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

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

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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) $.

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


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