Abstract
We study three‐dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. 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.
Highlights
In the present paper some boundary value problems BVPs formulated by M
Protter arrived at the multidimensional problems for hyperbolic equations while examining BVPs for mixed type equations, starting with planar problems with strong connection to transonic flow phenomena
For the Gellerstedt equation of mixed type, Protter 2 proposes a 3D analogue to the two-dimensional Guderley-Morawetz problem
Summary
In the present paper some boundary value problems BVPs formulated by M. From the results in 6 it follows that for n ∈ N there exists a smooth right-hand side function f ∈ Cn 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. To all functions Wkn,i x, t , To avoid these infinite which are solutions number necessary conditions in the framework of classical solvability, one needs to introduce some generalized solutions of Problems P2 with possible singularities on the characteristic cone S2, or only at its vertex O. For right-hand side functions f ∈ C10 Ω in 33 the necessary and sufficient conditions for the existence of bounded solution are found They involve infinite number of orthogonality conditions for f that comes from the fact that this is not a Fredholm problem. Ii Is it possible to find some orthogonality conditions for the function f, as here, under which only bounded solutions exist?
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