phenomena.

4. Existence of Generalized Solutions

In the previous section we discussed the structure of spaces of generalized functions which are obtained as the completion of suitable uniform convergence spaces, the elements of which are nearly finite normal lower semicontinuous functions. This construction is an extension of that given in [6] for spaces ๐’ฉ โ„’ ๐‘š ( ฮฉ ) of generalized functions which admit only generalized partial derivatives of an arbitrary but fixed finite order ๐‘š , to the case of infinitely differentiable functions.

It is shown in [6] that a large class of systems of nonlinear PDEs admit solutions, in a suitable generalized sense, in the spaces ๐’ฉ โ„’ ๐‘š ( ฮฉ ) . In this section we discuss the existence of such generalized solutions in the space ๐’ฉ โ„’ โˆž ( ฮฉ ) . In this regard, consider nonlinear PDE of order ๐‘š of the form ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) , ๐‘ฅ โˆˆ ฮฉ . ( 4 . 1 ) Here the right hand term ๐‘“ is supposed to be ๐’ž โˆž -smooth on ฮฉ , while the nonlinear partial differential operator ๐‘‡ ( ๐‘ฅ , ๐ท ) is defined by a ๐’ž โˆž -smooth mapping ๐น โˆถ ฮฉ ร— โ„ ๐พ โŸถ โ„ ( 4 . 2 ) through ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ( ๐‘ฅ ) = ๐น ( ๐‘ฅ , โ€ฆ , ๐ท ๐›ผ ๐‘ข ( ๐‘ฅ ) , โ€ฆ ) , | ๐›ผ | โ‰ค ๐‘š ( 4 . 3 ) for any sufficiently smooth function ๐‘ข defined on ฮฉ . For each ๐›ฝ โˆˆ โ„• ๐‘› , we denote by ๐น ๐›ฝ the mapping ๐น ๐›ฝ โˆถ ฮฉ ร— โ„ ๐‘€ ๐›ฝ โŸถ โ„ ( 4 . 4 ) such that ๐ท ๐›ฝ ( ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ( ๐‘ฅ ) ) = ๐น ๐›ฝ ( ๐‘ฅ , โ€ฆ , ๐ท ๐›ผ | | ๐›ฝ | | ๐‘ข ( ๐‘ฅ ) , โ€ฆ ) , | ๐›ผ | โ‰ค ๐‘š + ( 4 . 5 ) for all functions ๐‘ข โˆˆ ๐’ž โˆž ( ฮฉ ) . Consider the mapping ๐น โˆž โˆถ ฮฉ ร— โ„ โ„• ๐‘› โˆ‹ ๎€ท ๎€ท ๐œ‰ ๐‘ฅ , ๐›ผ ๎€ธ ๐›ผ โˆˆ โ„• ๐‘› ๎€ธ โŸผ ๎€ท ๐น ๐›ฝ ๎€ท ๐‘ฅ , โ€ฆ , ๐œ‰ ๐›ผ , โ€ฆ ๎€ธ ๎€ธ ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ โ„ โ„• ๐‘› . ( 4 . 6 ) We will assume that the nonlinear PDE (4.1) satisfies the condition โˆ€ ๐‘ฅ โˆˆ ฮฉ โˆถ โˆƒ ๐œ‰ ( ๐‘ฅ ) โˆˆ โ„ โ„• ๐‘› , ๐น โˆž ๎€ท ๐ท ( ๐‘ฅ , ๐œ‰ ( ๐‘ฅ ) ) = ๐›ฝ ๎€ธ ๐‘“ ( ๐‘ฅ ) ๐›ฝ โˆˆ โ„• ๐‘› โˆถ โˆƒ ๐‘‰ โˆˆ ๐’ฑ ๐‘ฅ , ๐‘Š โˆˆ ๐’ฑ ๐œ‰ ( ๐‘ฅ ) โˆถ ๐น โˆž โˆถ ๐‘‰ ร— ๐‘Š โŸถ โ„ โ„• ๐‘› o p e n , ( 4 . 7 ) where โ„ โ„• ๐‘› is equipped with the product topology.

With the nonlinear operator (4.3) we may associate a mapping ๐‘‡ โˆถ ๐’ž โˆž ( ฮฉ ) โŸถ ๐’ž โˆž ( ฮฉ ) . ( 4 . 8 ) This mapping may be extended so as to act on โ„ณ โ„’ โˆž ( ฮฉ ) . In this regard, we set ๐‘‡ โˆถ โ„ณ โ„’ โˆž ๎€ท ๐น ๎€ท ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ( ๐ผ โˆ˜ ๐‘† ) โ‹… , ๐‘ข , โ€ฆ , ๐’Ÿ ๐›ผ ๐‘ข , โ€ฆ ๎€ธ ๎€ธ โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) . ( 4 . 9 ) Furthermore, the partial derivatives ๐’Ÿ ๐›ฝ ( ๐‘‡ ๐‘ข ) of ๐‘‡ ๐‘ข , for ๐‘ข โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) , may be represented through the mappings (4.4). In particular, โˆ€ ๐›ฝ โˆˆ โ„• ๐‘› โˆถ โˆ€ ๐‘ข โˆˆ โ„ณ โ„’ โˆž ๐’Ÿ ( ฮฉ ) โˆถ ๐›ฝ ( ๐‘‡ ๐‘ข ) = ๐‘‡ ๐›ฝ ๐‘ข , ( 4 . 1 0 ) where the ๐‘‡ ๐›ฝ , with ๐›ฝ โˆˆ โ„• ๐‘› , are the mappings defined in terms of (4.4) as ๐‘‡ ๐›ฝ โˆถ โ„ณ โ„’ โˆž ๎€ท ๐น ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ( ๐ผ โˆ˜ ๐‘† ) ๐›ฝ ๎€ท โ‹… , โ€ฆ , ๐’Ÿ ๐›ผ ๐‘ข , โ€ฆ ๎€ธ ๎€ธ โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) โŠ‚ โ„ณ โ„’ 0 ( ฮฉ ) , ( 4 . 1 1 ) where ๐›ผ โ‰ค ๐‘š + | ๐›ฝ | . We denote by ๐‘‡ โˆž the mapping ๐‘‡ โˆž โˆถ โ„ณ โ„’ โˆž ๎€ท ๐‘‡ ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ๐›ฝ ๐‘ข ๎€ธ ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ โ„ณ โ„’ 0 ( ฮฉ ) โ„• ๐‘› . ( 4 . 1 2 ) From (4.10) to (4.12) if follows that the diagramEq3(4.13) commutes, with ๐ƒ the mapping (3.12).

Through the mapping (4.9) we obtain a first extension of the nonlinear PDE (4.1). Namely, the equation ๐‘‡ ๐‘ข = ๐‘“ , ( 4 . 1 4 ) where ๐‘‡ is the mapping (4.9) and the unknown ๐‘ข is supposed to belong to โ„ณ โ„’ โˆž ( ฮฉ ) . Equation (4.14) generalizes (4.1) in the sense that any solution ๐‘ข โˆˆ ๐’ž โˆž ( ฮฉ ) of (4.1) is also a solution of (4.14). Conversely, any solution ๐‘ข โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) of (4.14) satisfies (4.1) everywhere except on the closed nowhere dense set ฮ“ associated with ๐‘ข through (3.1). That is, ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) , ๐‘ฅ โˆˆ ฮฉ โงต ฮ“ . ( 4 . 1 5 ) A further generalization of (4.1) is obtained by extending the mapping (4.9) to the completion ๐’ฉ โ„’ โˆž ( ฮฉ ) of โ„ณ โ„’ โˆž ( ฮฉ ) . In order for the concept of generalized solution of (4.1) obtained through such an extension to be a sensible one, the extension of (4.9) to ๐’ฉ โ„’ โˆž ( ฮฉ ) must be constructed in a canonical way. In this regard, the following is the fundamental result.

Theorem 4.1. The mapping ๐‘‡ โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) โ†’ โ„ณ โ„’ โˆž ( ฮฉ ) associated with the nonlinear partial differential operator (4.3) through (4.9) is uniformly continuous.

