Abstract
For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertexOof the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance toO. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.
Highlights
In the present paper, boundary value problems for the wave equation in R4ux1x1 + ux2x2 + ux3x3 − utt = f (x, t), (1)with points (x, t) = (x1, x2, x3, t), are studied in the domain Ω = {(x, t) : < t, t
A necessary condition for the existence of classical solution for the Problem P2 is the orthogonality with respect to the L2(Ω) inner product, of the right-hand side function f to all functions Wkn,m(x, t) from Lemma 1
To avoid an infinite number of necessary conditions in the framework of classical solvability, we introduce generalized solutions for the Problem P2 (see the similar definition for the (2+1)-D case in [5])
Summary
Abstract and Applied Analysis exists a right-hand side function f ∈ Cn(Ω) of the wave equation, for which the uniquely determined generalized solution of Problem P2 has a strong power-type singularity like r−n at the origin O. A necessary condition for the existence of classical solution for the Problem P2 is the orthogonality with respect to the L2(Ω) inner product, of the right-hand side function f to all functions Wkn,m(x, t) from Lemma 1. To avoid an infinite number of necessary conditions in the framework of classical solvability, we introduce generalized solutions for the Problem P2 (see the similar definition for the (2+1)-D case in [5]). Suppose that there is a bounded generalized solution of the Protter Problem P2 with right-hand side function f(x, t) ∈ C(Ω). The long and technical proof of Theorem 22 is postponed to Section 7
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