The Existence for RDE with Small Delay

摘要

In this article, we prove that for the classical reaction-diffusion equations (RDE) u_t - △u = f(u(t),u(t - τ)) with a small delay, there exists a weak solution. Our method is Galerkin approximations which is one of the most important methods in proving the existence of weak solution. This method can be found in many works, for example [4]. It is to build a weak solution by first constructing solutions of certain finite-dimensional approximations, and then building energy estimate and last passing to limits. Imposing some condition on the nonlinear f , we first make use of Galerkin approximations to prove the local existence and uniqueness theorem for weak solutions of the Initial-Bounded Value Problem.