Proof. In view of the commutative diagram (4.13) it is sufficient to show that the mapping (4.12) is uniformly continuous. In this regard, we claim that each of the mappings ๐‘‡ ๐›ฝ โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) โŸถ โ„ณ โ„’ 0 ( ฮฉ ) ( 4 . 1 6 ) is uniformly continuous. To see that this is so, we represent each mapping ๐‘‡ ๐›ฝ through the diagram Eq4(4.17) where ๐ƒ | ๐›ฝ | โˆถ โ„ณ โ„’ โˆž ๎€ท ๐’Ÿ ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ๐›ผ ๐‘ข ๎€ธ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | โˆˆ โ„ณ โ„’ 0 ( ฮฉ ) ๐‘€ | ๐›ฝ | , ( 4 . 1 8 ) ๐น ๐›ฝ โˆถ โ„ณ โ„’ 0 ( ฮฉ ) ๐‘€ ๐›ฝ ๎€ท ๐‘ข โˆ‹ ๐ฎ = ๐›ผ ๎€ธ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๎€ท ๐น ๎€ท โŸผ ( ๐ผ โˆ˜ ๐‘† ) โ‹… , โ€ฆ , ๐‘ข ๐›ผ , โ€ฆ ๎€ธ ๎€ธ โˆˆ โ„ณ โ„’ 0 ( ฮฉ ) . ( 4 . 1 9 ) Clearly the mapping (4.18) is uniformly continuous. As such, it suffices to show that (4.19) is uniformly continuous. In this regard, for each | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | consider a sequence ( ๐ผ ๐›ผ ๐‘› ) of order intervals that satisfies (1) and (2) of (3.5). For each ๐‘› โˆˆ โ„• there is an order interval ๐ผ ๐‘› in โ„ณ โ„’ 0 ( ฮฉ ) such that ๐น ๐›ฝ ๎ƒฉ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐‘› ๎ƒช โŠ† ๐ผ ๐‘› . ( 4 . 2 0 ) Indeed, there exists a closed nowhere dense set ฮ“ โŠ‚ ฮฉ such that โˆ€ ๐พ โŠ‚ ฮฉ โงต ฮ“ c o m p a c t โˆถ โˆƒ ๐‘€ ๐พ > 0 โˆถ โˆ€ ๐‘› โˆˆ โ„• , ๐‘ฅ โˆˆ ๐พ , ๐ฎ ๐‘› = ๎€ท ๐‘ข ๐‘› , ๐›ผ ๎€ธ โˆˆ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ผ ๐‘› โˆถ โˆ’ ๐‘€ ๐พ < ๐‘ข ๐›ผ ( ๐‘ฅ ) < ๐‘€ ๐พ , | | ๐›ฝ | | . | ๐›ผ | โ‰ค ๐‘š + ( 4 . 2 1 ) The inclusion (4.20) now follows by the continuity of the mapping (4.4) and the definitions of the operators ๐ผ and ๐‘† , respectively. For each ๐‘› โˆˆ โ„• , set ๐œ† ๐‘› = i n f ๐น ๐›ฝ ๎ƒฉ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ผ ๐‘› ๎ƒช ๐œ‡ โˆˆ ๐’ฉ โ„’ ( ฮฉ ) , ๐‘› = s u p ๐น ๐›ฝ ๎ƒฉ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ผ ๐‘› ๎ƒช โˆˆ ๐’ฉ โ„’ ( ฮฉ ) . ( 4 . 2 2 ) Clearly the sequence ( ๐œ† ๐‘› ) is increasing, while ( ๐œ‡ ๐‘› ) is decreasing and ๐œ† ๐‘› โ‰ค ๐œ‡ ๐‘› , ๐‘› โˆˆ โ„• . ( 4 . 2 3 ) We show that ๎€ฝ ๐œ† ๐‘ข = s u p ๐‘› ๎€พ ๎€ฝ ๐œ‡ โˆถ ๐‘› โˆˆ โ„• = i n f ๐‘› ๎€พ โˆถ ๐‘› โˆˆ โ„• = ๐‘ข . ( 4 . 2 4 ) From condition (2) of (3.5) it follows that โˆ€ ๐œ– > 0 , ๐‘‰ โŠ† ฮฉ n o n e m p t y a n d o p e n โˆถ โˆƒ ๐‘ ๐œ– , ๐‘‰ โˆˆ โ„• , ๐‘ฅ โˆˆ ๐‘‰ โˆถ โˆ€ ๐‘› โ‰ฅ ๐‘ ๐œ– , ๐‘‰ โˆถ ๐‘ข ๐›ผ ( ๐‘ฅ ) โˆ’ ๐œ– < ๐‘ข ๐‘› , ๐›ผ ( ๐‘ฅ ) < ๐‘ข ๐›ผ ( ๐‘ฅ ) + ๐œ– , ๐ฎ ๐‘› = ๎€ท ๐‘ข ๐‘› , ๐›ผ ๎€ธ โˆˆ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ผ ๐‘› , ( 4 . 2 5 ) where ๐‘ข ๐›ผ โˆˆ ๐’ฉ โ„’ ( ฮฉ ) is the function associated with ( ๐ผ ๐›ผ ๐‘› ) through (2) of (3.5). In fact, due to (2.12), the inequality in (4.25) holds on a nonempty, open subset of ๐‘‰ . That is, โˆƒ ๐‘‰ ๎…ž โŠ† ๐‘‰ n o n e m p t y a n d o p e n โˆถ โˆ€ ๐‘ฅ โˆˆ ๐‘‰ ๎…ž โˆถ ๐‘ข ๐›ผ ( ๐‘ฅ ) โˆ’ ๐œ– < ๐‘ข ๐‘› , ๐›ผ ( ๐‘ฅ ) < ๐‘ข ๐›ผ ( ๐‘ฅ ) + ๐œ– , ๐ฎ ๐‘› = ๎€ท ๐‘ข ๐‘› , ๐›ผ ๎€ธ โˆˆ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ผ ๐‘› . ( 4 . 2 6 ) Therefore the continuity of the mapping (4.4) implies โˆ€ ๐œ– > 0 , ๐‘‰ โŠ† ฮฉ n o n e m p t y a n d o p e n โˆถ โˆƒ ๐‘ ๐œ– , ๐‘‰ โˆˆ โ„• , ๐‘‰ ๎…ž โŠ† ๐‘‰ n o n e m p t y a n d o p e n โˆถ โˆ€ ๐‘ฅ โˆˆ ๐‘‰ ๎…ž , ๐‘› โ‰ฅ ๐‘ ๐œ– , ๐‘‰ โˆถ | | | ๐น ๐›ฝ ๎€ท ๐ฎ ๐‘› ๎€ธ ( ๐‘ฅ ) โˆ’ ๐น ๐›ฝ ๎€ท ๐ฏ ๐‘› ๎€ธ | | | ( ๐‘ฅ ) < ๐œ– , ๐ฎ ๐‘› , ๐ฏ ๐‘› โˆˆ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ผ ๐‘› ( 4 . 2 7 ) so that โˆ€ ๐œ– > 0 , ๐‘‰ โŠ† ฮฉ n o n e m p t y a n d o p e n โˆถ โˆƒ ๐‘ ๐œ– , ๐‘‰ โˆˆ โ„• , ๐‘‰ ๎…ž โŠ† ๐‘‰ n o n e m p t y a n d o p e n โˆถ โˆ€ ๐‘ฅ โˆˆ ๐‘‰ ๎…ž , ๐‘› โ‰ฅ ๐‘ ๐œ– , ๐‘‰ โˆถ 0 < ๐œ† ๐‘› ( ๐‘ฅ ) โˆ’ ๐œ‡ ๐‘› ( ๐‘ฅ ) < ๐œ– ( 4 . 2 8 ) which verifies (4.24). From the sequential order denseness (2.28) of โ„ณ โ„’ 0 ( ฮฉ ) in ๐’ฉ โ„’ ( ฮฉ ) we obtain โˆƒ ๎€ท ๐œ† โˆ€ ๐‘› โˆˆ โ„• โˆถ ๐‘› , ๐‘š ๎€ธ , ๎€ท ๐œ‡ ๐‘› , ๐‘š ๎€ธ โŠ‚ โ„ณ โ„’ 0 ( ฮฉ ) โˆถ ( 1 ) ๐œ† ๐‘› , ๐‘š โ‰ค ๐œ† ๐‘› , ๐‘š + 1 โ‰ค ๐œ‡ ๐‘› , ๐‘š + 1 โ‰ค ๐œ‡ ๐‘› , ๐‘š ๎€ฝ ๐œ† , ๐‘š โˆˆ โ„• , ( 2 ) s u p ๐‘› , ๐‘š ๎€พ โˆถ ๐‘š โˆˆ โ„• = ๐œ† ๐‘› ๎€ฝ ๐œ‡ , i n f ๐‘› , ๐‘š ๎€พ โˆถ ๐‘š โˆˆ โ„• = ๐œ‡ ๐‘› . ( 4 . 2 9 ) By [23, Lemma 36] it follows that โˆƒ ๎€ท ๐œ† ๎…ž ๐‘› ๎€ธ , ๎€ท ๐œ‡ ๎…ž ๐‘› ๎€ธ โŠ‚ โ„ณ โ„’ 0 ( ฮฉ ) โˆถ ( 1 ) ๐œ† ๎…ž ๐‘› โ‰ค ๐œ† ๎…ž ๐‘› + 1 โ‰ค ๐œ‡ ๎…ž ๐‘› + 1 โ‰ค ๐œ‡ ๎…ž ๐‘› , ๐‘› โˆˆ โ„• , ( 2 ) ๐œ† ๎…ž ๐‘› โ‰ค ๐œ† ๐‘› โ‰ค ๐œ‡ ๐‘› โ‰ค ๐œ‡ ๎…ž ๐‘› ๎€ฝ ๐œ† , ๐‘› โˆˆ โ„• , ( 3 ) s u p ๎…ž ๐‘› ๎€พ โˆถ ๐‘› โˆˆ โ„• = ๐‘ข = ๎€ฝ ๐œ‡ ๐‘ข = i n f ๎…ž ๐‘› ๎€พ . โˆถ ๐‘› โˆˆ โ„• ( 4 . 3 0 ) Therefore the sequence of order intervals ( ๐ผ ๐‘› ) = ( [ ๐œ† ๎…ž ๐‘› , ๐œ‡ ๎…ž ๐‘› ] ) satisfies (1) and (2) of (3.5) and โˆ€ ๐‘› โˆˆ โ„• โˆถ ๐น ๐›ฝ ๎ƒฉ ๎‘ | ๐›ผ | โ‰ค ๐‘š + | ๐›ฝ | ๐ผ ๐›ฝ ๐‘› ๎ƒช โŠ† ๐ผ ๐‘› ( 4 . 3 1 ) which shows that ๐น ๐›ฝ is uniformly continuous. Therefore, according to the diagram (4.17) each of the mappings ๐‘‡ ๐›ฝ is also uniformly continuous.
The uniform continuity of the mapping (4.12) now follows by the commutative diagram Eq5(4.32) This completes the proof.

