Electron. J. Differential Equations, Vol. 2018 (2018), No. 132, pp. 1-21.

Renormalized solutions for nonlinear parabolic equations with general measure data

Mohammed Abdellaoui, Elhoussine Azroul

Abstract:
We prove the existence of parabolic initial boundary value problems of the type
$$\displaylines{ u_t-\text{div}(a_{\epsilon}(t,x,u_{\epsilon},\nabla u_{\epsilon})) =\mu_{\epsilon}\quad\text{in }Q:=(0,T)\times\Omega,\cr u_{\epsilon}=0\quad\text{on }(0,T)\times\partial \Omega,\quad u_{\epsilon}(0)=u_{0,\epsilon}\quad\text{in }\Omega, }$$
with respect to suitable convergence of the nonlinear operators $a_{\epsilon}$ and of the measure data $\mu_{\epsilon}$. As a consequence, we obtain the existence of a renormalized solution for a general class of nonlinear parabolic equations with right-hand side measure.

Submitted July 15, 2017. Published June 27, 2018.
Math Subject Classifications: 35R06, 41A30, 35B45, 37K45, 32U20.
Key Words: Nonlinear parabolic problems; p-capacity; renormalized solution; stability; general measure.

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Mohammed Abdellaoui
University of Fez, Faculty of Sciences Dhar El Mahraz
Laboratory LAMA, Department of Mathematics, B.P. 1796
Atlas Fez, Morocco
email: mohammed.abdellaoui3@usmba.ac.ma
Elhoussine Azroul
University of Fez, Faculty of Sciences Dhar El Mahraz
Laboratory LAMA, Department of Mathematics, B.P. 1796
Atlas Fez, Morocco
email: elhoussine.azroul@usmba.ac.ma

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