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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 278542, 33 pages
Exact Asymptotic Expansion of Singular Solutions for the ()-D Protter Problem
1Faculty of Technology, Narvik University College, Lodve Langes Gate 2, 8505 Narvik, Norway
2Faculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria
Received 29 March 2012; Accepted 24 June 2012
Academic Editor: Valery Covachev
Copyright © 2012 Lubomir Dechevski 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.
We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in . In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right-hand side functions there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.
In the present paper some boundary value problems (BVPs) formulated by M. H. Protter for the wave equation with two space and one time variables are studied as a multidimensional analogue of the classical Darboux problem in the plane. While the Darboux BVP in is well posed the Protter problem is not and its cokernel is infinite dimensional. Therefore the problem is not Fredholm and the orthogonality of the right-hand side function to the cokernel is one necessary condition for existence of classical solution. Alternatively, to avoid infinite number of conditions the notion of generalized solution is introduced that allows the solution to have singularity on a characteristic part of the boundary. It is known that for smooth right-hand side functions there is unique generalized solution and it may have a strong power-type singularity that is isolated at one boundary point. In the present paper we prove asymptotic expansion formula for the generalized solutions in negative powers of the distance to the singular point in the case when is trigonometric polynomial. We leave for the next section the precise formulation of the paper’s main results and the comparisons with recent publications concerning Protter problems, including a semi-Fredholm solvability result in the general case of smooth but for somewhat easier ()-D wave equation problem. First we give here a short historical survey.
Protter arrived at the multidimensional problems for hyperbolic equations while examining BVPs for mixed type equations, starting with planar problems with strong connection to transonic flow phenomena. In the plane, the problems of Tricomi, Frankl, and Guderley-Morawetz are the classical boundary-value problems that appear in hodograph plane for 2D transonic potential flows (see, e.g., the survey of Morawetz ). The first two of these problems are relevant to flows in nozzles and jets, and the third problem occurs as an approximation to a respective “exact” boundary-value problem in the study of flows around airfoils. For the Gellerstedt equation of mixed type, Protter  proposes a 3D analogue to the two-dimensional Guderley-Morawetz problem. At the same time, he formulates boundary value problems in the hyperbolic part of the domain, which is bounded by two characteristics and one noncharacteristic surfaces of the equation. The planar Guderley-Morawetz mixed-type problem is well studied. Existence of weak solutions and uniqueness of strong solutions in weighted Sobolev spaces were first established by Morawetz by reducing the problem to a first order system which then gives rise to solutions to the scalar equation in the presence of sufficient regularity. The availability of such sufficient regularity follows from the work of Lax and Phillips  who also established that the weak solutions of Morawetz are strong. On the other hand, for the 3D Protter mixed-type problems a general understanding of the situation is not at hand—even the question of well posedness is surprisingly subtle and not completely resolved. One has uniqueness results for quasiregular solutions, a class of solutions introduced by Protter, but there are real obstructions to existence in this class. To investigate the situation, we study a simpler problem—the Protter problems in the hyperbolic part of the domain for the mixed-type problem. For the wave equation this is the set It is bounded, see Figure 1, by two characteristic cones of (1.1) and the disk , centered at the origin .
One could think of the Protter problems in as three-dimensional variant of the planar Darboux problem. The classic Darboux problem involves a hyperbolic equation in a characteristic triangle bounded by two characteristic and one noncharacteristic segments. The data are prescribed on the noncharacteristic part of the boundary and one of the characteristics. Actually, the set could be produced via rotation around the -axis in of the flat triangle —a characteristic triangle for the corresponding string equation As mentioned before, the classical Darboux problem for (1.4) is to find solution in with data prescribed on and , for example. In conformity with this planar BVP, Protter [2, 4] formulated and studied the following problems.
