Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem

Volume 2012, Article ID 278542,33pages doi:10.1155/2012/278542

Research Article

Exact Asymptotic Expansion of Singular Solutions for the (2 1)-D Protter Problem

Lubomir Dechevski,

Nedyu Popivanov,

and Todor Popov

1Faculty of Technology, Narvik University College, Lodve Langes Gate 2, 8505 Narvik, Norway

2Faculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria

Correspondence should be addressed to Nedyu Popivanov,[email protected] Received 29 March 2012; Accepted 24 June 2012

Academic Editor: Valery Covachev

Copyrightq2012 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 inR². 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.

1. Introduction

In the present paper some boundary value problemsBVPsformulated 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 R² 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 functionf 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 whenfis 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 f but for somewhat easier 3 1-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 flowssee, e.g., the survey of Morawetz1. 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, Protter2proposes 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 Phillips3who 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 partΩof the domain for the mixed-type problem. For the wave equation

u≡ux1x1ux2x2−utt fx, t, 1.1

this is the set

Ω:

x1, x2, t: 0< t < 1 2, t <

x²₁x²₂ <1−t

. 1.2

It is bounded, seeFigure 1, by two characteristic cones of1.1

S1

x1, x2, t: 0< t < 1 2,

x²₁x²₂ 1−t

,

S₂

x1, x₂, t: 0< t < 1 2,

x²₁x₂² t

,

1.3

and the diskS0 {x1, x2, t: t 0, x²₁x₂²<1}, centered at the originO0,0,0.

One could think of the Protter problems in Ω 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

S2

t

S1

x2

x1

O

S0

Figure 1: The domainΩ.

the characteristics. Actually, the set Ωcould be produced via rotation around thet-axis in R³ of the flat triangleΩ2 : {x1, t : 0 < t < 1/2; t < x1 < 1−t} ⊂ R²—a characteristic triangle for the corresponding string equation

u_x1x1−u_tt gx1, t. 1.4

As mentioned before, the classical Darboux problem for1.4is to find solution inΩ2with data prescribed on{t 0}and{t 1−x₁}, for example. In conformity with this planar BVP, Protter2,4formulated and studied the following problems.

ProblemsP1andP2

Find a solution of the wave equation1.1in whichΩsatisfies one of the following boundary conditions:

u|S0 0, u|S1 0, P1

or

u_t|S0 0, u|S1 0. P2

Nowadays, it is known that the Protter ProblemsP1and P2are not well posed, in contrast to the planar Darboux problem. In fact, in 1957 Tong 5proved the existence of infinite number nontrivial classical solutions to the corresponding homogeneous adjoint problemP1^∗. The adjoint BVPs to ProblemsP1andP2were also introduced by Protter.

ProblemsP1^∗andP2^∗

Find a solution of the wave equation1.1inΩwhich satisfies the boundary conditions:

u|S0 0, u|S2 0 adjoint to ProblemP1, P1^∗

or

u_t|S0 0, u|S2 0 adjoint to ProblemP2. P2^∗

Since5, for each of the homogeneous ProblemsP1^∗andP2^∗i.e.,f ≡0 in1.1, an infinite number of classical solutions has been foundsee Popivanov, Schneider6, Khe 7. According to this fact, a necessary condition for classical solvability of ProblemP1or P2is the orthogonality inL₂Ωof the right-hand side functionfx, tto all the solutions of the corresponding homogenous adjoint problemP1^∗orP2^∗. Although Garabedian proved 8the uniqueness of a classical solution of ProblemP1for its analogue inR⁴, generally, ProblemsP1and P2 are not classically solvable. Instead, Popivanov and Schneider6 introduced the notion of generalized solution. It allows the solution to have singularity on the inner coneS2and by this the authors avoid the infinite number of necessary conditions in the frame of the classical solvability. In6some existence and uniqueness results for the generalized solutions are proved and some singular solutions of Protter ProblemsP1and P2are constructed.

In the present paper we study the properties of the generalized solution for Protter ProblemP2 inR³. From the results in6it follows that for n ∈ Nthere exists a smooth right-hand side functionf ∈CⁿΩ, such that the corresponding unique generalized solution of ProblemP2has a strong power-type singularity at the originOand behaves liker⁻ⁿP, O there. This feature deviates from the conventional belief that such BVPs are classically solvable for very smooth right-hand side functionsf. Another interesting aspect is that the singularity is isolated only at a single point the vertex O of the characteristic light cone, and does not propagate along the bicharacteristics which makes this case different from the traditional case of propagation of singularitysee, e.g., H ¨ormander9, 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 6 and references therein. For further publications in this area see 7, 10–16. On the other hand, Bazarbekov gives inΩanother analogue of the classical Darboux problem see17and analogously inR⁴ see18in the corresponding four-dimensional domain Ω. Some different statements of Darboux type problems in R³ or connected with them Protter problems for mixed type equationsalso studied in 2 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 in26. For recent existence results concerning closed boundary-value problems for mixed type equations see for example 27, and also 28 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 inR³ and R⁴, as well as for the Euler-Poisson-Darboux equation has been studied in7,13–16,29.

