A necessary and sufficient condition for the existence of global solutions to reaction-diffusion equations on bounded domains

초록

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations u(t) = Delta u + psi(t)f (u), in Omega x (0, infinity), under the mixed boundary condition on a bounded domain Omega. In fact, this has remained an open problem for a few decades, even for the case f (u) = u(p). As a matter of fact, we prove: there is no global solution for any initial data if and only if integral(infinity)(0) psi(t) f (parallel to S(t)u(0)parallel to(infinity))/parallel to S(t)u(0)parallel to(infinity) dt = infinity for every nonnegative nontrivial initial data u(0) is an element of C-0(Omega). Here, (S(t))(t >= 0) is the heat semigroup with the mixed boundary condition.

키워드