Find a solution of the wave equation (1.1) in which satisfies one of the following boundary conditions: or
Nowadays, it is known that the Protter Problems and are not well posed, in contrast to the planar Darboux problem. In fact, in 1957 Tong  proved the existence of infinite number nontrivial classical solutions to the corresponding homogeneous adjoint problem . The adjoint BVPs to Problems and were also introduced by Protter.
Find a solution of the wave equation (1.1) in which satisfies the boundary conditions: or
Since , for each of the homogeneous Problems and (i.e., in (1.1)), an infinite number of classical solutions has been found (see Popivanov, Schneider , Khe ). According to this fact, a necessary condition for classical solvability of Problem or is the orthogonality in of the right-hand side function to all the solutions of the corresponding homogenous adjoint problem or . Although Garabedian proved  the uniqueness of a classical solution of Problem (for its analogue in ), generally, Problems and are not classically solvable. Instead, Popivanov and Schneider  introduced the notion of generalized solution. It allows the solution to have singularity on the inner cone and by this the authors avoid the infinite number of necessary conditions in the frame of the classical solvability. In  some existence and uniqueness results for the generalized solutions are proved and some singular solutions of Protter Problems and are constructed.
In the present paper we study the properties of the generalized solution for Protter Problem in . From the results in  it follows that for there exists a smooth right-hand side function , such that the corresponding unique generalized solution of Problem has a strong power-type singularity at the origin and behaves like there. This feature deviates from the conventional belief that such BVPs are classically solvable for very smooth right-hand side functions . Another interesting aspect is that the singularity is isolated only at a single point the vertex of the characteristic light cone, and does not propagate along the bicharacteristics which makes this case different from the traditional case of propagation of singularity (see, e.g., Hörmander , Chapter 24.5).
The Protter problems have been studied by different authors using various types of techniques like Wiener-Hopf method, special Legendre functions, a priori estimates, nonlocal regularization, and others. For recent known results concerning Protter’s problems see the paper  and references therein. For further publications in this area see [7, 10–16]. On the other hand, Bazarbekov gives in another analogue of the classical Darboux problem (see ) and analogously in (see ) in the corresponding four-dimensional domain . Some different statements of Darboux type problems in or connected with them Protter problems for mixed type equations (also studied in ) can be found in [19–25]. Some results concerning the nonexistence principle for nontrivial solution of semilinear mixed type equations in multidimensional case, can be found in . For recent existence results concerning closed boundary-value problems for mixed type equations see for example , and also  that studies an elliptic-hyperbolic equation which arises in models of electromagnetic wave propagation through zero-temperature plasma. The existence of bounded or unbounded solutions for the wave equation in and , as well as for the Euler-Poisson-Darboux equation has been studied in [7, 13–16, 29].
Further, we aim to find some exact a priori estimates for the singular solutions of Problem and to outline the exact structure and order of singularity. For some other Protter problems necessary and sufficient conditions for existence of solutions with fixed order of singularity were found (see  in and  in and an asymptotic formula for the solution of Problem in was obtained in .
Considering Protter Problems, Popivanov and Schneider  proved the existence of singular solutions for both wave and degenerate hyperbolic equation. First a priori estimates for singular solutions of Protter Problems, involving the wave equation in , were obtained in . In  Aldashev mentioned the results of  and, for the case of the wave equation in , he notes the existence of solutions in the domain ( and approximates if , which blows up on the cone like , when . It is obvious that for this results can be compared to the estimates in Corollary 2.4 here. Finally, we point out that in the case of an equation, which involves the wave operator and nonzero lower terms, Karatoprakliev  obtained a priori estimates, but only for the sufficiently smooth solutions of Protter Problem.
Regarding the ill-posedness of the Protter Problems, there have appeared some possible regularization methods in the case of the wave equation, involving either lower order terms ([11, 31]), or some other type perturbations, like integrodifferential term, or nonlocal one ().