Further, we aim to find some exact a priori estimates for the singular solutions of Problem P2 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 foundsee15inR³and16inR⁴and an asymptotic formula for the solution of ProblemP1inR⁴was obtained in30.

Considering Protter Problems, Popivanov and Schneider6proved 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 inR³, were obtained

in6. In10Aldashev mentioned the results of6and, for the case of the wave equation in R^m1, he notes the existence of solutions in the domainΩεΩε → ΩandS_2,εapproximates S₂ifε → 0, which blows up on the coneS_2,εlikeε^−nm−2, whenε → 0. It is obvious that for m 2 this results can be compared to the estimates inCorollary 2.4here. Finally, we point out that in the case of an equation, which involves the wave operator and nonzero lower terms, Karatoprakliev24obtained 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 one12.

In Section 2 the result of the existence of infinite number of classical solutions to the homogeneous Problem P2^∗ Lemma 2.1 and the definition of generalized solution of ProblemP2are given. The main results of the paper, concerning the asymptotic expansion of the unique generalized solutionux, tof ProblemP2Theorem 2.3are formulated and discussed. The expansion of uP is given in negative powers of the distance rP, O to the pointO 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 inCorollary 2.4. InSection 3, the auxiliary 2D boundary value Problems P2.1 and P2.2, which correspond to the 2 1-D Problem P2, are considered. Actually, these 2D problems are transferred to an integral Volterra equation, which is invertible. Using the special Legendre functions Pν, some exact formulas for the solution of the ProblemP2.2are derived inLemma 3.4. Some figures showing the effects appearing near the singularity point are also presented.Section 4contains the most technical part of the paper. In this section the results concerning the asymptotic expansions of the generalized solution of the 2D ProblemP2.1are proved and the proof of the main Theorem 2.3is given.

2. Main Results on (2 1)-D Protter’s Problem P2

Define the functions

Eⁿ_kx, t ^k

i 0

B^k_i

x²₁x₂²−t²_{n−1/2−k−i}

x²₁x₂²_n−i , n, k∈N∪ {0}, 2.1

where the coefficients are

B^k_i : −1ⁱk−i1_in1/2−k−i_i

i!n−i_i , B₀^k 1, 2.2

witha_i: aa1· · ·ai−1,a₀: 1. Then for the functions W_k,1ⁿ x, t : Eⁿ_kx, tRe

x1ix₂ⁿ , W_k,2ⁿ x, t: Eⁿ_kx, tIm

x1ix₂ⁿ , 2.3

we have the following lemma.

Lemma 2.1 see29. Letn ∈ N,n ≥ 4. For k 0, . . .,n−3/2and i 1,2 the functions W_k,iⁿ x, tare classicalC²Ω∩C^∞Ωsolutions to the homogeneous ProblemP2^∗.

A necessary condition for the existence of classical solution for ProblemP2 is the orthogonality of the right-hand side functionfto all functionsW_k,iⁿ x, t, which are solutions of the homogeneous adjoint Problem P2^∗. To avoid these infinite number necessary conditions in the framework of classical solvability, one needs to introduce some generalized solutions of ProblemsP2with possible singularities on the characteristic coneS2, or only at its vertexO. Popivanov and Schneider in6give the following definition.

Definition 2.2. A functionu ux1, x2, tis called a generalized solution of the ProblemP2 inΩif:

1u∈C¹Ω\O,u_t|S0\O 0,u|S1 0, 2the identity

Ω

utwt−ux1wx1−ux2wx2−fw

dx1dx2dt 0 2.4

holds for allw∈C¹Ω, wt 0 onS₀, andw 0 in a neighborhood ofS₂.

The uniqueness of the generalized solution of ProblemP2and existence results for f∈C¹Ωcan be found in6.

