Homogenization of a nonlinear parabolic problem corresponding to a Leray–Lions monotone operator with right-hand side measure

Abstract

In this paper we deal with asymptotic behaviour of renormalized solutions $$u_{n}$$ to the nonlinear parabolic problems whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} (u_{n})_{t}-\text {div}(a_{n}(t,x,\nabla u_{n}))=\mu _{n}&{}\text { in }Q=(0,T)\times \Omega ,\\ u_{n}(t,x)=0&{}\text { on }(0,T)\times \partial \Omega ,\\ u_{n}(0,x)=u_{0}^{n}&{}\text { in }\Omega , \end{array}\right. } \end{aligned}$$where $$\Omega $$ is a bounded open set of $$\mathbb {R}^{N}$$, $$N\ge 1$$, $$T>0$$ and $$u_{0}^{n}\in C^{\infty }_{0}(\Omega )$$ that approaches $$u_{0}$$ in $$L^{1}(\Omega )$$. Moreover $$(\mu _{n})_{n\in \mathbb {N}}$$ is a sequence of Radon measures with bounded variation in Q which converges to $$\mu $$ in the narrow topology of measures. The main result states that, under the assumption of G-convergence of the operators $$A_{n}(v)=-\text {div}(a_{n}(t,x,\nabla v_{n}))$$, defined for $$v_{n}\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))$$ for $$p>1$$, to the operator $$A_{0}(v)=-\text {div}(a_{0}(t,x,\nabla v))$$ and up to subsequences, $$(u_{n})$$ converges a.e. in Q to the renormalized solution u of the problem $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\text {div}(a_{0}(t,x,\nabla u))=\mu &{}\text { in }Q=(0,T)\times \Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega ,\\ u(0,x)=u_{0}&{}\text { in }\Omega . \end{array}\right. } \end{aligned}$$The proposed renormalized formulation differs from the usual one by the fact that truncated function $$T_{k}(u_{n})$$ (which depend on the solutions) are used in place of the solutions $$u_{n}$$. We prove existence of such a limit-solution and we discuss its main properties in connection with G-convergence, we finally show the relationship between the new approach and the previous ones and we extend this result using capacitary estimates and auxiliary test functions.