Singular solutions of protter's problem for the 3 + 1;-D wave equation

Abstract

For the wave equation we study boundary value problems, which are four-dimensional analogues of Darboux problems on the plane. It is shown that for n in N there exists a right hand side smooth function from C n , for which the corresponding unique generalized solution has a strong power-type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone and does not propagate along the cone. The present article describes the exact behavior of the singular solutions at the point O. We give some necessary and sufficient conditions for existence of solutions with fixed order of singularity. It states some exact a priori estimates for the singular solutions

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

“…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].…”

“…According to the results from [10] we know that the generalized solution of Problem 2 may have a power type singularity at the origin : = 0, = 0. In the present paper we study more accurately the exact behavior of the solution of Problem 2 at .…”

“…Without any additional conditions imposed on the right-hand side function ∈ (Ω), it is known (see [3,10]) that the generalized solution may have power type singularity. Alternatively, we will prove the following necessary conditions for the existence of bounded solutions.…”

“…Lemma 1 (see [10] A necessary condition for the existence of classical solution for the Problem 2 is the orthogonality with respect to the 2 (Ω) inner product, of the right-hand side function to all functions , ( , ) from Lemma 1. To avoid an infinite number of necessary conditions in the framework of classical solvability, we introduce generalized solutions for the Problem 2 (see the similar definition for the (2+1)-D case in [5]).…”

“…Definition 2 (see [10]). A function = ( , ) is called a generalized solution of the Problem 2 in Ω, if the following conditions are satisfied:…”