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

Abstract

We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in R 2 . 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.Problems P1 * and P2 * Find a solution of the wave equation 1.1 in Ω which satisfies the boundary conditions: u| S 0 0, u| S 2 0 adjoint to Problem P1 , P1 * 4 Abstract and Applied Analysis or u t | S 0 0, u| S 2 0 adjoint to Problem P2 . P2 * Since 5 , for each of the homogeneous Problems P1 * and P2 * i.e., f ≡ 0 in 1.1 , an infinite number of classical solutions has been found see Popivanov, Schneider 6 , Khe 7 . According to this fact, a necessary condition for classical solvability of Problem P1 or P2 is the orthogonality in L 2 Ω of the right-hand side function f x, t to all the solutions of the corresponding homogenous adjoint problem P1 * or P2 * . Although Garabedian proved 8 the uniqueness of a classical solution of Problem P1 for its analogue in R 4 , generally, Problems P1 and P2 are not classically solvable. Instead, Popivanov and Schneider 6 introduced the notion of generalized solution. It allows the solution to have singularity on the inner cone S 2 and by this the authors avoid the infinite number of necessary conditions in the frame of the classical solvability. In 6 some existence and uniqueness results for the generalized solutions are proved and some singular solutions of Protter Problems P1 and P2 are constructed.In the present paper we study the properties of the generalized solution for Protter Problem P2 in R 3 . From the results in 6 it follows that for n ∈ N there exists a smooth right-hand side function f ∈ C n Ω , such that the corresponding unique generalized solution of Problem P2 has a strong power-type singularity at the origin O and behaves like r −n P, O there. This feature deviates from the conventional belief that such BVPs are classically solvable for very smooth right-hand side functions f. 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 singularity see, e.g., Hörmander 9 , Chapter 24.5 .The Protter problems have been studied by different authors using variou...

Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

“…which shows that = , , according to their definitions (96) and (13). Now we can apply Theorem 22 for the functions ( , ) and ( , ).…”

“…Necessary and sufficient conditions for the existence of solutions with fixed order of singularity were obtained in [10]. Similarly, for the R 3 -analogues of Protter problems, some results are presented in [13,14]. For the problem with Dirichlet type boundary condition on Σ 0 , a formula for the asymptotic expansion of the singular solution can be found in [15], and the semi-Fredholm solvability is discussed in [16] for ∈ 10 (Ω).…”

“…For the problem with Dirichlet type boundary condition on Σ 0 , a formula for the asymptotic expansion of the singular solution can be found in [15], and the semi-Fredholm solvability is discussed in [16] for ∈ 10 (Ω). A comparison of various recent results for Protter problems is made in [13].…”

“…Here, Theorem 4 describes the effect of the parameters , on the behavior of the generalized solution of Problem 2. The constants , are defined by (13) and are obviously related to the orthogonality conditions (10). When the right-hand function is a harmonic polynomial (11), the asymptotic expansion in Theorem 4 shows that the generalized solution could be bounded only if all , involved are zero.…”

“…3 (as treated in [13,14]) and R 4 here (see also [10,15]). In both cases the study of these BVPs is based on the properties of the special Legendre functions.…”