Existence and Uniqueness of Renormalized Solutions to Nonlinear Parabolic Equations with Lower Order Term and Diffuse Measure Data

Bouajaja, A.; Redwane, Hicham; Marah, Amine

doi:10.1007/s00009-018-1223-8

Cited by 6 publications

(1 citation statement)

References 15 publications

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“…The case of possibly diffuse measure as data in (5.1) is, by far, are limited. If µ s = 0 and b(u) = b(x, u) we refer to [MR1] (see also [BRM,MR2,AA3]) for a complete account on generalized porous medium equations in the context of renormalized solutions. Moreover the case of unbounded nonlinearities has been treated in [BGR,BR2] and parabolic equations with natural growth terms have been obtained in [AR] (see also [BP1] and references therein for primary results).…”

Section: Equation With Absorptionmentioning

confidence: 99%

On some parabolic problems with measure sources

Abdellaoui

2019

Moroccan Journal of Pure and Applied Analysis

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One of the recent advances in the investigation of nonlinear parabolic equations with a measure as forcing term is a paper by F. Petitta in which it has been introduced the notion of renormalized solutions to the initial parabolic problem in divergence form. Here we continue the study of the stability of renormalized solutions to nonlinear parabolic equations with measures but from a different point of view: we investigate the existence and uniqueness of the following nonlinear initial boundary value problems with absorption term and a possibly sign-changing measure datawhere Ω is an open bounded subset of R N , N ≥ 2, T > 0 and Q is the cylinder (0, T) × Ω, Σ = (0, T) × ∂Ω being its lateral surface, the operator is modeled on the p−Laplacian with p > 2 − 1 N+1 , µ is a Radon measure with bounded total variation on Q, b is a C 1 −increasing function which satisfies 0 < b 0 ≤ b (s) ≤ b 1 (for positive constants b 0 and b 1 ). We assume that b(u 0 ) is an element of L 1 (Ω) and h : R → R is a continuous function such that h(s)s ≥ 0 for every |s| ≥ L and L ≥ 0 (odd functions for example). The existence of a renormalized solution is obtained by approximation as a consequence of a stability result. We provide a new proof of this stability result, based on the properties of the truncations of renormalized solutions. The approach, which does not need the strong convergence of the truncations of the solutions in the energy space, turns out to be easier and shorter than the original one.

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Section: Equation With Absorptionmentioning

confidence: 99%