In Section 2 the result of the existence of infinite number of classical solutions to the homogeneous Problem (Lemma 2.1) and the definition of generalized solution of Problem are given. The main results of the paper, concerning the asymptotic expansion of the unique generalized solution of Problem (Theorem 2.3) are formulated and discussed. The expansion of is given in negative powers of the distance to the point of singularity. An estimate for the remainder term and the exact behavior of the singularity under the orthogonality conditions imposed on the right-hand side function of the wave equation is found. Necessary and sufficient conditions for the existence of only bounded solutions are given in Corollary 2.4. In Section 3, the auxiliary 2D boundary value Problems and , which correspond to the ()-D Problem , are considered. Actually, these 2D problems are transferred to an integral Volterra equation, which is invertible. Using the special Legendre functions , some exact formulas for the solution of the Problem are derived in Lemma 3.4. Some figures showing the effects appearing near the singularity point are also presented. Section 4 contains the most technical part of the paper. In this section the results concerning the asymptotic expansions of the generalized solution of the 2D Problem are proved and the proof of the main Theorem 2.3 is given.
2. Main Results on ()-D Protter’s Problem 6
Define the functions where the coefficients are with , . Then for the functions we have the following lemma.
Lemma 2.1 (see ). Let , . For and the functions are classical solutions to the homogeneous Problem .
A necessary condition for the existence of classical solution for Problem is the orthogonality of the right-hand side function to all functions , which are solutions of the homogeneous adjoint Problem . To avoid these infinite number necessary conditions in the framework of classical solvability, one needs to introduce some generalized solutions of Problems with possible singularities on the characteristic cone , or only at its vertex . Popivanov and Schneider in  give the following definition.
The uniqueness of the generalized solution of Problem and existence results for can be found in .
Further, we fix the right-hand side function as a trigonometric polynomial of order with respect to the polar angle: with some complex-valued function-coefficients . For and , denote by the constants
Note that actually and in cases of and , due to the special form of the functions and the fact that in the representation (2.5) of the function the sum starts from .
The main result is as follows.
Theorem 2.3. Suppose that the function is a trigonometric polynomial (2.5). Then there exist functions , , with the following properties:(i)the unique generalized solution of Problem exists, belongs to and has the asymptotic expansion at the origin : (ii)for the coefficient functions the representation holds, where the functions are bounded and independent of ,(iii)if in the expression (2.8) for at least one of the constants is different from zero (i.e., the corresponding orthogonality condition is not fulfilled), then there exists a direction with for , such that ,(iv)if in the expression (2.8) for at least one of the constants is different from zero (i.e., the corresponding orthogonality condition is not fulfilled), then the generalized solution is not continuous at ,(v)for the function the estimate holds with a constant independent of .
As a consequence of Theorem 2.3 one gets the following results that highlight the two extreme cases of the assertion. The first part gives rough estimate of the expansion (2.7) and describes the “worst” possible singularity. The second part shows that one could control the solution by making some of the defined by (2.6) constants in (2.8) to be zero, that is, by taking to be orthogonal in to the corresponding functions defined in (2.3).
Corollary 2.4. Suppose that has the form (2.5).(i)Without any orthogonality conditions imposed, the unique generalized solution of Problem satisfies the a priori estimate (ii)Let the orthogonality conditions, be fulfilled for all and . Then the generalized solution belongs to , is bounded and the a priori estimate holds.(iii)In addition to (ii), if the conditions (2.11) are fulfilled for also, then is a classical solution and .
Let us point out that in the case (ii), the generalized solution is bounded if and only if the conditions (2.11) are fulfilled for due to Theorem 2.3(iii). In addition, if all the conditions (2.11) are fulfilled for , but for some the corresponding orthogonality condition is not satisfied, then is not continuous at , according to Theorem 2.3 (iv). Such a solution is illustrated in Figure 4.
Notice that some of the functions involved in the orthogonality conditions in Corollary 2.4(ii) and (iii) are not classical solutions of the homogenous adjoint Problem in view of Lemma 2.1, although they satisfy the homogenous wave equation in . In fact, for some , or their derivatives may be discontinuous at . For example when is an odd number and , the functions are not continuous at the origin . On the other hand, when is even and , are singular on the cone and do not satisfy the homogeneous adjoint boundary condition there. However, this singularity is integrable in the domain .
