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Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data

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Abstract

An existence result is established for a class of quasilinear parabolic problem which is a diffusion type equations having continuous coefficients blowing up for a finite value of the unknown, a second hand \(\mu \in \mathcal {M}_{b}(Q)\) and an initial data \(u_{0}\in L^{1}(\Omega )\). We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some compactness results are also shown under additional hypotheses.

Résumé

Un résultat d’existenece est établi pour une classe de problèmes paraboliques quasilinéaires sous forme d’équations de type diffusion à coefficients continues qui explosent pour une valeur finie du variable, la deuxième partie \(\mu \in \mathcal {M}_{b}(Q)\) et la donnée initiale \(u_{0}\in L^{1}(\Omega )\). Nous developpons une technique basée sur la notion de solution renormalisée et une régularisation adéquate en temps pour certaines fonctions troncatures. Quelques résultas de compacité sont aussi démontrés sous des hypthèses supplémentaires.

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  1. Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University of Fez, B.P. 1796, Atlas Fez, Morocco

    Mohammed Abdellaoui & Elhoussine Azroul

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  1. Mohammed Abdellaoui
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  2. Elhoussine Azroul
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Correspondence to Mohammed Abdellaoui.

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Abdellaoui, M., Azroul, E. Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data. Ricerche mat 68, 745–767 (2019). https://doi.org/10.1007/s11587-019-00436-0

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  • DOI: https://doi.org/10.1007/s11587-019-00436-0

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