A nonlinear parabolic problem with singular terms and nonregular data

Abstract

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$\{\begin{array}{ll}{u}_t-{\Delta }_{p}u=h(u)f+\mu & \text{in}\Omega \times (0,T),\\ u=0& \text{on}\partial \Omega \times (0,T),\\ u=u_0& \text{in}\Omega \times \{0\},\end{array}$$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$), $u_0$ is a nonnegative integrable function, ${\Delta }_p$ is the $p$-laplace operator, $\mu$ is a nonnegative bounded Radon measure on $\Omega \times (0,T)$ and $f$ is a nonnegative function of $L^1(\Omega \times (0,T))$. The term $h$ is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing $h$.