As a consequence of Theorem 4.1 we obtain a canonical extension of the mapping (4.9) to a mapping ๐‘‡ โ™ฏ โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) โŸถ โ„ณ โ„’ โˆž ( ฮฉ ) . ( 4 . 3 3 ) Indeed, since ๐‘‡ is uniformly continuous, there exists a unique uniformly continuous extension of (4.9). This extension of the nonlinear partial differential operator to the space of generalized functions gives rise to a concept of generalized solution of (4.1). Namely, any solution ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) of the extended equation ๐‘‡ โ™ฏ ๐‘ข โ™ฏ = ๐‘“ ( 4 . 3 4 ) corresponding to the nonlinear PDE (4.1) is interpreted as a generalized solution of (4.1).

In proving the uniform continuity of the mapping (4.9) in Theorem 4.1, we also showed that the mappings (4.11) and (4.12) are uniformly continuous. Since the mapping (3.12) is a uniformly continuous embedding, the diagram (4.13) may be extended toEq6(4.35) where ๐‘‡ โˆž โ™ฏ is the uniformly continuous extension of (4.12). Furthermore, each of the mappings (4.11) extend uniquely to uniformly continuous mappings ๐‘‡ ๐›ฝ โ™ฏ โˆถ ๐’ฉ โ„’ โˆž ( ฮฉ ) โŸถ ๐’ฉ โ„’ ( ฮฉ ) , ๐›ฝ โˆˆ โ„• ๐‘› . ( 4 . 3 6 ) Since (4.10) coincides with ๐’Ÿ ๐›ฝ โ™ฏ โˆ˜ ๐‘‡ โ™ฏ on the dense subspace โ„ณ โ„’ โˆž ( ฮฉ ) of ๐’ฉ โ„’ โˆž ( ฮฉ ) , it follows [24] that โˆ€ ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ๐‘‡ ( ฮฉ ) โˆถ ๐›ฝ โ™ฏ ๐‘ข โ™ฏ = ๐’Ÿ ๐›ฝ โ™ฏ ๎‚€ ๐‘‡ โ™ฏ ๐‘ข โ™ฏ ๎‚ . ( 4 . 3 7 ) From the commutative diagram (4.35), and the identity (3.15) it follows that the mapping ๐‘‡ โˆž โ™ฏ may be represented as ๐‘‡ โˆž โ™ฏ โˆถ ๐’ฉ โ„’ โˆž ๎‚€ ๐‘‡ ( ฮฉ ) โˆ‹ ๐‘ข โŸผ ๐›ฝ โ™ฏ ๐‘ข โ™ฏ ๎‚ ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ ๐’ฉ โ„’ ( ฮฉ ) โ„• ๐‘› . ( 4 . 3 8 ) The meaning of this is that the usual situation encountered when dealing with classical, ๐’ž โˆž -smooth solution of (4.1), namely, โˆ€ ๐‘ข โˆˆ ๐’ž โˆž ( ฮฉ ) a s o l u t i o n o f ( 4 . 1 ) , ๐›ฝ โˆˆ โ„• ๐‘› โˆถ ๐ท ๐›ฝ ( ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ) ( ๐‘ฅ ) = ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) , ๐‘ฅ โˆˆ ฮฉ , ( 4 . 3 9 ) remains valid, in a generalized sense, for any generalized solution ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) of (4.34). That is, โˆ€ ๐›ฝ โˆˆ โ„• ๐‘› โˆถ ๐‘‡ ๐›ฝ โ™ฏ ๐‘ข โ™ฏ = ๐’Ÿ ๐›ฝ โ™ฏ ๎‚€ ๐‘‡ โ™ฏ ๐‘ข โ™ฏ ๎‚ = ๐ท ๐›ฝ ๐‘“ . ( 4 . 4 0 ) The main result of this paper, concerning the existence of solutions of (4.34), is the following.

Theorem 4.2. Consider a nonlinear PDE of the form (4.1). If the nonlinear operator (4.9) satisfies (4.7), then there exists some ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) that satisfies (4.34).

