ISRN Applied Mathematics

Volume 2014 (2014), Article ID 305718, 13 pages

http://dx.doi.org/10.1155/2014/305718

## On Second-Order Differential Equations with Nonsmooth Second Member

^{1}DM, UEPB, Campina Grande-PB, Brazil^{2}IM, UFRJ, Rio de Janeiro, RJ, Brazil

Received 15 January 2014; Accepted 27 February 2014; Published 24 March 2014

Academic Editors: A. Bellouquid and C. Join

Copyright © 2014 M. Milla Miranda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In an abstract framework, we consider the following initial value problem: *u*′′ + *μ**A**u* + *F*(*u*)*u* = *f* in (0,*T*), , where is a positive function and *f* a nonsmooth function. Given *u*^{0}, *u*^{1}, and *f* we determine in order to have a solution *u* of the previous equation. We analyze two cases of . In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).

#### 1. Introduction

Let and be two real separable Hilbert spaces with dense in and continuously embedding in . The scalar product and norms of and are represented, respectively, by

Let be the self-adjoint operator of defined by the triplet . Consider , . We denote by the Hilbert space equipped with the scalar product (cf. Lions [1]).

Consider the following initial value problem: where is a positive function and a nonsmooth function.

The objective of this work is to study the following inverse problem: given , , and ( dual space of ) to determine such that Problem (4) has a solution . We analyze two cases of , more precisely, the cases where is an appropriate Hilbert space.

In Lions [2, Page 284], the following problem is formulated where is an open bounded set of with boundary , , and being the Dirac mass supported at . He says not to know if this problem admits a solution. He say also that one of the difficulties in the study of existence of solutions of the nonlinear equations lies in the difficulty in defining weak solutions, since the transposition method is essentially a linear method. This is ultimately connected to the fact that one cannot multiply distributions.

Problem (4) with of the form (6) is an abstract formulation of Problem (7) with a slight modification of the nonlinear term. Theorem 3 gives the existence of solutions of this problem. In applications we give examples of Problem (7), with the modification of the nonlinear term, for an open bounded set of , .

In Grotta Ragazzo [3] the following equation is studied: This equation is considered as a first approximation of the Klein-Gordon equation Observe that (8) with and is the meson equation of Schiff [4] (cf. also Jörgens [5]).

The physical motivation of (8) with can be seen in Lourêdo et al. [6].

Problem (4) with of the form (5) generalizes (8) when . The existence of solutions of this problem is studied in Theorem 1.

In Louredo et al., loc.cit., is analyzed the equation with nonlinear boundary condition. The given in (5) is different from the of this equation. The term is related to the nonsmoothness of .

#### 2. Main Results

We use the notation , , . Identifying with , we have Here and in what follows the notation means that the space in dense in the space and the embedding of in are continuous. Note that . Also, if , with , we have Assume that

First we analyze Problem (2) with , that is, the problem

Theorem 1. *Assume condition (13). Let and be real numbers with . Consider
**
Then there exists a function in the class
**
such that is solution of the equation
**
and satisfies the initial conditions
*

*Remark 2. *When it is possible to obtain a solution of Problem (14) by using the Theory of Semigroups (cf. Pazy [7]).

*To formulate the second problem, we introduce some notations. In fact, let us define
where
By (12) we have
*

*Consider Problem (4) with and , that is, the problem
*

*Theorem 3. Assume that , , and satisfy the hypotheses of Theorem 1 and . Then there exists a function in the class (16) such that is solution of the problem
*

*Remark 4. *Note that if belongs to class (16) then (cf. Lions and Magenes [8]).

*Corollary 5. Under the same hypotheses of Theorem 1, there exists a function in the class (16) such that is solution of the problem
*

*We analyze the uniqueness of solutions. Consider in Theorem 1. Then the solution gives by this theorem when satisfies
*

*Theorem 6. Let be a real number. Consider
Then there exists a unique solution of Problem (26) in the class (25).*

*We do not know if there is uniqueness of solutions for Theorem 3, even when .*

*In what follows we prove the above results.*

*3. Proof of Theorem 1*

*3. Proof of Theorem 1*

*Before proving the theorem, we make some considerations on the operator . Recall hypothesis (13). By solving the spectral problem , for all , we determine the eigenfunctions and eigenvalues, respectively, and of the operator , that is,
Note that is a Hilbert basis of (cf. Brezis [9] and Komornik [10]).*

*Let be , . Then the linear operator
is continuous, bijective, and
Also,
is given by
These results can be found in Lions [1] and Medeiros and Milla Miranda [11].*

*Introduce the adjoint operator of , that is,
where . Note that is identified with . By the properties of , we have that
Thus, the linear operator
is continuous and bijective.*

*Proposition 7. Let and . Then one has the following.(i) converges in ,
and(ii) converges in ,
andwhere .*

*Proof. *We prove . As , we have
Consider . Then noting that , we obtain
On the other hand, by (30) we derive
The last two expressions give
This and (38) provide that
So is proved. Taking the limit in (41) and observing (33)_{2}, we obtain (36).