Both functions , are located in column number and row number in Table 1. Thus, form the rightmost diagonal, the next one is empty—we put in these cells “diamonds” , constitute the third one, and so on. The row number designates the order of singularity of the generalized solution.
Corollary 2.4 shows that the generalized solution is bounded, when the right-hand side function is orthogonal to the functions in Table 1, except the ones in row number . If is orthogonal to all the functions in the Table 1 (including the row ), then is continuous in . When the right-hand side satisfies orthogonal conditions (2.11) for all the functions from the rows in Table 1 with row-number larger then , , but there is a function with from th row which is not orthogonal to (i.e., ), then the solution behaves like at the origin, according to the expansion (2.7). If there are no orthogonality conditions, then the worst case with singularity appears.
Figures 2–5 are created using MATLAB and represent some numerical computations for singular solutions of Problem (actually the behaviour in -domain , not including the terms and ). They illustrate different cases according to the main results for the existence of a singularity at the origin depending on orthogonality conditions. Figure 2 is related to Corollary 2.4(i)—it gives the graph of the solution for the worst case without any orthogonality conditions fulfilled and the solution is going to at the singular point . In Figure 3, only one of orthogonality conditions (2.11) for is not fulfilled and the solution tends to . Figures 4 and 5 are connected to Corollary 2.4(ii) and (iii): Figure 4 presents the case when all the orthogonality conditions (2.11) for are satisfied and the solution is bounded but not continuous at , while Figure 5 concerns the last part (iii) from Corollary 2.4, when conditions (2.11) are additionally fulfilled for and the solution is continuous.
Remark 2.5. We mention some differences between the results given here for the Problem and some other results in , but for the Problem , like that from . (i)In , assuming the right-hand side function is smooth enough (i.e., ) only the behavior of the singularities was studied using some weighted norms (analogous to the weighted Sobolev norms in corner domains). In the present paper we need only and find in addition the explicit asymptotic expansion of the generalized solution. The bounded but not continuous at the origin solutions are also studied here. (ii)Comparing the power of singularity of the generalized solution for Problem here and for Problem in  for the worst case without any orthogonality conditions one can see that the power in the estimate (2.10) from Corollary 2.4(i) is , while in the analogous estimate in Conclusion 1  it is .(iii)It is interesting to compare the results [14, 15], published in 2002. Going in a different way in both cases the authors asked for singular solutions of Problem in . However, in  there are absent any analogues to the orthogonality conditions presented in , and in contrast to  in  the dependence of the exact order of singularity on the data is not clarified.
Remark 2.6. Let us also compare the present expansion and the results in , where an asymptotic expansion of Problem is found for somewhat easier four-dimensional case.(i)Both for Problem in here and Problem in as in , the study is based on the properties of the special Legendre functions. Instead of Legendre functions with non-integer indices here, in the four-dimensional case one has to deal with integer indices , that is, simply with the Legendre polynomials . One can easily modify both these techniques to obtain similar results for the -dimensional problems: for even (analogous to the present case ) or for odd (similarly to case). Some different kind of results for the Protter problems in are presented in [10, 11]. (ii)For the four-dimensional Problem in , the Corollary 3.3 gives only that the solution is bounded, it could be discontinuous at the origin. On the other hand, here Theorem 2.3 gives us also the control over the bounded but not continuous parts of the generalized solution (through the coefficient for in the expansion formula (2.7)). As a sequence, Corollary 2.4(iii) guarantees that the solution is continuous.(iii)Based on the formulae and the computations from , the general case in is also treated, when the right-hand side is smooth enough, but not a finite harmonic polynomial analogous to (2.5). The results are announced and published in [32, 33]. For right-hand side functions in  the necessary and sufficient conditions for the existence of bounded solution are found. They involve infinite number of orthogonality conditions for that comes from the fact that this is not a Fredholm problem. On the other hand, the results from  show that the linear operator mapping the generalized solution into is a semi-Fredholm operator in . Let us recall that a semi-Fredholm operator is a bounded operator that has a finite dimensional kernel or cokernel and closed range. Additionally, in  a right-hand side function is constructed such that the unique generalized solution of Protter Problem in has exponential type singularity. One expects that similar results could also be obtained for the Problem in studied here. These questions correspond to the Open Problem (1) below.
