On classes of well-posedness for quasilinear diffusion equations in the whole space

Boris Andreianov; Mohamed Maliki

doi:10.3934/dcdss.2020361

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2021, Volume 14, Issue 2: 505-531. Doi: 10.3934/dcdss.2020361

This issue Previous Article A generalization of a criterion for the existence of solutions to semilinear elliptic equations Next Article The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model

On classes of well-posedness for quasilinear diffusion equations in the whole space

Boris Andreianov^1,2, , and
Mohamed Maliki^3,

1.
Institut Denis Poisson CNRS UMR7013, Université de Tours, Université d'Orléans, Parc Grandmont, 37200 Tours, France
2.
Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
3.
Équipe Modélisation, EDP et Analyse Numérique, FST Mohammédia, B.P. 146, Mohammédia, Morocco

^* Corresponding author
^* Corresponding author

Dedicated to Michel Pierre on the occasion of his 70th birthday

Received: September 30, 2019

Early access: May 2020

Published: February 2021

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Well-posedness classes for degenerate elliptic problems in $ {\mathbb R}^N $ under the form $ u = \Delta {{\varphi}}(x,u)+f(x) $, with locally (in $ u $) uniformly continuous nonlinearities, are explored. While we are particularly interested in the $ L^\infty $ setting, we also investigate about solutions in $ L^1_{loc} $ and in weighted $ L^1 $ spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of $ u\mapsto {{\varphi}}(x,u) $. Under additional restrictions on the dependency of $ {{\varphi}} $ on $ x $, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity $ {{\varphi}}: {\mathbb R}\mapsto {\mathbb R} $, the space $ L^\infty $ (endowed with the $ L^1_{loc} $ topology) is a well-posedness class for the problem $ u = \Delta {{\varphi}}(u)+f(x) $.

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Mathematics Subject Classification: Primary: 35J62, 35A02, 37L05; Secondary: 35J70, 35D30.

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