Proof. Let us express ฮฉ as ๎š ฮฉ = ๐œˆ โˆˆ โ„• ๐ถ ๐œˆ , ( 4 . 4 1 ) where, for ๐œˆ โˆˆ โ„• , the compact sets ๐ถ ๐œˆ are ๐‘› -dimensional intervals ๐ถ ๐œˆ = ๎€บ ๐‘Ž ๐œˆ , ๐‘ ๐œˆ ๎€ป ( 4 . 4 2 ) with ๐‘Ž ๐œˆ = ( ๐‘Ž ๐œˆ , 1 , โ€ฆ , ๐‘Ž ๐œˆ , ๐‘› ) , ๐‘ ๐œˆ = ( ๐‘ ๐œˆ , 1 , โ€ฆ , ๐‘ ๐œˆ , ๐‘› ) โˆˆ โ„ ๐‘› and ๐‘Ž ๐œˆ , ๐‘— โ‰ค ๐‘ ๐œˆ , ๐‘— for every ๐‘— = 1 , โ€ฆ , ๐‘› . We assume that { ๐ถ ๐œˆ โˆถ ๐œˆ โˆˆ โ„• } is locally finite, that is, ๎€ฝ โˆ€ ๐‘ฅ โˆˆ ฮฉ โˆถ โˆƒ ๐‘‰ โŠ† ฮฉ a n e i g h b o r h o o d o f ๐‘ฅ โˆถ ๐œˆ โˆˆ โ„• โˆถ ๐ถ ๐œˆ ๎€พ โˆฉ ๐‘‰ โ‰  โˆ… i s ๏ฌ n i t e . ( 4 . 4 3 ) Such a partition of ฮฉ exists, see for instance [25].
Fix ๐œˆ โˆˆ โ„• . To each ๐‘ฅ 0 โˆˆ ๐ถ ๐œˆ we apply (4.7) so that we obtain โˆ€ ๐‘ฅ 0 โˆˆ ๐ถ ๐œˆ โˆถ ๎€ท ๐‘ฅ โˆƒ ๐œ‰ 0 ๎€ธ = ๎€ท ๐œ‰ ๐›ผ ๎€ท ๐‘ฅ 0 ๎€ธ ๎€ธ โˆˆ โ„ โ„• ๐‘› , ๐น โˆž ๎€ท ๐‘ฅ 0 ๎€ท ๐‘ฅ , ๐œ‰ 0 = ๎€ท ๐ท ๎€ธ ๎€ธ ๐›ฝ ๐‘“ ๎€ท ๐‘ฅ 0 ๎€ธ ๎€ธ ๐›ฝ โˆˆ โ„• ๐‘› โˆถ โˆƒ ๐›ฟ ๐‘ฅ 0 , ๐œ– ๐‘ฅ 0 > 0 โˆถ ( 1 ) ๐น โˆž โˆถ ๐ต ๐›ฟ ๐‘ฅ 0 ๎€ท ๐‘ฅ 0 ๎€ธ ร— ๐‘Š 1 ๐‘ฅ 0 โŸถ โ„ โ„• ๎€ท ๐ท o p e n ( 2 ) ๐›ฝ ๎€ธ ๐‘“ ( ๐‘ฅ ) ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ ๐น โˆž ๎€ท { ๐‘ฅ } ร— ๐‘Š 1 ๐‘ฅ 0 ๎€ธ , ๐‘ฅ โˆˆ ๐ต ๐›ฟ ๐‘ฅ 0 ๎€ท ๐‘ฅ 0 ๎€ธ , ( 4 . 4 4 ) where ๐‘Š 1 ๐œ– ๐‘ฅ 0 is the neighborhood of ๐œ‰ ( ๐‘ฅ 0 ) defined as ๐‘Š 1 ๐‘ฅ 0 = ๎‘ | ๐›ผ | โ‰ค ๐‘€ ๐‘ฅ 0 , 1 ๐ต ๐œ– ๐‘ฅ 0 ๎€ท ๐œ‰ ๐›ผ ๎€ท ๐‘ฅ 0 ๎€ธ ๎€ธ ร— โ„ โ„• ๐‘› ( 4 . 4 5 ) for ๐‘€ 1 > ๐‘š a sufficiently large integer. Note that our choice of integer ๐‘€ 1 does not depend on the set ๐ถ ๐œˆ or the point ๐‘ฅ 0 โˆˆ ๐ถ ๐œˆ . Consider now a function ๐‘ˆ ๐‘ฅ 0 โˆˆ ๐’ž โˆž ( ฮฉ ) such that โˆ€ ๐›ผ โˆˆ โ„• ๐‘› โˆถ ๐ท ๐›ผ ๐‘ˆ ๐‘ฅ 0 ๎€ท ๐‘ฅ 0 ๎€ธ = ๐œ‰ ๐›ผ ๎€ท ๐‘ฅ 0 ๎€ธ . ( 4 . 4 6 ) From the continuity of ๐‘ˆ ๐‘ฅ 0 and (4.44) it follows that โˆ€ ๐‘ฅ 0 โˆˆ ๐ถ ๐œˆ โˆถ โˆƒ ๐›ฟ ๐‘ฅ 0 > 0 โˆถ โˆ€ ๐‘ฅ โˆˆ ๐ต ๐›ฟ ๐‘ฅ 0 ๎€ท ๐‘ฅ 0 ๎€ธ , | | ๐›ฝ | | โ‰ค ๐‘€ 1 โˆ’ ๐‘š โˆถ ( 1 ) ๐น โˆž โˆถ ๐ต ๐›ฟ ๐‘ฅ 0 ๎€ท ๐‘ฅ 0 ๎€ธ ร— ๐‘Š 1 ๐‘ฅ 0 โŸถ โ„ โ„• ( o p e n , 2 ) ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) โˆ’ ๐œ– ๐‘ฅ 0 < ๐‘‡ ๐›ฝ ๐‘ˆ ๐‘ฅ 0 ( ๐‘ฅ ) < ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) + ๐œ– ๐‘ฅ 0 , ๎€ท ๐ท ( 3 ) ๐›ผ ๐‘ˆ ๐‘ฅ 0 ๎€ธ ( ๐‘ฅ ) ๐›ผ โˆˆ โ„• ๐‘› โˆˆ ๐‘Š 1 ๐‘ฅ 0 , ๎€ท ๐ท ( 4 ) ๐›ฝ ๐‘“ ๎€ธ ( ๐‘ฅ ) ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ ๐น โˆž ๎€ท { ๐‘ฅ } ร— ๐‘Š 1 ๐‘ฅ 0 ๎€ธ . ( 4 . 4 7 ) Since ๐ถ ๐œˆ is compact (4.47) may be strengthened to โˆƒ ๐›ฟ ๐œˆ > 0 โˆถ โˆ€ ๐‘ฅ 0 โˆˆ ๐ถ ๐œˆ , ๐‘ฅ โˆˆ ๐ต ๐›ฟ ๐œˆ ๎€ท ๐‘ฅ 0 ๎€ธ , | | ๐›ฝ | | โ‰ค ๐‘€ 1 โˆ’ ๐‘š โˆถ ( 1 ) ๐น โˆž โˆถ ๐ต ๐›ฟ ๐œˆ ๎€ท ๐‘ฅ 0 ๎€ธ ร— ๐‘Š 1 ๐‘ฅ 0 โŸถ โ„ โ„• o p e n , ( 2 ) ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) โˆ’ ๐œ– ๐‘ฅ 0 < ๐‘‡ ๐›ฝ ๐‘ˆ ๐‘ฅ 0 ( ๐‘ฅ ) < ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) + ๐œ– ๐‘ฅ 0 , ๎€ท ๐ท ( 3 ) ๐›ผ ๐‘ˆ ๐‘ฅ 0 ๎€ธ ( ๐‘ฅ ) ๐›ผ โˆˆ โ„• ๐‘› โˆˆ ๐‘Š 1 ๐‘ฅ 0 , ๎€ท ๐ท ( 4 ) ๐›ฝ ๎€ธ ๐‘“ ( ๐‘ฅ ) ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ ๐น โˆž ๎€ท { ๐‘ฅ } ร— ๐‘Š 1 ๐‘ฅ 0 ๎€ธ . ( 4 . 4 8 ) Now subdivide ๐ถ ๐œˆ into locally finite, compact ๐‘› -dimensional intervals ๐ถ ๐œˆ , 1 , โ€ฆ , ๐ถ ๐œˆ , ๐พ ๐œˆ with pairwise disjoint interiors such that each ๐ถ ๐œˆ , ๐‘– has diameter not exceeding ๐›ฟ ๐œˆ . Let ๐‘Ž ๐œˆ , ๐‘– denote the midpoint of ๐ถ ๐œˆ , ๐‘– . Then, from (4.48), it follows that โˆƒ ๐‘ˆ ๐œˆ , ๐‘– โˆˆ ๐’ž โˆž ( ฮฉ ) , ๐œ– ๐œˆ , ๐‘– > 0 โˆถ โˆ€ ๐‘ฅ โˆˆ ๐ถ ๐œˆ , ๐‘– , | | ๐›ฝ | | โ‰ค ๐‘€ 1 ( โˆ’ ๐‘š โˆถ 1 ) ๐น โˆž โˆถ i n t ๐ถ ๐œˆ , ๐‘– ร— ๐‘Š 1 ๐œˆ , ๐‘– โŸถ โ„ โ„• ( o p e n , 2 ) ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) โˆ’ ๐œ– ๐œˆ , ๐‘– < ๐‘‡ ๐›ฝ ๐‘ˆ ๐œˆ ( ๐‘ฅ ) < ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) + ๐œ– ๐œˆ , ๐‘– , ๎€ท ๐ท ( 3 ) ๐›ผ ๐‘ˆ ๐œˆ ๎€ธ ( ๐‘ฅ ) ๐›ผ โˆˆ โ„• ๐‘› โˆˆ ๐‘Š 1 ๐œˆ , ๐‘– , ๎€ท ๐ท ( 4 ) ๐›ฝ ๐‘“ ๎€ธ ( ๐‘ฅ ) ๐›ฝ โˆˆ โ„• ๐‘› โˆˆ ๐น โˆž ๎€ท { ๐‘ฅ } ร— ๐‘Š 1 ๐œˆ , ๐‘– ๎€ธ , ( 4 . 4 9 ) where ๐‘Š 1 ๐œˆ , ๐‘– = ๎‘ | ๐›ผ | โ‰ค ๐‘€ ๐œˆ , 1 ๐ต ๐œ– ๐œˆ ๎€ท ๐œ‰ ๐›ผ ๎€ท ๐‘Ž ๐œˆ , ๐‘– ๎€ธ ๎€ธ ร— โ„ โ„• ๐‘› . ( 4 . 5 0 ) Now consider the function ๐‘‰ 1 = ๎“ ๐œˆ โˆˆ โ„• โŽ› โŽœ โŽœ โŽ ๐พ ๐œˆ ๎“ ๐‘– = 1 ๐œ’ ๐œˆ , ๐‘– ๐‘ˆ ๐œˆ , ๐‘– โŽž โŽŸ โŽŸ โŽ  , ( 4 . 5 1 ) with ๐œ’ ๐œˆ , ๐‘– the characteristic function of i n t ๐ถ ๐œˆ , ๐‘– , the interior of ๐ถ ๐œˆ , ๐‘– . Clearly we have ๐‘‰ 1 โˆˆ ๐’ž โˆž ( ฮฉ โงต ฮ“ 1 ) where ฮ“ 1 โŠ‚ ฮฉ is the closed nowhere dense set ฮ“ 1 โŽ› โŽœ โŽœ โŽ ๎š = ฮฉ โงต ๐œˆ โˆˆ โ„• โŽ› โŽœ โŽœ โŽ ๐พ ๐œˆ ๎š ๐‘– = 1 i n t ๐ถ ๐œˆ , ๐‘– โŽž โŽŸ โŽŸ โŽ  โŽž โŽŸ โŽŸ โŽ  . ( 4 . 5 2 ) Upon application of (4.49) we find โˆ€ ๐œˆ โˆˆ โ„• , ๐‘– = 1 , โ€ฆ , ๐พ ๐œˆ โˆถ โˆ€ ๐‘ฅ โˆˆ i n t ๐ถ ๐œˆ , ๐‘– , | | ๐›ฝ | | โ‰ค ๐‘€ 1 ๐ท โˆ’ ๐‘š โˆถ ๐›ฝ ๐‘“ ( ๐‘ฅ ) โˆ’ ๐œ– ๐œˆ , ๐‘– < ๐ท ๐›ฝ ๎€ท ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘‰ 1 ๎€ธ ( ๐‘ฅ ) < ๐ท ๐›ฝ ๐‘“ ( ๐‘ฅ ) + ๐œ– ๐œˆ , ๐‘– . ( 4 . 5 3 ) Furthermore, โˆ€ ๐œˆ โˆˆ โ„• , ๐‘– = 1 , โ€ฆ , ๐พ ๐œˆ โˆถ โˆ€ ๐‘ฅ โˆˆ i n t ๐ถ ๐œˆ , ๐‘– , | ๐›ผ | โ‰ค ๐‘€ 1 โˆถ ๐œ† ๐›ผ 1 ( ๐‘ฅ ) < ๐ท ๐›ผ ๐‘‰ 1 ( ๐‘ฅ ) < ๐œ‡ ๐›ผ 1 ( ๐‘ฅ ) , ( 4 . 5 4 ) where ๐œ† ๐›ผ 1 , ๐œ‡ ๐›ผ 1 โˆˆ ๐’ž 0 ( ฮฉ ) are defined as ๐œ† ๐›ผ 1 ( ๐‘ฅ ) = ๐œ‰ ๐›ผ ๎€ท ๐‘Ž ๐œˆ , ๐‘– ๎€ธ โˆ’ ๐œ– ๐œˆ , ๐‘– i f ๐‘ฅ โˆˆ ๐ถ ๐œˆ , ๐‘– , ๐œ‡ ๐›ผ 1 ( ๐‘ฅ ) = ๐œ‰ ๐›ผ ๎€ท ๐‘Ž ๐œˆ , ๐‘– ๎€ธ + ๐œ– ๐œˆ , ๐‘– i f ๐‘ฅ โˆˆ ๐ถ ๐œˆ , ๐‘– . ( 4 . 5 5 ) Clearly the functions ๐œ† ๐›ผ 1 and ๐œ‡ ๐›ผ 1 satisfy ๐œ‡ ๐›ผ 1 ( ๐‘ฅ ) โˆ’ ๐œ† ๐›ผ 1 ( ๐‘ฅ ) = 2 ๐œ– ๐œˆ , ๐‘– i f ๐‘ฅ โˆˆ ๐ถ ๐œˆ , ๐‘– . ( 4 . 5 6 ) Continuing in this way, we may construct a sequence ( ฮ“ ๐‘› ) of closed nowhere dense subsets of ฮฉ such that ฮ“ ๐‘› โŠ† ฮ“ ๐‘› + 1 for each ๐‘› โˆˆ โ„• , a strictly increasing sequence of integers ( ๐‘€ ๐‘› ) and functions ๐‘‰ ๐‘› โˆˆ ๐’ž โˆž ( ฮฉ โงต ฮ“ ๐‘› ) such that โˆ€ ๐œˆ โˆˆ โ„• , ๐‘– = 1 , โ€ฆ , ๐พ ๐œˆ โˆถ โˆ€ ๐‘ฅ โˆˆ i n t ๐ถ ๐œˆ , ๐‘– โงต ฮ“ ๐‘› , | | ๐›ฝ | | โ‰ค ๐‘€ ๐‘› ๐ท โˆ’ ๐‘š โˆถ ๐›ฝ ๐œ– ๐‘“ ( ๐‘ฅ ) โˆ’ ๐œˆ , ๐‘– ๐‘› < ๐‘‡ ๐›ฝ ๐‘‰ ๐‘› ( ๐‘ฅ ) < ๐ท ๐›ฝ ๐œ– ๐‘“ ( ๐‘ฅ ) + ๐œˆ , ๐‘– ๐‘› . ( 4 . 5 7 ) Furthermore, for each ๐‘› โˆˆ โ„• we have โˆ€ | ๐›ผ | โ‰ค ๐‘€ ๐‘› โˆถ โˆƒ ๐œ† ๐›ผ ๐‘› , ๐œ‡ ๐›ผ ๐‘› โˆˆ ๐’ž 0 ๎€ท ฮฉ โงต ฮ“ ๐‘› ๎€ธ โˆถ โˆ€ ๐‘ฅ โˆˆ ฮฉ โงต ฮ“ ๐‘› , ๐œˆ โˆˆ โ„• , ๐‘– = 1 , โ€ฆ , ๐พ ๐œˆ โˆถ ( 1 ) ๐œ† ๐›ผ ๐‘› ( ๐‘ฅ ) < ๐ท ๐›ผ ๐‘‰ ๐‘› ( ๐‘ฅ ) < ๐œ‡ ๐›ผ ๐‘› ( ๐‘ฅ ) , ( 2 ) ๐œ‡ ๐›ผ ๐‘› ( ๐‘ฅ ) โˆ’ ๐œ† ๐›ผ ๐‘› ๐œ– ( ๐‘ฅ ) < ๐œˆ , ๐‘– ๐‘› , ๐‘ฅ โˆˆ i n t ๐ถ ๐œˆ , ๐‘– โงต ฮ“ ๐‘› , ( 3 ) ๐œ‡ ๐›ผ ๐‘› โ‰ค ๐œ‡ ๐›ผ ๐‘› + 1 โ‰ค ๐œ† ๐›ผ ๐‘› + 1 โ‰ค ๐œ† ๐›ผ ๐‘› . ( 4 . 5 8 ) Consider now the functions ๐‘ข ๐‘› = ๎€ท ๐‘‰ ( ๐ผ โˆ˜ ๐‘† ) ๐‘› ๎€ธ โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) . ( 4 . 5 9 ) From (4.57) as well as the monotonicity and idempotency of the operator ๐ผ โˆ˜ ๐‘† , it follows that the sequence ( ๐‘‡ ๐‘ข ๐‘› ) converges to ๐‘“ in โ„ณ โ„’ โˆž ( ฮฉ ) . Furthermore, (4.58) implies that the sequence ( ๐‘ข ๐‘› ) is a Cauchy sequence in โ„ณ โ„’ โˆž ( ฮฉ ) . As such, it follows by Theorem 4.1 that there is some ๐‘ข โ™ฏ โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) that satisfies (4.34).