Further, we fix the right-hand side functionfas a trigonometric polynomial of orderl with respect to the polar angle:

fx1, x2, t Re _l

n 2

fn|x|, tx1ix2ⁿ

, 2.5

with some complex-valued function-coefficients fn|x|, t. Forn 0, . . . , l; k 0, . . . ,n/2 andi 1,2, denote byβⁿ_k,ithe constants

βⁿ_k,i:

ΩW_k,iⁿx, tfx, tdxdt. 2.6

Note that actuallyβ⁰_k,i 0 andβ_k,i¹ 0 in cases ofn 0 andn 1, due to the special form of the functionsW_k,iⁿ and the fact that in the representation2.5of the functionf the sum starts fromn 2.

The main result is as follows.

Theorem 2.3. Suppose that the functionfx, t∈C¹Ωis a trigonometric polynomial2.5. Then there exist functionsFⁿx, t,Fⁿ_k,ix, t,Fx, t∈C²Ω\Owith the following properties:

ithe unique generalized solutionux, tof ProblemP2exists, belongs toC²Ω\Oand has the asymptotic expansion at the originO:

ux, t ^l

m 0

|x|²t²_−m/2

F^mx, t

|x|²t²1/4

Fx, tln

|x|²t²

, 2.7

iifor the coefficient functionsF^mx, tthe representation

F^mx, t ^l−m/2

k 0

2 i 1

β^m2k_k,i F^m2k_k,i x, t, m 0, . . . , l, 2.8

holds, where the functionsF_k,iⁿ x, tare bounded and independent off,

iiiif in the expression2.8forF^mx, tat least one of the constantsβ^m2k_k,i is different from zero (i.e., the corresponding orthogonality condition is not fulfilled), then there exists a direction α1, α2,1withα1, α2,1t∈S2 for 0< t <1/2, such that lim_t→0F^mα1t, α2t, t cm

const /0,

ivif in the expression 2.8 forF⁰x, t at least one of the constants β^2k_k,i is different from zero (i.e., the corresponding orthogonality condition is not fulfilled), then the generalized solution is not continuous atO,

vfor the functionFx, tthe estimate

|Fx, t| ≤C

maxΩ

fx, tmax

Ω

f_tx, t

, x, t∈Ω, 2.9

holds with a constantCindependent off.

As a consequence ofTheorem 2.3one gets the following results that highlight the two extreme cases of the assertion. The first part gives rough estimate of the expansion2.7and describes the “worst” possible singularity. The second part shows that one could control the solution by making some of the defined by2.6constantsβⁿ_k,iin2.8to be zero, that is, by takingfto be orthogonal inL2Ωto the corresponding functionsW_k,iⁿ defined in2.3.

Corollary 2.4. Suppose thatf∈C¹Ωhas the form2.5.

iWithout any orthogonality conditions imposed, the unique generalized solution u of ProblemP2satisfies the a priori estimate

|ux, t| ≤C

|x|²t²_−l/2f

C¹Ω, x, t∈Ω. 2.10

iiLet the orthogonality conditions,

βⁿ_k,i≡

ΩW_k,iⁿx, tfx, tdx dt 0, 2.11 be fulfilled for alln 2, . . . , l; k 0, . . . ,n−1/2andi 1,2. Then the generalized solutionux, tbelongs toC²Ω\O, is bounded and the a priori estimate

sup

Ω |u| ≤Cf

CΩft

CΩ

2.12

holds.

iiiIn addition to (ii), if the conditions2.11are fulfilled fork n/2also, thenu∈CΩis a classical solution anduO 0.

Let us point out that in the case ii, the generalized solution uis bounded if and only if the conditions 2.11 are fulfilled for k ≤ n −1/2 due to Theorem 2.3iii. In addition, if all the conditions2.11are fulfilled fork ≤n−1/2, but for somek n/2 the corresponding orthogonality condition is not satisfied, then u is not continuous at O, according toTheorem 2.3iv. Such a solution is illustrated inFigure 4.

Notice that some of the functionsW_k,iⁿx, tinvolved in the orthogonality conditions in Corollary 2.4iiandiiiare not classical solutions of the homogenous adjoint ProblemP2^∗ in view ofLemma 2.1, although they satisfy the homogenous wave equation inΩ. In fact, for somek,W_k,iⁿ or their derivatives may be discontinuous atS₂. For example whennis an odd number andk n−1/2, the functionsW_k,iⁿ are not continuous at the originO. On the other hand, whennis even andk n/2,W_n/2,iⁿ are singular on the coneS2and do not satisfy the homogeneous adjoint boundary condition there. However, this singularity is integrable in the domainΩ.

To explain the results in Theorem 2.3 and Corollary 2.4 we construct Table 1. It illustrates the connection between the singularity of the generalized solution and the functionsW_k,iⁿ.

