Local boundary value problems for a loaded equation of parabolic-hyperbolic type degenerating inside the domain

Abstract

In the beginning of 21st century, boundary value problems for non-degenerating equations of hyperbolic, parabolic, hyperbolic-parabolic and elliptic-hyperbolic types were studied. Recently this direction is intensively developed since rather important problems in mathematical physics and biology lead to boundary value problems for non-degenerate loaded partial differential equations. Boundary value problems for second order degenerating equation of a mixed type were not studied before. This is first of all because of the fact that there is no representation for the general solution to this equations. On the other hand, such problems are reduced to poorly studied integral equations with a shift. The present work is devoted to formulating and studying local boundary value problems for loaded equation of parabolic-hyperbolic type degenerating inside the domain. In the present work we find a new approach for obtaining a representation for the general solution to a degenerating loaded equation of a mixed type. The uniqueness of the formulated problem is proved by the methods of energy integrals. The existence of solutions to the formulated problems is equivalently reduced to a second order integral Fredholm and Volterra equations with a shift. We prove the unique solvability of the obtained integral equations.