Verified computations for solutions of semilinear heat equations using an analytic semigroup generated by a self-adjoint operator

摘要

A computer-assisted method is proposed for verifying existence and uniqueness of a solution for a semilinear heat equation. The method is based on a sufficient condition for the solution existence is proposed. A fixed-point form of the heat equation is defined by using an analytic semigroup generated by a self-adjoint operator. Banach's fixed-point theorem with the fixed point form gives the sufficient condition. By numerically checking whether the sufficient condition holds, existence and uniqueness of the solution are verified.