A necessary and sufficient condition for the existence of global solutions to discrete semilinear parabolic equations on networks

초록

A necessary-sufficient condition for the existence and non-existence of global solutions to the following semilinear parabolic equations u(t) = delta u + psi(t)u(p), in omega x (0, t*), has remained as an open problem for a few decades. This paper discuss the necessary-sufficient condition for the existence and non-existence of global solutions of the above equations on networks for more general source term psi(t)f(u) under various boundary conditions. The purpose of this paper is to prove that there is no global solution for any initial data if and only if the function f satisfies integral(infinity)(0)psi(t)f epsilon?S(t)u0?(infinity/)?S(t)u(0)?(infinity) dt = infinity for every epsilon > 0, where (S(t))(t >= 0) is the heat semigroup on networks with the corresponding boundary condition. (C) 2022 Elsevier Ltd. All rights reserved.

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