Solution of the multidimensional problem for the parabolic equation with incompatible initial and boundary data in the Hölder and weighted spaces

Abstract

Studying the solutions of the boundary value problems for the parabolic equations in the Hölder spaces we should require the fulfilment of the compatibility conditions of the initial and boundary data of all necessary orders, they provide the continuity of the solution and its derivatives of all acceptable orders up to the boundary and boundedness of the Hölder constants of the highest derivatives in the closure of a domain. Such problems describe the physical processes which go continuously all the time since the beginning of the processes. If we consider the processes (for instance, heating or cooling), which are not continuous at the initial moment, then compatibility conditions of the initial and boundary data of the problems modeling this processes can not be fulfilled, but processes go, that is the problems with incompatible initial and boundary data have physical sense and they can have the solutions.There is considered a multidimensional first boundary value problem for the parabolic equation with incompatible initial and boundary data of the zero and first orders. It is proved that the solution of the problem may be represented as a sum of a Hölder solution and two singular ones corresponding these two incompatible initial and boundary conditions. The singular solutions belong to the weighted space with parabolic weights and space of the functions, the highest derivatives of which are not continuous, but bounded. These regular and singular solutions are unique, the estimates for them are obtained.