Existence and uniqueness of renormalized solutions for initial boundary value parabolic problems with possibly very singular right-hand side

Abstract

We study the existence and uniqueness of renormalized solutions for initial boundary value problems of the type

$$\begin{aligned} \left( {\mathcal {P}}_{b}^{1}\right) \quad \left\{ \begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))=H(u)\mu \text { in }Q:=(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$

where \(u_{0}\in L^{1}(\Omega )\), \(\mu \in {\mathcal {M}}_{b}(Q)\) is a general Radon measure on Q and \(H\in C_{b}^{0}({\mathbb {R}})\) is a continuous positive bounded function on \({\mathbb {R}}\). The difficulties in the study of such problems concern the possibly very singular right-hand side that forces the choice of a suitable formulation that ensures both existence and uniqueness of solution. Using similar techniques, we will prove existence/nonexistence results of the auxiliary problem

$$\begin{aligned} \left( {\mathcal {P}}_{b}^{2}\right) \quad \left\{ \begin{aligned}&u_{t}-\text {div}(a(t,x,\nabla u))+g(x,u)|\nabla u|^{2}=\mu \text { in }Q:=(0,T)\times \Omega ,\\&u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$

under the assumption that g satisfies a sign condition and the nonlinear term depends on both x, u and its gradient. Thus, our results improve and complete the previous known existence results for problems \(\left( {\mathcal {P}}_{b}^{1,2}\right) \).