Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation

Abstract

In this paper, we study the existence of local/global solutions to the Cauchy problem \begin{eqnarray}\rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\u(x,0)=u_{0}(x)\ge 0, x \in R^N\end{eqnarray}with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.