Remark 2.7. Let us mention one obvious consequence of Theorem 2.3 and all the arguments above, concerning construction of functions orthogonal to the solutions of the homogeneous adjoint Problem . Take an arbitrary function satisfying the boundary conditions . Then the function with the wave operator , is orthogonal to all the functions , ..
Finally, we formulate some still open questions, that naturally arise from the previous works on the Protter problem and the discussions above.
Open Problems. (1) To study the more general case when the right-hand side function , for an appropriate . The smooth function could be represented as a Fourier series rather than, the finite trigonometric polynomial (2.5) in the discussions here.(i)Find some appropriate conditions for the function under which there exists a generalized solution of the Protter problem . (ii)What kind of singularity can the generalized solution have? The a priori estimates, obtained in [6, 31], which indicate that the generalized solutions of Problem (including the singular ones), can have at most an exponential growth as . The natural question is as follows: is there a singular solution of these problems with exponential growth as or do all such solutions have only polynomial growth?(iii)Is it possible to prove some a priori estimates for generalized solutions of the Problem with smooth function which is not a harmonic polynomial? (iv)Find some appropriate conditions for the function under which the Problem has only regular, bounded solutions, or even classical solutions.
(2) To study the Protter problems for degenerate hyperbolic equations. Up to now it is only known that some singular solutions exist.(i)We do not know what is the exact behavior of the singular solution even when the right-hand side function is a finite sum like (2.5). Can we prove some a priori estimates for generalized solutions?(ii)Is it possible to find some orthogonality conditions for the function , as here, under which only bounded solutions exist?
(3) Why does there appear a singularity for such smooth right-hand side even for the wave equation? Can we numerically model this phenomenon?
(4) What happens with the ill-posedness of the Protter problems in a more general domain (as in [2, 4]) when the maximal symmetry is lost if the cone is replaced by another light characteristic one with the vertex away from the origin.
We have a relation between the functions and the Legendre functions . For , the functions could be defined by the equality (Section 3.7, formula (6), from Erdélyi et al. ), where for in this formula .
Let , be the cylindrical coordinates in , that is, , . For simplicity, define the function . The following result is in connection with Lemma 5.1 from . Actually, to prove this result one could formally follows the arguments of Lemma 2.3 from , where the four-dimensional analogue of Problem is treated.
Lemma 3.1. For and define the functions for . Then in the equality holds for with some constants .
Proof. Lemma 5.1 from  for gives where and according to Lemma 2.2 from  Therefore the equality (3.3) holds for . We have to prove it for . In the proof of Lemma 5.1 from  the integrals were calculated using the Mellin transform. In order to compute in the same way let us first introduce the variables and : As a consequence after some calculations, formulas (2.2.(4)), (1.10), and (1.4) from Samko et al. , show that where is the Riemann-Liouville fractional integral (for its properties see e.g., [34, 35]); in our case we have . As usual, we denote also for for . The substitution of (3.6) in (3.7) shows that and thus
Lemma 3.2. Let , and the functions satisfy . Then all solutions of the Volterra integral equation of first kind are
One could use the Mellin transform to calculate the following integral.
Lemma 3.3 (see ). Let , , then
According to the existence and uniqueness results in , it is sufficient to study Problem when the right-hand side of the wave equation is simply Then we seek solutions for the wave equation of the same form: Thus Problem reduces to the following one.
