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.