Existence and non-existence of global solutions for a heat equation with degenerate coefficients

Abstract

In this paper, we will study the following parabolic problem $u_t-div(\omega (x)\nabla u)=h(t)f(u)+l(t)g(u)$ with non-negative initial conditions pertaining to $C_b({\mathbb{R}}^N)$, where the weight $\omega$ is an appropriate function that belongs to the Munckenhoupt class $A_{1+\frac{2}{N}}$ and the functions $f$, $g$, $h$ and $l$ are non-negative and continuous. The main goal is to establish of global and non-global existence of non-negative solutions. In addition, to present the particular case when $h(t)\sim t^r(r>-1)$, $l(t)\sim t^s(s>-1)$, $f(u)=u^p$ and $g(u)=(1+u)[\ln (1+u){]}^p,$ we obtain both the so-called Fujita's exponent and the second critical exponent in the sense of Lee and Ni \cite{Lee-Ni}. Our results extend those obtained by Fujishima et al. \cite{Fujish} who worked when $h(t)=1$, $l(t)=0$ and $f(u)=u^p$.