Parabolic equations with singular and supercritical reaction terms

Abstract

We prove that, for every $\lambda \gt 0 $ and $u_0 \geq 0$, the following evolution problem has a solution for small value of $T$, \begin{equation} \label{zbep1} \begin{cases} u > 0 \text{ in }\;\Omega_T = \Omega \times (0,T) \;, \\ \displaystyle \; u_t - \div (M(x)\nabla u)= \frac{\lambda}{u^\gamma}+ \mu \, u^{p} \; \text{ in }\;\Omega_T, \\ u=0 \text{ on }\;\partial\Omega \times (0,T),\\ u(0)=u_0 \text{ in }\; \Omega, \end{cases} \end{equation} where $\gamma > 0$, $\mu\geq0$ and $p > 0$. Moreover, we show the existence of a solution for every value of $T$ for suitable small data $\lambda$ and $u_0$ if $p > 1$ and for every data if $0 < p < 1$.