BOUNDARY VALUE PROBLEMS FOR A FOURTH ORDER PARABOLIC TYPE EQUATION

Abstract

The existence and uniqueness of a solution to a boundary value problem in a rectangle for a fourth-order parabolic equation with a variable coefficient at the lowest term are proved. The straight line y=const is a quadruple characteristic of the given equation. By the method of lowering the order of the equation, the problem under consideration is reduced to the first boundary value problem for the heat equation and the boundary value problem for the second order ordinary differential equation. With the help of the Green's function, representations of the solution of the problems under consideration are obtained. To prove the uniqueness of the solution of the second boundary value problem, the method of energy integrals is used. An example is given in which the explicit form of the constructed Green's function is indicated.