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

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 vertex of 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 to . Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

Nonexistence of nontrivial generalized solutions for 2-D and 3-D BVPs with nonlinear mixed type equations

“…Even in the linear case, the question of well-posedness is surprisingly subtle and not completely resolved (see [32] , [1] and [2] [32] it was shown that the homogeneous adjoint problem admits infinitely many nontrivial classical solutions v n ∈ C n (Ω − ), n ∈ N. [31,30].…”

“…Actually, if in the definition of the generalized solution (see (31) [31,30]. Currently, we have constructed a singular solution for the same linear Protter problem for wave equation (m=0) with exponential growth of singularity at the point O(see [13] [28], [29].…”

Analysis of singularities of solutions to Protter problems for the wave equation

“…This differs the conventional case of propagation of singularities, like in Hörmander [10, Chapter 24.5], since the point O here lies both on the characteristic part of the boundary Σ 2 and on the non-characteristic part Σ 0 . In the special case when the right-hand side function f is a harmonic polynomial, the exact behavior of the generalized solution of Problem P2 is found in [24]. Garabedian [7] proved the uniqueness of a classical solution for a four-dimensional Proter problem.…”

“…On the other hand, in the more general case, when f ∈ C 6 (Ω) the necessary and sufficient conditions for existence of bounded solutions are given in [24,Theorem 1.3]. These are an infinite number of orthogonality conditions for f .…”

“…These are an infinite number of orthogonality conditions for f . According to [24,Theorem 1.2], each orthogonality condition "removes" one power type singularity. Therefore, if only finite number of the orthogonality conditions are not fulfilled, the solution may have at most power type singularity.…”