Generalized solutions for a class of nonlinear parabolic problems with irregular data in unbounded domains

Abstract

We consider the following nonlinear parabolic problem with singular data whose model is

$$\begin{aligned} ({\mathcal {P}}_{b})\ \left\{ \begin{aligned}&u_t-\text {div}(a(t,x,\nabla u))+a_{0}(t,x, u)=\mu \text{ in } Q:=(0,T)\times \varOmega ,\\&u(0,x)=u_{0}(x) \text{ in } \varOmega ,\ u(t,x)=0\text { on }(0,T)\times \partial \varOmega , \end{aligned}\right. \end{aligned}$$

where $\varOmega$ is an open, possibly unbounded, subset of ${\mathbb {R}}^{N}$, $N\ge 2$, $T>0$, $u_{0}\in L^{1}(\varOmega )$, and $\mu$ is a Radon measure with bounded variation on Q. The function $u\mapsto -\text {div}(a(t,x,\nabla u))+a_{0}(t,x,u)$ is a continuous, bounded and monotone operator acting in $L^{p}(0,T;W^{1,p}_{0}(\varOmega ))$, $1<p\le N$, and satisfying some growth conditions. The originality of this paper is to study the existence and uniqueness of problem $({\mathcal {P}}_{b})$ when $\mu$ is a general measure with additional decomposition property $\mu =\mu _{d}+\mu _{c}$, where $\mu _{d}$ is the “diffuse” part of $\mu$, and $\mu _{c}$ is “concentrated” on a set of zero parabolic “p-capacity”. The study of problem $({\mathcal {P}}_{b})$ will be splitted in two different cases according to the boundedness of the domain $\varOmega$ (bounded or not), and to the comportment of the solution when the singular part appears or disappears ($\mu _{c}\ne 0$ or $\mu _{c}=0$): the first one consists on proving the existence of generalized (“renormalized”) solutions, and the second one is the uniqueness where the main obstacle relies on the presence of the term $\mu _{c}$. Moreover, if $\mu _{c}\equiv 0$, the uniqueness result is proved by assuming a strictly monotonicity property.