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Non-stability result of entropy solutions for nonlinear parabolic problems with singular measures

  • Mohammed AbdellaouiEmail author
  • Elhoussine Azroul
Article
  • 13 Downloads

Abstract

In this paper, we study the nonlinear parabolic equation given by
$$\begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))+|u|^{q-1}u=f+\lambda ,\quad \text { in }(0,T)\times \Omega , \end{aligned}$$
where \(1<p<N\), \(q>1\), \(f\in L^{1}(Q)\), \(\lambda \) is a measure concentrated on a set of zero parabolic r-capacity and \( u\mapsto -\text {div}(a(t,x,\nabla u))\) is a pseudo-monotone operator. We also consider the corresponding bilateral obstacle problem with measure data concentrated on a set of zero parabolic p-capacity whose model is
$$\begin{aligned} \langle u_{t}-\text {div}(a(t,x,\nabla u))-\lambda , v-u\rangle \ge 0, \end{aligned}$$
with \(u\in K=\lbrace w\in L^{p}(0,T;W^{1,p}_{0}(\Omega )): |w|\le 1\rbrace \) for every \(v\in K\). We define a notion of entropy solutions, we give convergence properties essential to our proofs and we establish a non-stability result.

Keywords

Entropy solutions Non-stability Parabolic inequalities p-capacity Singular measures 

Mathematics subject classification

35K86 37K45 32U20 32D20 

Notes

Funding

Funding information is not applicable / No funding was received.

Compliance with ethical standards

Conflict of Interest

The authors would like to thank Pr. Francesco Petitta, Università di Roma, and anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. On behalf of all authors, the corresponding author states that there is no conflict of interest.

Informed consent

Informed consent was obtained from all individual participants included in the study.

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Copyright information

© Orthogonal Publisher and Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El MahrazSidi Mohamed Ben Abdellah University of FezAtlas FezMorocco

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