Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time

Abstract : The paper deals with two nonlinear problems for parabolic equations. The first, problem A, is an initial-boundary value problem for the heat equation where the nonlinearity is in the boundary condition. The second, problem B, is a final value problem for the porous medium equation. It is shown that if the nonlinearity and initial data in A satisfy certain restrictions then no classical (or weak) solution of A can exist for all time. It is further shown that no weak solution of B can have existed for all previous time. An indication is given of how the methods used in A can be used to obtain (under reasonable hypotheses) the same type of nonexistence result for nonlinear problems associated with certain systems of parabolic and hyperbolic equations. (Author)