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

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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.

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Correspondence to Mohammed Abdellaoui.

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The authors would like to thank Pr. Alexander Pankov, the reviewers for their thoughtful comments towards improving our manuscript. On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Abdellaoui, M., Azroul, E. Homogenization of a nonlinear parabolic problem corresponding to a Leray–Lions monotone operator with right-hand side measure. SeMA 77, 1–26 (2020). https://doi.org/10.1007/s40324-019-00197-8

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Keywords

  • Nonlinear parabolic problems
  • Homogenization (G-convergence)
  • Measure data

Mathematics Subject Classification

  • 35R06
  • 32U20
  • 80M40