# A nonlinear parabolic problem with singular terms and nonregular data

## TL;DR

This paper investigates the existence and uniqueness of nonnegative solutions to a nonlinear parabolic PDE with singular lower order terms and measure data, extending the understanding of such equations with potentially unbounded nonlinearities.

## Contribution

It establishes existence and uniqueness results for solutions to a complex nonlinear parabolic problem with singularities and measure data, under broad conditions.

## Key findings

- Existence of nonnegative solutions under general conditions.
- Uniqueness of finite energy solutions with nonincreasing nonlinear terms.
- Handling of singular lower order terms and measure data in parabolic equations.

## 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 $$ \begin{cases} \displaystyle u_t - \Delta_p u = h(u)f+\mu & \text{in}\ \Omega \times (0,T),\\ u=0 &\text{on}\ \partial\Omega \times (0,T),\\ u=u_0 &\text{in}\ \Omega \times \{0\}, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $u_0$ 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 $h$.