EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR A SEMILINEAR HEAT EQUATION WITH MEMORY

Abstract

The first problem posed when studying the classes of PDEs is well-posedness (as V.P.
Maslov is impressed that a PDE of practical significance then it will definitely be solutions,
some kind of solutions). The well-posedness of a problem refers to whether the problem has
a solution, a unique solution and continuous dependence on the initial data of solution. In
this paper we prove the well-posedness of weak solutions to a semilinear heat equation with
memory and the nonlinearity f of exponential type by the Galerkin approximation and
compactness method. The main novelty of our result is that no restriction on the growth of
the nonlinearities is imposed.