Find a solution of the equation in the domain with the following boundary conditions:
Problems and were introduced in , although the change of coordinates and was used there instead of (3.17). Of course, because the solution of Problem may be singular, the same is true for the solutions of and . For that reason, Popivanov and Schneider  defined and proved the existence and uniqueness of generalized solutions of Problems and , which correspond to the generalized solution of Problem . Further, by “solution” of Problem or we mean exactly this unique generalized solution.
Proof. Notice that the function is a Riemann function for (3.19) (Copson ). Therefore, following Aldashev , we can construct the function as a solution of a Goursat problem in with boundary conditions and with some unknown function , which will be determined later: Now, the boundary condition gives an integral equation for . For that reason, let us define the function : Obviously, . The condition (3.25) leads us to the following equation: Then, using , we have A necessary solvability condition for the unknown function is: . One could solve this Volterra integral equation of the first kind, using Lemma 3.2. The result is Integrate, we find Now, using Lemma 3.3 and the equality for one finds that the coefficient of in (3.30) is zero. Using again (3.31) for , is simply Obviously, and . Finally, the solution of Problem is given by the formulae (3.20), (3.21), and (3.22).
4. Proofs of Main Results
In order to study the behavior of the generalized solution of Problem , in view of relations (3.18) and Lemma 3.4, we will examine the function defined by the formulae (3.20), (3.21), and (3.22). It is not hard to see that the part “responsible” for the singularity is the integral in (3.21) for the function . In fact, blows up at , since the argument and thus the values of the Legendre function in (3.21) go to infinity when . Actually, grows like at infinity. In the next lemma we find the dependance of the exact order of singularity of on the function . It is governed by the constants Actually, these numbers are closely connected to the constants from Theorem 2.3. We will clarify this relation later in Lemma 4.1 and the proof of Theorem 2.3.
Lemma 4.1. Let , where , , and let the function . Then the function belongs to and satisfies the representation where the function , and the nonzero constants and are independent of .
Proof. The argument of the Legendre function in (4.2) satisfies the inequality , which allows us to apply the representation (3.1):
We will study the expansion at of the function
Let us define the functions
for . Then, obviously
First, we will examine the properties of the functions and their derivatives with respect to . We start with the equality
Further, the index will be less than . Notice that for the integrals are convergent. Then, for we have the equality
and the inequality
Differentiating with respect to one finds
Therefore, for the derivatives of we find by induction where the coefficients are positive constants. We want to evaluate these derivatives of at . Let us estimate the terms in the last sum for :(i)when is such that the inequality (4.10) gives the estimate (ii)when and , we have Hence for . Therefore, at the point the only one nonzero term in the sum (4.12) is for , that is, The last observation is that (4.8) and (4.12) imply where are constants.
Now, we go back to the function . We want to differentiate times and evaluate at . Differentiating (4.7) we find the following: Next, since the assertion for is only , instead of we will differentiate the function Notice that it belongs to and the derivative is In the same way, after denoting , define for the functions with the constants from (4.16). Then, using (4.16), it follows by induction that is continuous in and On the other hand, Hence, according to (4.15), for ,
The next step is to evaluate the integral . Using (4.12), one could rewrite it in the form For all the terms in the last sum, except one, the estimate is straightforward.
(1) When is such that , for the corresponding terms we have and, therefore, where .
(2) For the last term in (4.24) with there are two cases:(2a) when is an even number this is the integral (2b) when is an odd number, the integral is For simplicity let us define some constants related to the constants given by (4.1). Indeed, due to (4.9) the equality holds. Let us begin with the following case.(2a):. We will evaluate the difference: For the first integral, using the estimate (4.10), we calculate and, therefore, For the second integral From the last two inequalities we get the estimate Therefore, in the case , , where .(2b) When , we have to study the integral (4.28). Obviously, For the last integral we have Now, to estimate the first term in the right-hand side of (4.36), there are two cases:(i) when , we have and similarly to the previous case (2a) we can apply inequality (4.31). Thus, we estimate the difference: (ii) when , denote for simplicity , then and directly from the definition (4.6) of the functions , we get