Asymptotic expansions of singular solutions for (3+1)-D Protter problems

Abstract

Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. The smoothness of the right-hand side function of the wave equation is decisive for the behavior of the solution of the boundary value problem. It is shown that for each n ∈ N there exists such a right-hand side function from C n , for which the uniquely determined generalized solution has a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. The present article describes asymptotic expansions of the generalized solutions in negative powers of the distance to this singularity point. Some necessary and sufficient conditions for existence of regular solutions, or solutions with fixed order of singularity, are derived and additionally some a priori estimates for the singular solutions are given.

On the Riemann Function

“…and 2 F 1 is the hypergeometric function (see the Appendix A for details of all the hypergeometric functions used in this article). Equation (15) was derived via a Fourier cosine transform. 2.…”

“…Some applications include solving electromagnetic problems exhibiting rotational symmetry [1], finding existence criteria for the eigenvalues of the solution of focal point problems [2], solving for the solution of transient plane waves [3], and the inverse problem of scattering theory [4][5][6][7][8][9][10][11][12]. More recently, Riemann's method has been applied to the solution of coupled Korteweg-de Vries equations [13], to boundary value problems for the non-homogeneous wave equation [14][15][16][17][18], to the solution of the non-linear Schrödinger equation [19,20], and modelling hyperbolic quasi-linear equations [21][22][23].…”

New singular solutions for the (3+1)-D Protter problem

“…They also give a priori estimates for the singularity of the solution. In fact, the behavior of the generalized solution depends strongly on the L 2 (Ω)-inner product of the right-hand side function f (x, t) with the functions v n k,m (x, t) from Lemma 1 (see also [20,3]). Accordingly, we denote by β n k,m the parameters…”

“…Definition 1 [3]. A function u = u(x, t) is called a generalized solution of the problem P 1 in Ω, if the following conditions are satisfied: 1) u ∈ C 1 Ω\O , u| Σ0\O = 0, u| Σ1 = 0, and 2) the identity…”

“…In the special case when the right-hand side function f is a harmonic polynomial, the exact behavior of the generalized solution of problem P 1 is found in [3]. In [7] the semi-Fredholm solvability of problem P 1 is discussed.…”