| Abstract: |
| From the classical works of Keldysh and Fichera, it is well known that BVPs for multidimensional linear second-order equations with a non-negative characteristic form are well understood in the sense that boundary conditions should not be imposed on the whole boundary. In this talk analogous statements of multidimensional BVPs for weakly hyperbolic equations of Keldysh type are studied. These equations are considered in a domain bonded by two characteristic surfaces and a ball in the hyperplane of the parabolic degeneration. A data is prescribed only on a part of the characteristic boundary, while the parabolic part of the boundary is free of data and the normal derivative of the solution can have singularity on it. In the frame of the classical solvability these problems are not Fredholm, since they have infinite-dimensional co-kernels. Alternatively, a notion of a generalized solution with possible singularity is given. Results on the existence and uniqueness of such solution are obtained under special condition on the lower-order terms. Further, orthogonality conditions on the smooth right-hand side functions are presented, which are necessary and sufficient for the existence of generalized solutions with fixed order of singularity. |
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