It should be noted that the concept of generalized solution of nonlinear PDEs introduced here is similar to many of those that are typical in the literature, at least as far as the way in which the concept of a generalized solution is arrived at. In this regard, we may recall a construction of generalized solutions of PDEs that is representative of many of those methods that are customary in the study of PDEs. In order to construct a generalized solution of a nonlinear PDE ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) , ๐‘ฅ โˆˆ ฮฉ โŠ† โ„ ๐‘› ( 4 . 6 0 ) one considers some relatively small space ๐‘‹ of usual, sufficiently smooth functions on ฮฉ , and a space ๐‘Œ of functions on ฮฉ such that ๐‘“ โˆˆ ๐‘Œ . With the partial differential operator ๐‘‡ ( ๐‘ฅ , ๐ท ) one associates a mapping ๐‘‡ โˆถ ๐‘‹ โŸถ ๐‘Œ ( 4 . 6 1 ) in the usual way, namely, for every ๐‘ข โˆˆ ๐‘‹ one has ๐‘‡ ๐‘ข ( ๐‘ฅ ) = ๐‘‡ ( ๐‘ฅ , ๐ท ) ๐‘ข ( ๐‘ฅ ) , ๐‘ฅ โˆˆ ฮฉ . ( 4 . 6 2 ) The spaces ๐‘‹ and ๐‘Œ are equipped with uniform topologies, in fact, usually metrizable locally convex linear space topologies. The mapping (4.61) is assumed to be suitably compatible with the topologies on ๐‘‹ and ๐‘Œ . In particular, ๐‘‡ is supposed to be uniformly continuous, which enables one to extend the mapping (4.61) in canonical way to the completions ๐‘‹ โ™ฏ and ๐‘Œ โ™ฏ of the spaces ๐‘‹ and ๐‘Œ with respect to their respective uniform topologies. In this case, one ends up with a mapping ๐‘‡ โ™ฏ โˆถ ๐‘‹ โ™ฏ โŸถ ๐‘Œ โ™ฏ . ( 4 . 6 3 ) A solution ๐‘ข โ™ฏ โˆˆ ๐‘‹ โ™ฏ of the generalized equation ๐‘‡ โ™ฏ ๐‘ข โ™ฏ = ๐‘“ ( 4 . 6 4 ) is now interpreted as a generalized solution of (4.60). Showing that such a generalized solution exists is often a rather difficult task, and may involve highly nontrivial ideas from function analysis and topology. Furthermore, a method that applies to a particular equation, may fail completely if the equation is changed slightly. This is in contradistinction with the generality and type independence of the solution method presented here. Moreover, one may note that the sequence of approximating solutions obtained in the proof of Theorem 4.2 is constructed using only basic properties of continuous, real-valued functions and elementary topology of Euclidean space.

5. An Application

In this section we show how the general theory developed in Sections 3 and 4 may be applied to particular equations, in fact, systems of equations. Furthermore, it is also demonstrated how the techniques of the preceding sections may be adapted so as to also incorporate initial and/or boundary conditions that may be associated with a given system of PDEs. In this way, we come to appreciate yet another advantage of solving linear and nonlinear PDEs by the methods introduced in this paper, as well as in [5โ€“7]. Namely, and in contradistinction with the customary linear functional analytic methods, initial and/or boundary value problems are solved by essentially the same techniques that apply to the free problem. Indeed, the basic theory need only be adjusted in a minimal way in order to incorporate such additional conditions.

