Existence of a solution for the problem with a concentrated source in a subdiffusive medium

Abstract

By using Green's function, the problem is converted into an integral equation. It is shown that there exists a $t_b$ such that, for $0\leq t\lt t_b$, the integral equation has a unique nonnegative continuous solution $u$; if $t_b$ is finite, then $u$ is unbounded in $[0, t_b)$. Then, $u$ is proved to be the solution of the original problem.