Both functionsW_k,iⁿ,i 1,2 are located in column numbernand row numbern−2k inTable 1. Thus,W_0,iⁿ form the rightmost diagonal, the next one is empty—we put in these cells “diamonds”,W_1,iⁿ constitute the third one, and so on. The row number designates the order of singularity of the generalized solution.

Corollary 2.4shows that the generalized solutionux, tis bounded, when the right- hand side functionfis orthogonal to the functions inTable 1, except the ones in row number 0. If f is orthogonal to all the functions in the Table 1 including the row 0, then u is continuous in Ω. When the right-hand side f satisfies orthogonal conditions2.11 for all the functions from the rows inTable 1with row-number larger thenm, 0< m < l, but there is a functionW_q,i^p withp−2q mfrommth row which is not orthogonal tofi.e.,β_q,i^p /0, then the solution behaves liker^−mat the origin, according to the expansion2.7. If there are no orthogonality conditions, then the worst case with singularityr^−lappears.

Figures2–5are created using MATLAB and represent some numerical computations for singular solutions of Problem P2 actually the behaviour in r, t-domain D₁, not including the terms sinnϕ and cosnϕ. They illustrate different cases according to the main results for the existence of a singularity at the originO depending on orthogonality

Table 1: The order of singularity of the solution and the functionsW_k,iⁿ.

l l−1 l−2 l−3 · · · p · · · 4 3 2

0 · · · W_2,i⁴ W_1,i²

1 · · · W_1,i³

2 · · · W_1,i⁴ W_0,i²

3 · · · W_0,i³

4 · · · W_0,i⁴

... · · · ·

p−2q · · · W_q,i^p · · ·

· · ·

... · · · ·

l−3 W_1,i^l−1 W_0,i^l−3 l−2 W_1,i^l W_0,i^l−2

l−1 W_0,i^l−1 l W_0,i^l

−20

−15

−10

−5 0 5

0.10 0.30.2 0.50.4

0 0.2 0.4 0.6 0.8 1 −10

−8

−6

−4

−2 0 2 4 6 8 10

Figure 2: No orthogonality conditions.

conditions. Figure 2 is related to Corollary 2.4i—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 O. In Figure 3, only one of orthogonality conditions 2.11 for k ≤ n−1/2is not fulfilled and the solution tends to±∞. Figures4and5are connected toCorollary 2.4iiandiii:Figure 4presents the case when all the orthogonality conditions 2.11for k ≤ n−1/2are satisfied and the solution is bounded but not continuous at 0,0, whileFigure 5concerns the last partiiifrom Corollary 2.4, when conditions 2.11 are additionally fulfilled fork n/2and the solution is continuous.

Remark 2.5. We mention some differences between the results given here for the ProblemP2 and some other results inR³, but for the ProblemP1, like that from15.

iIn15, assuming the right-hand side functionf is smooth enoughi.e.,f ∈ C^l only the behavior of the singularities was studied using some weighted norms

−10

−5 0 5

−8

−6

−4

−2 0 2 4 6 8 10

0.1 0 0.3 0.2 0.5 0.4

0 0.2 0.4 0.6 0.8 1

Figure 3: One orthogonality condition is not fulfilled.

−10

−5 0 5 10 15

0

−4 1

−2101214161802468

0.1 0 0.3 0.2

0.5 0.4 0.5

Figure 4: Orthogonality conditions fulfilled fork≤n−1/2.

0 1 2 3 4 5 6 7

01 23 45 67 8

0.1 0 0.3 0.2

0.5 0.4

0 0.20.40.60.81

−1 Figure 5: All orthogonality conditions fulfilled fork≤n/2.

analogous to the weighted Sobolev norms in corner domains. In the present paper we need onlyf ∈C¹and find in addition the explicit asymptotic expansion of the generalized solution. The bounded but not continuous at the origin solutions are also studied here.

iiComparing the power of singularity of the generalized solution for Problem P2 here and for Problem P1 in 15for the worst case without any orthogonality conditions one can see that the power in the estimate2.10fromCorollary 2.4i is|x|²t^2−l/2, while in the analogous estimate in Conclusion 115 it is|x|² t^2−l−1/2.

iiiIt 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 P1inR³. However, in14there are absent any analogues to the orthogonality conditions presented in15, and in contrast to15in14the 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 30, where an asymptotic expansion of ProblemP1is found for somewhat easier four-dimensional case.

