Singular parabolic problems with possibly changing sign data

Abstract

We show the existence of bounded solutions $u\in L^2(0,T;H^1_0(\Omega))$ for a class of parabolic equations having a lower order term $b(x,t,u,\nabla u)$ growing quadratically in the $\nabla u$-variable and singular in the $u$-variable on the set $\{u=0\}$. We refer to the model problem$$\left\{\begin{array}{ll}u_t - \Delta u = b(x,t) \frac{|\nabla u|^2}{|u|^k} + f(x,t) & in \Omega \times (0,T)\\u(x,t) = 0 & on \partial\Omega\times(0,T)\\u(x,0) = u_0 (x) & &nbsp; &nbsp in \Omega\end{array}\right.$$where $\Omega$ is a bounded open subset of $\mathbb{R}^N, N \geq 2, 0 < T < + \infty$ and $0 < k < 1$. The data $f(x,t), u_0(x)$ can change their sign, so that the possible solution $u$ can vanish inside $Q_T=\Omega\times(0,T)$ even in a set of positive measure. Therefore, we have to carefully define the meaning of solution. Also $b(x,t)$ can have a quite general sign.