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 $$ \begin{cases} \displaystyle u_t - Δ_p u = h(u)f+μ& \text{in}\ Ω\times (0,T),\\ u=0 &\text{on}\ \partialΩ\times (0,T),\\ u=u_0 &\text{in}\ Ω\times \{0\}, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $u_0$ is a nonnegative integrable function, $Δ_p$ is the $p$-laplace operator, $μ$ is a nonnegative bounded Radon measure on $Ω\times (0,T)$ and $f$ is a nonnegative function of $L^1(Ω\times (0,T))$. 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 $h$.