iBoth for Problem P2 in R³ here and Problem P1 in R⁴ as in 30, the study is based on the properties of the special Legendre functions. Instead of Legendre functions P_ν with non-integer indices ν n−1/2 here, in the four-dimensional case one has to deal with integer indicesν n, that is, simply with the Legendre polynomials P_n. One can easily modify both these techniques to obtain similar results for them1-dimensional problems: for evenmanalogous to the present caseR³or for oddmsimilarly toR⁴case. Some different kind of results for the Protter problems inR^m1are presented in10,11.

iiFor the four-dimensional ProblemP1in30, the Corollary 3.3 gives only that the solution is bounded, it could be discontinuous at the origin. On the other hand, hereTheorem 2.3gives us also the control over the bounded but not continuous parts of the generalized solution through the coefficient F⁰x, t for m 0 in the expansion formula2.7. As a sequence,Corollary 2.4iiiguarantees that the solution is continuous.

iiiBased on the formulae and the computations from30, the general case inR⁴is also treated, when the right-hand sidef is smooth enough, but not a finite harmonic polynomial analogous to 2.5. The results are announced and published in32, 33. For right-hand side functionsf ∈C¹⁰Ωin33the necessary and sufficient conditions for the existence of bounded solution are found. They involve infinite number of orthogonality conditions forf that comes from the fact that this is not a Fredholm problem. On the other hand, the results from33show that the linear operator mapping the generalized solution uintof is a semi-Fredholm operator inC¹⁰Ω. Let us recall that a semi-Fredholm operator is a bounded operator that has a finite dimensional kernel or cokernel and closed range. Additionally, in32 a right-hand side function is constructed such that the unique generalized solution of Protter ProblemP1 inR⁴ has exponential type singularity. One expects that similar results could also be obtained for the ProblemP2inR³studied here. These questions correspond to the Open Problem1below.

Remark 2.7. Let us mention one obvious consequence ofTheorem 2.3and all the arguments above, concerning construction of functions orthogonal to the solutions W_k,iⁿ of the homogeneous adjoint Problem P2^∗. Take an arbitrary C²Ω function Ux, t satisfying the boundary conditions P2. Then the function F : U with the wave operator , is orthogonal to all the functionsW_k,iⁿ ,n 1,2, . . ..

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 f ∈ C^kΩ, for an appropriatek. The smooth functionf could be represented as a Fourier series rather than, the finite trigonometric polynomial2.5in the discussions here.

iFind some appropriate conditions for the function f under which there exists a generalized solution of the Protter problemP2.

iiWhat kind of singularity can the generalized solution have? The a priori estimates, obtained in 6, 31, which indicate that the generalized solutions of ProblemP2 including the singular ones, can have at most an exponential growth asρ → 0.

The natural question is as follows: is there a singular solution of these problems with exponential growth asρ → 0 or do all such solutions have only polynomial growth?

iiiIs it possible to prove some a priori estimates for generalized solutions of the Problem P2with smooth functionfwhich is not a harmonic polynomial?

ivFind some appropriate conditions for the functionfunder which the ProblemP2 has only regular, bounded solutions, or even classical solutions.

2To study the Protter problems for degenerate hyperbolic equations. Up to now it is only known that some singular solutions exist.

iWe do not know what is the exact behavior of the singular solution even when the right-hand side functionf is a finite sum like2.5. Can we prove some a priori estimates for generalized solutions?

iiIs it possible to find some orthogonality conditions for the functionf, as here, under which only bounded solutions exist?

3Why 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 domainas in2,4when the maximal symmetry is lost if the coneS2is replaced by another light characteristic one with the vertex away from the origin.

3. Preliminaries

We have a relation between the functionsW_k,iⁿ and the Legendre functionsP_ν. Forν >−1/2, the functionsP_νcould be defined by the equalitySection 3.7, formula6, from Erd´elyi et al.

34,

Pνz 1 π

_π

0

z

z²−1 cost_ν

dt, z≥0, 3.1

where forz <1 in this formula√

z²−1 : i√ 1−z².

Letρ,ϕ, tbe the cylindrical coordinates inR³, that is,x₁ ρcosϕ,x₂ ρsinϕ. For simplicity, define the functionEⁿ_kρ, t : E_kⁿx, t|x|ⁿ. The following result is in connection with Lemma 5.1 from 15. Actually, to prove this result one could formally follows the arguments of Lemma 2.3 from16, where the four-dimensional analogue of ProblemP2 is treated.