In this regard, we consider the one-dimensional parametrically driven, damped nonlinear Schrรถdinger equation ๐‘– ๐œ“ ๐‘ก + ๐œ“ ๐‘ฅ ๐‘ฅ | | ๐œ“ | | + 2 2 ๐œ™ = โ„Ž ๐œ“ โˆ— ๐‘’ 2 ๐‘– ๐‘ก โˆ’ ๐‘– ๐›พ ๐œ“ , ( 5 . 1 ) with ๐œ“ โˆ— denoting complex conjugation, which, upon setting ๐œ“ = ๐‘ข + ๐‘– ๐‘ฃ , may be written as a system of equations โˆ’ ๐‘ฃ ๐‘ก + ๐‘ข ๐‘ฅ ๐‘ฅ + 2 ๐‘ข 3 + 2 ๐‘ฃ 2 ๐‘ข ๐‘ข = โ„Ž ๐‘ข c o s ( 2 ๐‘ก ) + โ„Ž ๐‘ฃ s i n ( 2 ๐‘ก ) + ๐›พ ๐‘ฃ , ๐‘ก + ๐‘ฃ ๐‘ฅ ๐‘ฅ + 2 ๐‘ฃ 3 + 2 ๐‘ข 2 ๐‘ฃ = โˆ’ โ„Ž ๐‘ฃ c o s ( 2 ๐‘ก ) + โ„Ž ๐‘ข s i n ( 2 ๐‘ก ) โˆ’ ๐›พ ๐‘ข , ( 5 . 2 ) subject to the initial condition ๐‘ข ( ๐‘ฅ , 0 ) = ๐‘ข 0 ๐‘ฃ ( ๐‘ฅ ) , ( ๐‘ฅ , 0 ) = ๐‘ฃ 0 ( ๐‘ฅ ) , ๐‘ฅ โˆˆ โ„ , ( 5 . 3 ) where ๐‘ข 0 , ๐‘ฃ 0 โˆˆ ๐’ž โˆž ( โ„ ) . We will show that the initial value problem (5.2) and (5.3) admits a generalized solution ( ๐‘ข โ™ฏ , ๐‘ฃ โ™ฏ ) โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 , where ฮฉ = โ„ ร— [ 0 , โˆž ) . Furthermore, this solution is shown to be a strongly generic weak solution of (5.2) in the following sense.

Definition 5.1. A pair of functions ( ๐‘ข , ๐‘ฃ ) โˆˆ ๐ฟ โˆž l o c ( ฮฉ ) 2 is a strongly generic weak solution of (5.2) if there exists a closed nowhere dense set ฮ“ โŠ‚ โ„ ร— ( 0 , โˆž ) so that ( ๐‘ข , ๐‘ฃ ) satisfies (5.2) weakly on ( โ„ ร— ( 0 , โˆž ) ) โงต ฮ“ .

The motivation for this definition comes from systems theory. In this regard, recall [26] that a property of a system defined on an open subset ฮฉ of โ„ ๐‘› is strongly generic if it holds on an open and dense subset of ฮฉ .

We may note that a large variety of nonlinear resonant phenomena in various physical media is described by the system of equations (5.1). Among these, we may count the Faraday resonance in fluid dynamics [27], instabilities in plasma [28], oscillons in granular materials [29] and anisotropic XY model of ferromagnetism [30โ€“33]. In these applications, it is often the so-called soliton solutions that are of interest, and a lot of work has been carried out on the analysis of such solutions, see for instance [34โ€“36]. Here we are concerned with just the basic existence and regularity results for solutions of the initial value problem (5.2) to (5.3), for a large class of initial conditions. Of course, in case the initial data in (5.3) is analytic, the Cauchy-Kovalevskaia Theorem guarantees that a solution exists, at least locally, while the global version of that theorem [21] gives existence of a generalized solution, in a suitable algebra of generalized functions, which is analytic everywhere except possibly on a closed nowhere dense subset of ฮฉ . However, in the case of arbitrary ๐’ž โˆž -smooth initial values, we are not aware of any general existence or regularity results.

Before we consider the problem of existence of generalized solutions of (5.2) and (5.3), let us express the problem in the notation of Section 4. In particular, we write the system of equations (5.2) in the form ๐“ ( ๐‘ฅ , ๐‘ก , ๐ท ) ( ๐‘ข , ๐‘ฃ ) ( ๐‘ฅ , ๐‘ก ) = ๐ŸŽ ( ๐‘ฅ , ๐‘ก ) , ( ๐‘ฅ , ๐‘ก ) โˆˆ ฮฉ , ( 5 . 4 ) where ๐ŸŽ denotes the two-dimensional vector valued function which is identically ๐ŸŽ on ฮฉ . As in Section 4, the operator ๐“ ( ๐‘ฅ , ๐‘ก , ๐ท ) is defined through a jointly continuous, ๐’ž โˆž -smooth mapping ๐… โˆถ ฮฉ ร— โ„ 1 0 โŸถ โ„ 2 ( 5 . 5 ) as ๐“ ( ๐‘ฅ , ๐‘ก , ๐ท ) ( ๐‘ข , ๐‘ฃ ) ( ๐‘ฅ , ๐‘ก ) = ๐… ( ๐‘ฅ , ๐‘ก , โ€ฆ , ๐ท ๐›ผ ๐‘ข ( ๐‘ฅ , ๐‘ก ) , โ€ฆ , ๐ท ๐›ผ ๐‘ฃ ( ๐‘ฅ , ๐‘ก ) , โ€ฆ ) , | ๐›ผ | โ‰ค 2 . ( 5 . 6 ) In particular, the mapping ๐… takes the form ๎ƒฉ ๐… ( ๐‘ฅ , ๐‘ก , ๐œ‰ ) = โˆ’ ๐œ‰ 4 + ๐œ‰ 9 + 2 ๐œ‰ 3 1 + 2 ๐œ‰ 1 ๐œ‰ 2 2 โˆ’ โ„Ž ๐œ‰ 1 c o s ( 2 ๐‘ก ) โˆ’ โ„Ž ๐œ‰ 2 s i n ( 2 ๐‘ก ) โˆ’ ๐›พ ๐œ‰ 2 โˆ’ ๐œ‰ 3 + ๐œ‰ 1 0 + 2 ๐œ‰ 3 2 + 2 ๐œ‰ 2 1 ๐œ‰ 2 + โ„Ž ๐œ‰ 2 c o s ( 2 ๐‘ก ) โˆ’ โ„Ž ๐œ‰ 1 s i n ( 2 ๐‘ก ) + ๐›พ ๐œ‰ 1 ๎ƒช . ( 5 . 7 ) With the nonlinear operator ๐“ ( ๐‘ฅ , ๐‘ก , ๐ท ) we may associate a mapping ๐“ โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 โŸถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 , ( 5 . 8 ) the components of which are defined as ๐‘‡ ๐‘— โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎€ท ๐น โˆ‹ ( ๐‘ข , ๐‘ฃ ) โŸผ ( ๐ผ โˆ˜ ๐‘† ) ๐‘— ๎€ท โ‹… , โ‹… , โ€ฆ , ๐’Ÿ ๐›ผ ๐‘ข , โ€ฆ , ๐’Ÿ ๐›ผ ๐‘ฃ , โ€ฆ ๎€ธ ๎€ธ โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) ( 5 . 9 ) for | ๐›ผ | โ‰ค 2 and ๐‘— = 1 , 2 , where ๐น 1 and ๐น 2 are the components of the mapping (5.5).

Furthermore, we express the partial derivatives of ๐“ ( ๐‘ข , ๐‘ฃ ) through โˆ€ ๐›ฝ โˆˆ โ„• 2 โˆถ โˆ€ ( ๐‘ข , ๐‘ฃ ) โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) 2 โˆถ ๐’Ÿ ๐›ฝ ๎€ท ๐‘‡ ๐‘— ๎€ธ ( ๐‘ข , ๐‘ฃ ) = ๐‘‡ ๐›ฝ ๐‘— ( ๐‘ข , ๐‘ฃ ) , ๐‘— = 1 , 2 . ( 5 . 1 0 ) Here, for ๐›ฝ โˆˆ โ„• 2 and ๐‘— = 1 , 2 , ๐‘‡ ๐›ฝ ๐‘— โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 โŸถ โ„ณ โ„’ โˆž ( ฮฉ ) ( 5 . 1 1 ) is defined as ๐‘‡ ๐›ฝ ๐‘— โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎‚€ ๐น โˆ‹ ( ๐‘ข , ๐‘ฃ ) โŸผ ( ๐ผ โˆ˜ ๐‘† ) ๐›ฝ ๐‘— ๎€ท โ‹… , โ‹… , โ€ฆ , ๐’Ÿ ๐›ผ ๐‘ข , โ€ฆ , ๐’Ÿ ๐›ผ ๎€ธ ๎‚ ๐‘ฃ , โ€ฆ โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) , ( 5 . 1 2 ) where ๐น ๐›ฝ ๐‘— โˆถ ฮฉ ร— โ„ 2 ๐‘€ ๐›ฝ โ†’ โ„ is the ๐’ž โˆž -smooth mapping such that ๐ท ๐›ฝ ๎€ท ๐น ๐‘— ( โ‹… , โ‹… , โ€ฆ . , ๐ท ๐›ผ ๐‘ข , โ€ฆ , ๐ท ๐›ผ ๎€ธ ๐‘ฃ , โ€ฆ ) ( ๐‘ฅ , ๐‘ก ) = ๐น ๐›ฝ ๐‘— ( ๐‘ฅ , ๐‘ก , โ€ฆ . , ๐ท ๐›ผ ๐‘ข ( ๐‘ฅ , ๐‘ก ) , โ€ฆ , ๐ท ๐›ผ ๐‘ฃ ( ๐‘ฅ , ๐‘ก ) , โ€ฆ ) , ( ๐‘ฅ , ๐‘ก ) โˆˆ ฮฉ ( 5 . 1 3 ) for every ๐‘ข โˆˆ ๐’ž โˆž ( ฮฉ ) . Just as is done in Section 4, we may use the mappings (5.11) to obtain a representation of the the operator ๐“ through the mappings ๐“ โˆž โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎‚€ ๐‘‡ โˆ‹ ( ๐‘ข , ๐‘ฃ ) โŸผ ๐›ฝ 1 ( ๐‘ข , ๐‘ฃ ) , ๐‘‡ ๐›ฝ 2 ๎‚ ( ๐‘ข , ๐‘ฃ ) ๐›ฝ โˆˆ โ„• 2 โˆˆ โ„ณ โ„’ 0 ( ฮฉ ) 2 โ„• 2 , ๐ƒ 2 โˆถ โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎€ท ๐’Ÿ โˆ‹ ( ๐‘ข , ๐‘ฃ ) โŸผ ๐›ฝ ๐‘ข , ๐’Ÿ ๐œŒ ๐‘ฃ ๎€ธ ๐›ฝ , ๐œŒ โˆˆ โ„• 2 โˆˆ โ„ณ โ„’ โˆž ( ฮฉ ) 2 โ„• 2 . ( 5 . 1 4 ) In particular, the diagramEq7(5.15) commutes.