We prove . We have that there exists a unique such that
By (33)_{2} and (30)_{1}, we have
Then
This implies that
Thus is proved. By (43) and (45), we obtain
This concludes the proof of the proposition.

*Motived by (37), we equip the space with the scalar product
where . This scalar product on yields a norm
which is equivalent to the usual norm of .*

*By similarity between expressions (30) and (36) and between (32) and (37), respectively, we introduce the notations
Also we use the notation
With these considerations and expressions (29) and (33), we obtain
and by expressions (35) and (31),
Also by (37) and (48), (49), respectively, we find
*

*Proposition 8. Consider , . Then the linear operator
defined by
is continuous.*

*Proof. *We obtain
where . Then
which proves the proposition.

*Proof of Theorem 1. *We use the Galerkin method (cf. Lions [12] and Vicente and Frota [13]). Thus consider an approximate solution of Problem (14); that is,
where and .

*Remark 9. *Note that

*Remark 10. *Observe that if then

*Multiply both sides of (61) _{1} by and add from up to . We obtain
where
Then,
By Proposition 8 with and and noting that , we obtain
Substituting this equality into (66) and integrating on , we obtain
Note that
Also
*

*Substituting (69)–(71) into (68), we find
where
Applying Gronwall inequality in (72), we deduce
*

*With this inequality, we determine a subsequence of , still denoted by , and a function such that
By (13) we have that
(cf. [1] and [11]). Then convergences (75) and Aubin-Lions Theorem (cf. [14]) imply
Therefore,
This convergence and convergence (75) _{1} provide
which implies
since , defined in (65).*

*In order to obtain an estimate for , we apply the method of projections to the approximate equation (61) (cf. Lions [12]). Thus, we consider the orthogonal projection
where is the subspace generated by .*

*By similar arguments employed to obtain (67) and by (54) or (30), we find
*

*Multiply both members of (61) _{1} by and add from up to . Then, applying to this result, expression (82), affirmation (54), or (30) and noting that , , belong to , we obtain
which gives
Then estimates (74) and (80) provide
Thus, there exists a subsequence of , still denoted by , such that
*

*Expressions (75) and (86) tell us that belongs to class (16). Convergences (75) _{1}, (80), and (86) allow us to pass to limit in (83) and to obtain
which provides (17). Initial conditions (18) follow from convergences (75) and (86).*

*4. Proof of Theorem 3*

*4. Proof of Theorem 3*

*The idea is to apply a fixed point argument to the problem
where .*

*We solve (88). Consider an approximate solution of (88) given by
*

*By similar arguments used to obtain (74), we derive
where
*

*The preceding inequality gives
By the projection method, we obtain, as in (83),
This and estimate (91) provide
*

*Estimates (93) and (95) allow us to find a subsequence of , still denoted by , and a function such that, by passing to limit in (94), we obtain
This, initial conditions (90) _{2}, and estimates (95) imply
*

*By taking the lim inf in both side of (91), we obtain
*

*As and the embedding in are compact , it follows from of Aubin-Lions Theorem (see Simon [14]) that
*

*Define the map
where is the solution of Problem (97). We will prove that has a fixed point. Consider only the case . The case is outside of our attention. We will prove the following results.(I) One has. *

*
In fact if , we have that is a solution of (97) with , but this a contradiction since .(II) One has is continuous on .*

*Let . Consider . By (94) and (90) _{2} we obtain
Use the notation . Then the preceding problems give
*

*Taking the scalar product of of both sides of this equation with , we find
We have
Integrating on both sides of the last two expressions, we derive
We obtain
As , we have
Also,
*

*Taking into account estimate (91) in (107) and (109), we find
Substituting the last two inequalities into (106), we obtain
Combining this inequality with (105), we derive
Considering and using the Gronwall inequality, this expression gives
where the constant is independent of and . Taking the lim inf in both sides of this inequality, we find
By Simon [14] and noting that the embedding of in is compact, we derive
Thus.
which proves the continuity of at . In similar way we prove the continuity of at .(III) One has as .*

*Let be a sequence of positive numbers with . It follows from (98) and the compactness of the embedding of in that there exists a subsequence of , still denoted by , and a function such that
This implies
By estimate (91), we obtain
Then
Convergences (118) and (120) provide
Thus by (117) we find
As the sequence was arbitrary and the limit of is always the same, we conclude that
Thus
which proves part (III).*

*By (I)–(III), we deduce that there exists such that
Considering this in (97), we obtain a solution of (22) that satisfies all conditions of the theorem.*

*The proof of Corollary 5 follows by defining the map
where is the solution of the problem
and applying similar arguments to those used in the proof of Theorem 3.*

*5. Proof of Theorem 6*

*5. Proof of Theorem 6*

*Let and be solutions of Problem (26) with and in class (25). Consider . Then by (26) we have
*

*Fix . Consider . Introduce the function
We have
Apply the operator given by the first member of (128) _{1} to . We obtain
where . By (130)_{2}, we find
Also by (130)_{3},
Integrate (131) on and use (128)_{1}, (130), (132), and (133). We deduce
*