Lemma 3.1. Forn∈Nandν n−1/2 define the functions

h^ν_k ξ, η

: _ξ

η

s^kP_ν

ξηs² s

ξη

ds, 3.2

for 0< η < ξ. Then in{ρ > t}the equality

ρ^−1/2 ∂

∂th^ν_ν−2k ρt

2 ,ρ−t 2

aⁿ_kEⁿ_k ρ, t

3.3

holds fork 0,1, . . . ,n/2with some constantsaⁿ_k/0.

Proof. Lemma 5.1 from15fork≥0 gives

ρ^−1/2h^ν_ν−2k−2 ρt

2 ,ρ−t 2

Cⁿ_kH_kⁿ ρ, t

, 3.4

whereCⁿ_k const/0 and according to Lemma 2.2 from29

∂

∂tH_kⁿ ρ, t

2n−k−1Eⁿ_k1 ρ, t

. 3.5

Therefore the equality3.3holds fork ≥ 1. We have to prove it fork 0. In the proof of Lemma 5.1 from15the integralsh^ν_kwere calculated using the Mellin transform. In order to computeh^ν_kξ, ηin the same way let us first introduce the variablesxandz:

ξ ρt

2 ; η ρ−t

2 ; x ρ²

ρ²−t²; z

ρ²−t²1/2

2 . 3.6

As a consequence after some calculations, formulas2.2.4,1.10, and1.4from Samko et al.35, show that

z^−ν−1h^ν_ν ξ, η

Cνx^ν1/2I₀¹

x^−ν−3/2x−1^1/2

x, 3.7

whereI₀^α us is the Riemann-Liouville fractional integralfor its properties see e.g.,34, 35; in our case we haveI₀¹us _s

0uτdτ. As usual, we denote alsoλs : λsfor s >0, λs: 0 fors≤0. The substitution of3.6in3.7shows that

∂

∂t

z^ν1x^ν1/2I₀¹

x^−ν−3/2x−1^−1/2 ∂

∂t

z^ν1x^ν1/2 _x

0

τ^−ν−3/2τ−1^−1/2 dτ

∂

∂t

ρ^ν12^−ν−1 _x

1

τ^−ν−3/2τ−1^−1/2dτ

2^−ν−1ρ^ν1x^−ν−3/2x−1^−1/2∂x

∂t

2^−ν−1ρ^ν1

ρ²−t²_ν3/2 ρ^2ν3

ρ²−t²1/2

t

2tρ²

ρ²−t²₂ 2^−ν

ρ²−t²ν

ρ^ν

3.8

and thus

ρ^−1/2 ∂

∂th^ν_ν ρt

2 ,ρ−t 2

aⁿ₀Eⁿ₀ ρ, t

. 3.9

The next result is crucial for construction of solutions of Problem P2 in the discussions later.

Lemma 3.2. Letν∈R, ν >1/2 and the functionsFξ∈C¹0,1/2satisfyF1/2 0. Then all solutionsλ∈C¹0,1/2of the Volterra integral equation of first kind

_ξ

1/2

λξ1Pν

ξ ξ₁

dξ₁ Fξ 3.10

are

λξ λ 1

2

Fξ _1/2

ξ

P_ν ξ1

ξ

Fξ1

ξ₁ dξ₁. 3.11

Proof. Formulas 35.17 and 35.28 from Samko et al. 35 state that the solution of the integral equation3.10is given by

λξ −ξd² dξ²

ξ

_1/2

ξ

P_ν ξ₁

ξ

Fξ1 ξ²₁ dξ₁

−d dξ

ξ² d

dξ _1/2

ξ

P_ν ξ₁

ξ

Fξ1 ξ₁² dξ₁

.

3.12

Then, using thatF1/2 0, an integration gives3.11.

One could use the Mellin transform to calculate the following integral.

Lemma 3.3see16. Letν∈R,ν >−1/2, then _1/2

ξ

Pν

ξ1

ξ

Pν2ξ1

ξ₁² dξ1 1−2ξ

ξ . 3.13

According to the existence and uniqueness results in 6, it is sufficient to study ProblemP2when the right-hand sidefof the wave equation is simply

f ρ, t, ϕ

f_n¹ ρ, t

cosnϕf_n² ρ, t

sinnϕ, n∈N∪ {0}. 3.14

Then we seek solutions for the wave equation of the same form:

u ρ, t, ϕ

u¹_n ρ, t

cosnϕu²_n ρ, t

sinnϕ. 3.15

Thus ProblemP2reduces to the following one.