We may associate with the mapping (5.5) the continuous mapping ๐… โˆž โˆถ ฮฉ ร— โ„ 2 โ„• 2 โŸผ โ„ 2 โ„• 2 ( 5 . 1 6 ) defined as ๐… โˆž ๎‚€ ๎€ท ๐œ‰ ๐‘ฅ , ๐‘ก , ๐›ผ 1 , ๐œ‰ ๐œŒ , 2 ๎€ธ ๐›ฝ โˆˆ โ„• 2 ๎‚ = โŽ› โŽœ โŽœ โŽ ๐น ๐›ฝ 1 ๎‚€ ๎€ท ๐œ‰ ๐‘ฅ , ๐‘ก , ๐›ผ , 1 , ๐œ‰ ๐œŒ , 2 ๎€ธ ๐›ผ , ๐œŒ โˆˆ โ„• 2 ๎‚ ๐น ๐›ฝ 2 ๎‚€ ๎€ท ๐œ‰ ๐‘ฅ , ๐‘ก , ๐›ผ , 1 , ๐œ‰ ๐œŒ , 2 ๎€ธ ๐›ผ , ๐œŒ โˆˆ โ„• 2 ๎‚ โŽž โŽŸ โŽŸ โŽ  . ( 5 . 1 7 ) Note that the mapping (5.8) is linear in the components corresponding to ๐‘ข ๐‘ก , ๐‘ฃ ๐‘ก , ๐‘ข ๐‘ฅ ๐‘ฅ and ๐‘ฃ ๐‘ฅ ๐‘ฅ . Furthermore, ๐น 1 does not depend on the components corresponding to ๐‘ข ๐‘ก and ๐‘ฃ ๐‘ฅ ๐‘ฅ , while ๐น 2 does not depend on the components corresponding to ๐‘ฃ ๐‘ก and ๐‘ข ๐‘ฅ ๐‘ฅ . From this it follows quite easily that ๐… โˆž satisfies the following two-dimensional version of (4.7): โˆ€ ( ๐‘ฅ , ๐‘ก ) โˆˆ ฮฉ โˆถ โˆƒ ๐‘‰ โˆˆ ๐’ฑ ( ๐‘ฅ , ๐‘ก ) , ๐œ‰ ( ๐‘ฅ , ๐‘ก ) โˆˆ โ„ 2 โ„• 2 , ๐… โˆž ( ๐‘ฅ , ๐‘ก , ๐œ‰ ( ๐‘ฅ , ๐‘ก ) ) = ๐ŸŽ โˆถ โˆƒ ๐‘Š โˆˆ ๐’ฑ ๐œ‰ ( ๐‘ฅ , ๐‘ก ) โˆถ ๐… โˆž โˆถ ๐‘‰ ร— ๐‘Š โ†’ โ„ โ„• ๐‘› o p e n . ( 5 . 1 8 )

By the same arguments used in the proof of Theorem 4.1, which applies so single equations, we obtain the following existence result for generalized solution of the system of equations (5.2).

Theorem 5.2. If โ„ณ โ„’ โˆž ( ฮฉ ) 2 is equipped with the product uniform convergence structure with respect to the uniform convergence structure (3.9) on โ„ณ โ„’ โˆž ( ฮฉ ) , then the mapping (5.8) is uniformly continuous.

In view of Theorem 5.2, it follows that the mapping (5.8) extends uniquely to a uniformly continuous mapping ๐“ โ™ฏ โˆถ ๎€ท โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎€ธ โ™ฏ โŸถ ๎€ท โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎€ธ โ™ฏ , ( 5 . 1 9 ) where ( โ„ณ โ„’ โˆž ( ฮฉ ) 2 ) โ™ฏ denotes the completion of โ„ณ โ„’ โˆž ( ฮฉ ) 2 . We may identify ( โ„ณ โ„’ โˆž ( ฮฉ ) 2 ) โ™ฏ in a canonical way with the product of the completion ๐’ฉ โ„’ โˆž ( ฮฉ ) of โ„ณ โ„’ โˆž ( ฮฉ ) . That is, there exists a unique bijective uniformly continuous mapping ๐œ„ โ™ฏ โˆถ ๎€ท โ„ณ โ„’ โˆž ( ฮฉ ) 2 ๎€ธ โ™ฏ โŸถ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 ( 5 . 2 0 ) which extends the identity on โ„ณ โ„’ โˆž ( ฮฉ ) 2 . We will throughout use this identification, and hence write ๐“ โ™ฏ โˆถ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 โŸถ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 ( 5 . 2 1 ) instead of (5.19). As in the general case considered in Section 4, we call any solution ( ๐‘ข โ™ฏ , ๐‘ฃ โ™ฏ ) โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 of the extended equation ๐“ โ™ฏ ๎‚€ ๐‘ข โ™ฏ , ๐‘ฃ โ™ฏ ๎‚ = ๐ŸŽ ( 5 . 2 2 ) a generalized solution of (5.1). Since the systems of PDEs (5.2) satisfies the two-dimensional version (5.18) of (4.7) we obtain, by the methods of Theorem 4.2, the following basic existence result.

Theorem 5.3. There exists some ( ๐‘ข โ™ฏ , ๐‘ฃ โ™ฏ ) โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 that satisfies (5.22).

In order to incorporate the initial condition (5.3) into the solution method, the way in which approximations are constructed must be altered near ๐‘ก = 0 . This can be done in a rather straightforward way, as is seen next.

Theorem 5.4. For any ๐‘ข 0 , ๐‘ฃ 0 โˆˆ ๐’ž โˆž ( โ„ ) , there exists a solution ( ๐‘ข โ™ฏ , ๐‘ฃ โ™ฏ ) โˆˆ ๐’ฉ โ„’ โˆž ( ฮฉ ) 2 of (5.22) that satisfies ๎€ท ๐›ผ โˆ€ ๐›ผ = 1 ๎€ธ , 0 โˆˆ โ„• 2 โˆถ โˆ€ ๐‘ฅ โˆˆ โ„ โˆถ ( 1 ) ๐’Ÿ ๐›ผ โ™ฏ ๐‘ข โ™ฏ ( ๐‘ฅ , 0 ) = ๐ท ๐›ผ ๐‘ฅ ๐‘ข 0 ( ๐‘ฅ ) , ( 2 ) ๐’Ÿ ๐›ผ โ™ฏ ๐‘ฃ โ™ฏ ( ๐‘ฅ , 0 ) = ๐ท ๐›ผ ๐‘ฅ ๐‘ฃ 0 ( ๐‘ฅ ) , ( 3 ) ๐’Ÿ ๐›ผ โ™ฏ ๐‘ข โ™ฏ , ๐’Ÿ ๐›ผ โ™ฏ ๐‘ฃ โ™ฏ a r e c o n t i n u o u s a t ( ๐‘ฅ , 0 ) . ( 5 . 2 3 )