ProblemP2.1 Solve the equation

un_ρρ 1

ρun_ρ−un_tt−n²

ρ²u_n f_n ρ, t

3.16

inD₁ {0< t <1/2; t < ρ <1−t} ⊂R²with the boundary conditions

un_t ρ,0

0 for 0< ρ≤1, u_n

ρ,1−ρ

0 for 1

2 ≤ρ≤1. P2.1 Let us now introduce new coordinates

ξ ρt

2 ; η ρ−t

2 , 3.17

and set

v ξ, η

ρ^1/2un

ρ, t

; g

ξ, η

ρ^1/2fn

ρ, t

. 3.18

Denotingν n−1/2, one transforms ProblemP2.1into the following.

ProblemP2.2

Find a solutionvξ, ηof the equation

v_ξη−νν1 ξη₂v g

ξ, η

3.19

in the domainD {0< ξ <1/2; 0< η < ξ}with the following boundary conditions:

vξ−vη

η, η

0, v

1 2, η

0 for η∈

0,1 2

. P2.2

ProblemsP2.1andP2.2were introduced in6, although the change of coordinates ξ 1−ρ−tandη 1−ρtwas used there instead of3.17. Of course, because the solution of ProblemP2may be singular, the same is true for the solutions ofP2.1andP2.2. For that reason, Popivanov and Schneider6defined and proved the existence and uniqueness of generalized solutions of ProblemsP2.1andP2.2, which correspond to the generalized solution of ProblemP2. Further, by “solution” of ProblemP2.1orP2.2we mean exactly this unique generalized solution.

Lemma 3.4. Letν ∈ R,ν > 1/2 andg ∈ C¹D. Then the solutionvξ, ηof ProblemP2.2is given by the following formula:

v ξ, η

τξ _1/2

ξ

τξ1 ∂

∂ξ1

P_ν

ξ−η

ξ12ξη ξ1

ξη

dξ₁

− _1/2

ξ

_η

0

P_ν

ξ−η ξ1−η1

2ξ1η12ξη ξ1η1

ξη

g

ξ₁, η₁ dη₁

dξ₁,

3.20

where

τξ _ξ

1/2

P_ν ξ₁

ξ

Gξ1dξ1, 3.21

Gξ _1/2

ξ

_ξ

0

Pν

ξ₁η₁ξ² ξ

ξ₁η₁ ∂

∂ξ₁ − ∂

∂η₁

g ξ1, η1

dη1dξ1

− _ξ

0

Pν

η₁2ξ² ξ

2η₁1

g 1

2, η1

dη1−

_1/2

ξ

Pν

ξ ξ₁

gξ1,0dξ1.

3.22

Proof. Notice that the function

R

ξ1, η1;ξ, η Pν

ξ−η ξ₁−η₁

2ξ₁η₁2ξη ξ₁η₁

ξη

3.23

is a Riemann function for 3.19 Copson 36. Therefore, following Aldashev 10, we can construct the function vξ, ηas a solution of a Goursat problem in D with boundary conditionsv1/2, η 0 andvξ,0 τξwith some unknown functionτξ ∈C²0,1/2, which will be determined later:

v ξ, η

τξ _1/2

ξ

τξ1 ∂

∂ξ1R

ξ1,0;ξ, η dξ1

− _1/2

ξ

_η

0

R

ξ₁, η₁;ξ, η g

ξ₁, η₁ dη₁dξ₁.

3.24

Now, the boundary condition

∂

∂ξ − ∂

∂η

v η ξ

0. 3.25

gives an integral equation forτξ. For that reason, let us define the functionGξ:

Gξ: ∂

∂ξ − ∂

∂η

1/2 ξ

_η

0

R

ξ₁, η₁;ξ, η g

ξ₁, η₁ dη₁

dξ₁

η ξ

_1/2

ξ

ξ 0

P_ν

ξ₁η₁ξ² ξ

ξ₁η₁

ξ₁−η₁ ξ

ξ₁η₁g ξ1, η1

dη1

dξ1

− _ξ

0

g ξ, η₁

dη₁− _1/2

ξ

gξ1, ξdξ1 − _ξ

0

g ξ, η₁

dη₁− _1/2

ξ

gξ1, ξdξ1

− _1/2

ξ

ξ 0

g ξ1, η1

∂

∂ξ₁ − ∂

∂η₁

Pν

ξ₁η₁ξ² ξ

ξ₁η₁

dη1

dξ1

_1/2

ξ

ξ 0

Pν

ξ₁η₁ξ² ξ

ξ₁η₁ ∂

∂ξ₁ − ∂

∂η₁

g ξ1, η1

dη1

dξ1

− _ξ

0

P_ν

η₁2ξ² ξ

2η₁1

g 1

2, η₁

dη₁− _1/2

ξ

P_ν ξ

ξ₁

gξ1,0dξ1.