Proof. We express ฮฉ as ๎š ฮฉ = ๐œˆ โˆˆ โ„ค ร— โ„• ๐ผ ๐œˆ , ( 5 . 2 4 ) where, for ๐œˆ = ( ๐œˆ 1 , ๐œˆ 2 ) โˆˆ โ„ค ร— โ„• ๐ผ ๐œˆ = ๎€บ ๐œˆ 1 , ๐œˆ 1 ๎€ป ร— ๎€บ ๐œˆ + 1 2 , ๐œˆ 2 ๎€ป + 1 . ( 5 . 2 5 ) Consider an arbitrary but fixed set ๐ผ ๐œˆ . Since the mapping ๐… โˆž satisfies (5.18), it follows that โˆ€ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ โˆˆ ๐ผ ๐œˆ ๎€ท ๐‘ฅ โˆƒ ๐œ‰ 0 , ๐‘ก 0 ๎€ธ = ๎€ท ๐œ‰ ๐›ผ ๐‘– ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ๎€ธ โˆˆ โ„ 2 โ„• 2 , ๐… โˆž ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ท ๐‘ฅ , ๐œ‰ 0 , ๐‘ก 0 ๎€ธ ๎€ธ = ๐ŸŽ โˆˆ โ„ 2 โ„• 2 โˆถ โˆƒ ๐›ฟ = ๐›ฟ ๐‘ฅ 0 , ๐‘ก 0 , ๐œ– = ๐œ– ๐‘ฅ 0 , ๐‘ก 0 > 0 โˆถ ( 1 ) ๐… โˆž โˆถ ๐ต ๐›ฟ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ร— ๐‘Š 1 ๐œ– ๎€ท ๐œ‰ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ๎€ธ โŸถ โ„ 2 โ„• 2 o p e n , ( 2 ) ๐ŸŽ โˆˆ ๐… โˆž ๎€ท ( ๐‘ฅ , ๐‘ก ) ร— ๐‘Š 1 ๐œ– ๎€ท ๐œ‰ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ๎€ธ ๎€ธ , ( ๐‘ฅ , ๐‘ก ) โˆˆ ๐ต ๐›ฟ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , ( 5 . 2 6 ) where ๐‘Š 1 ๐œ– ( ๐œ‰ ( ๐‘ฅ 0 , ๐‘ก 0 ) ) is the neighborhood of ๐œ‰ ( ๐‘ฅ 0 , ๐‘ก 0 ) defined as ๐‘Š 1 ๐œ– ๎€ท ๐œ‰ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 = ๎€ธ ๎€ธ ๐‘– = 1 , 2 ๎‘ | ๐›ผ | โ‰ค ๐‘€ 1 ๐ต ๐œ– ๎€ท ๐œ‰ ๐›ผ ๐‘– ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ๎€ธ ร— โ„ 2 โ„• 2 ( 5 . 2 7 ) for ๐‘€ 1 > 2 a sufficiently large integer. Note that our choice of the constant ๐‘€ 1 does not depend on the point ( ๐‘ฅ 0 , ๐‘ก 0 ) , or the set ๐ผ ๐œˆ . If ๐‘ก 0 = 0 , we may chose ๐œ‰ ( ๐‘ฅ 0 , ๐‘ก 0 ) in such a way that ๎€ท ๐›ผ โˆ€ ๐›ผ = 1 ๎€ธ , 0 โˆˆ โ„• 2 โˆถ ( 1 ) ๐œ‰ ๐›ผ 1 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ = ๐ท ๐›ผ 1 ๐‘ฅ ๐‘ข 0 ๎€ท ๐‘ฅ 0 ๎€ธ , ( 2 ) ๐œ‰ ๐›ผ 2 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ = ๐ท ๐›ผ 1 ๐‘ฅ ๐‘ฃ 0 ๎€ท ๐‘ฅ 0 ๎€ธ . ( 5 . 2 8 ) For each ( ๐‘ฅ 0 , ๐‘ก 0 ) , let us fix ๐œ‰ ( ๐‘ฅ 0 , ๐‘ก 0 ) โˆˆ โ„ 2 โ„• 2 in (5.26) so that (5.28) holds if ๐‘ก 0 = 0 . For ( ๐‘ฅ 0 , ๐‘ก 0 ) โˆˆ ๐ผ ๐œˆ consider ๐’ž โˆž -smooth functions ๐‘ˆ = ๐‘ˆ ๐‘ฅ 0 , ๐‘ก 0 and ๐‘‰ = ๐‘‰ ๐‘ฅ 0 , ๐‘ก 0 on ฮฉ such that โˆ€ ๐›ผ โˆˆ โ„• 2 โˆถ ( 1 ) ๐ท ๐›ผ ๐‘ˆ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ = ๐œ‰ ๐›ผ 1 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , ( 2 ) ๐ท ๐›ผ ๐‘‰ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ = ๐œ‰ ๐›ผ 2 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ . ( 5 . 2 9 ) In particular, if ๐‘ก 0 = 0 we may set ๐‘ˆ ( ๐‘ฅ , ๐‘ก ) = ๐‘ข 0 ( ๐‘ฅ ) + ๐œ‘ 1 ๐‘ฅ 0 , ๐‘ก 0 ( ๐‘ก ) + ๐›ผ 1 , ๐›ผ 2 โ‰  0 ๎“ ๎€ท ๐›ผ ๐›ผ = 1 , ๐›ผ 2 ๎€ธ โˆˆ โ„• 2 1 ๐›ผ 1 ! ๐›ผ 2 ! ๐‘ก ๐›ผ 2 ๎€ท ๐‘ฅ โˆ’ ๐‘ฅ 0 ๎€ธ ๐›ผ 1 ๐œ‰ ๐›ผ 1 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , ๐‘‰ ( ๐‘ฅ , ๐‘ก ) = ๐‘ฃ 0 ( ๐‘ฅ ) + ๐œ‘ 2 ๐‘ฅ 0 , ๐‘ก 0 ( ๐‘ก ) + ๐›ผ 1 , ๐›ผ 2 โ‰  0 ๎“ ๎€ท ๐›ผ ๐›ผ = 1 , ๐›ผ 2 ๎€ธ โˆˆ โ„• 2 1 ๐›ผ 1 ! ๐›ผ 2 ! ๐‘ก ๐›ผ 2 ๎€ท ๐‘ฅ โˆ’ ๐‘ฅ 0 ๎€ธ ๐›ผ 1 ๐œ‰ ๐›ผ 2 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , ( 5 . 3 0 ) where ๐œ‘ 1 ๐‘ฅ 0 , ๐‘ก 0 , ๐œ‘ 2 ๐‘ฅ 0 , ๐‘ก 0 โˆˆ ๐’ž โˆž ( โ„ ) satisfy ๎€ท โˆ€ ๐›ผ = 0 , ๐›ผ 2 ๎€ธ โˆˆ โ„• 2 โˆถ ( 1 ) ๐ท ๐›ผ 2 ๐‘ก ๐œ‘ 1 ๐‘ฅ 0 , ๐‘ก 0 ( 0 ) = ๐œ‰ ๐›ผ 1 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , ( 2 ) ๐ท ๐›ผ 2 ๐‘ก ๐œ‘ 2 ๐‘ฅ 0 , ๐‘ก 0 ( 0 ) = ๐œ‰ ๐›ผ 2 ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ . ( 5 . 3 1 ) From the continuity of ๐‘ˆ , ๐‘‰ and their derivatives, as well as the components ๐น ๐›ฝ ๐‘— of ๐… โˆž , and (5.26) it follows that โˆ€ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ โˆˆ ๐ผ ๐œˆ 1 , ๐œˆ 2 โˆถ โˆƒ ๐›ฟ = ๐›ฟ ๐‘ฅ 0 , ๐‘ก 0 > 0 , ๐œ– = ๐œ– ๐‘ฅ 0 , ๐‘ก 0 > 0 โˆถ โˆ€ ( ๐‘ฅ , ๐‘ก ) โˆˆ ๐ต ๐›ฟ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , | | ๐›ฝ | | โ‰ค ๐‘€ 1 โˆ’ 2 , ๐‘— = 1 , 2 โˆถ ( 1 ) ๐… โˆž โˆถ ๐ต ๐›ฟ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ร— ๐‘Š 1 ๐œ– ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ โŸถ โ„ 2 โ„• 2 o p e n , ( 2 ) โˆ’ 1 < ๐ท ๐›ฝ ๐‘‡ ๐‘— ( ๐‘ฅ , ๐‘ก , ๐ท ) ( ๐‘ˆ , ๐‘‰ ) ( ๐‘ฅ , ๐‘ก ) < 1 , ( 3 ) ( ๐ท ๐›ผ ๐‘ˆ ( ๐‘ฅ , ๐‘ก ) , ๐ท ๐œŒ ๐‘‰ ( ๐‘ฅ , ๐‘ก ) ) ๐›ผ , ๐œŒ โˆˆ โ„• 2 โˆˆ ๐‘Š 1 ๐œ– ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , ( 4 ) ๐ŸŽ โˆˆ ๐… โˆž ๎€ท ( ๐‘ฅ , ๐‘ก ) ร— ๐‘Š 1 ๐œ– ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ๎€ธ ( 5 . 3 2 ) Since ๐ผ ๐œˆ is compact, (5.32) may be strengthened to โˆƒ ๐›ฟ ๐œˆ โˆ€ ๎€ท ๐‘ฅ > 0 โˆถ 0 , ๐‘ก 0 ๎€ธ โˆˆ ๐ผ ๐œˆ โˆถ โˆƒ ๐œ– = ๐œ– ๐‘ฅ 0 , ๐‘ก 0 > 0 โˆถ โˆ€ ( ๐‘ฅ , ๐‘ก ) โˆˆ ๐ต ๐›ฟ ๐œˆ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ , | | ๐›ฝ | | โ‰ค ๐‘€ 1 โˆ’ 2 , ๐‘— = 1 , 2 โˆถ ( 1 ) ๐… โˆž โˆถ ๐ต ๐›ฟ ๐œˆ ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ ร— ๐‘Š 1 ๐œ– ๎€ท ๐‘ฅ 0 , ๐‘ก 0 ๎€ธ โŸถ โ„ 2 โ„• 2 o p e n , ( 2 ) โˆ’ 1 < ๐ท ๐›ฝ ๐‘‡ ๐‘— ( ๐‘ฅ , ๐‘ก , ๐ท ) ( ๐‘ˆ , ๐‘‰ ) ( ๐‘ฅ , ๐‘ก ) < 1 , ( 3 ) ( ๐ท ๐›ผ ๐‘ˆ ( ๐‘ฅ , ๐‘ก ) , ๐ท ๐œŒ ๐‘‰ ( ๐‘ฅ , ๐‘ก ) ) ๐›ผ , r h o โˆˆ โ„•