3.26

Obviously,G∈C²0,1/2. The condition3.25leads us to the following equation:

τξ−1

ξτξP_ν1− _1/2

ξ

τξ1 ξ²₁ P_ν

ξ ξ1

dξ₁ Gξ. 3.27

Then, usingτ1/2 v1/2,0 0, we have _ξ

1/2

d dξ1

ξ₁²τξ1 P_ν

ξ ξ1

dξ₁ ξ²Gξ−τ1/2

4 P_ν2ξ. 3.28

A necessary solvability condition for the unknown function τ ∈ C²0,1/2 is: τ1/2 G1/2. One could solve this Volterra integral equation of the first kind, usingLemma 3.2.

The result is

ξ²τξ−1 4τ

1 2

ξ²Gξ−1 4τ

1 2

P_ν2ξ _1/2

ξ

P_ν ξ₁

ξ

4ξ₁²Gξ1−τ1/2Pν2ξ1 4ξ1

dξ₁. 3.29

Integrate, we find

τξ _ξ

1/2

Gz 1

z² _1/2

z

P_ν ξ₁

z

ξ₁Gξ1dξ1

dz

τ1/2 4

_ξ

1/2

1

z² −P_ν2z z² − 1

z² _z

1/2

P_ν ξ₁

z

P_ν2ξ1ξ⁻¹₁ dξ₁

dz.

3.30

Now, usingLemma 3.3and the equality _ξ

1/2

1 z²

_1/2

z

P_ν ξ1

z

Fξ1dξ1

dz

_ξ

1/2

P_ν

ξ1

ξ

−1 Fξ1

ξ₁ dξ₁, 3.31

forFξ Pν2ξξ⁻¹ one finds that the coefficient of τ1/2 in3.30 is zero. Using again 3.31forFξ ξGξ,τis simply

τξ _ξ

1/2

Pν

ξ1

ξ

Gξ1dξ1. 3.32

Obviously,τ ∈C²0,1/2andτ1/2 0, τ1/2 G1/2. Finally, the solution of Problem P2.2is given by the formulae3.20,3.21, and3.22.

4. Proofs of Main Results

In order to study the behavior of the generalized solution of ProblemP2, in view of relations 3.18andLemma 3.4, we will examine the functionvξ, ηdefined by the formulae3.20, 3.21, and3.22. It is not hard to see that the part “responsible” for the singularity is the integral in3.21for the functionτξ. In fact,τξblows up atξ 0, since the argumentξ1/ξ and thus the values of the Legendre functionP_νin3.21go to infinity whenξ → 0. Actually,

Pνzgrows like|z|^νat infinity. In the next lemma we find the dependance of the exact order of singularity ofτξon the functionGξ. It is governed by the constants

γ_k: _1/2

0

ξ^ν−2kGξdξ fork 0, . . . , ν1

2

. 4.1

Actually, these numbers are closely connected to the constantsβⁿ_k,ifromTheorem 2.3. We will clarify this relation later inLemma 4.1and the proof ofTheorem 2.3.

Lemma 4.1. Letν n−1/2, wheren∈N,n≥2, and let the functionGξ∈C¹0,1/2. Then the function

τξ _ξ

1/2

P_ν ξ₁

ξ

Gξ1dξ1 4.2

belongs toC²0,1/2and satisfies the representation

τξ ^ν1/2

k 0

C^ν_kγ_kξ^−ν−2kψξ, ξ∈

0,1 2

, 4.3

where the functionψξ ∈ C²0,1/2,|ψξ| ≤ Cξmax{|Gξ| : ξ ∈ 0,1/2} and the nonzero constantsC^ν_kandCare independent ofGξ.

Proof. The argument of the Legendre function Pν in 4.2satisfies the inequality ξ1/ξ ≥ 1, which allows us to apply the representation3.1:

τξ 1 π

1 ξ^ν

_ξ

1/2

_π

0

ξ₁

ξ²₁−ξ²cost _ν

Gξ1dt dξ1. 4.4

We will study the expansion atξ 0 of the function

Fξ: _1/2

ξ

_π

0

ξ₁

ξ₁²−ξ²cost _ν

Gξ1dt dξ1. 4.5

Let us define the functions

M^ν_kξ1, ξ: −1^kν−2k1_2k 2^k1/2_k

_π

0

ξ₁

ξ²₁−ξ²cost _ν−2k

sin^2kt dt, 4.6

forξ≤ξ₁≤1/2. Then, obviously

Fξ _1/2

ξ

M^ν₀ξ1, ξGξ1dξ1. 4.7