Renormalized solutions of nonlinear parabolic equations with diffuse measure data

Blanchard, Dominique; Petitta, Francesco; Redwane, Hicham

doi:10.1007/s00229-012-0585-7

Cited by 25 publications

(22 citation statements)

References 27 publications

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“…Besides, they have potential applications in various fields such as continuous mechanics [L2], population dynamics [BM] and image processing [GSZW], etc. Perhaps for these reasons, such a field has attracted more and more attention and has a development in recent years, see [BPR,AA2,AA3] and the large amounts of references therein. Note that the interest of problems as (5.1) (with b = 1 and smooth data) is studied in [BCW] in connection with the study of conservation laws and in the theory of non-Newtonian fluids.…”

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%

On some parabolic problems with measure sources

Abdellaoui

2019

Moroccan Journal of Pure and Applied Analysis

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“…In the parabolic setting, renormalized solutions were studied first in [4,5,8,9,10] and further [6,21,22,50,51]. These studies are continued under weaker assumptions on the data [7,13,17]. Lately, generalising the setting, renormalized solutions to parabolic problems have been considered in the variable exponent setting [2,41,58] and in the model of thermoviscoelasticity [15].…”

Section: State Of Artmentioning

confidence: 99%

Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak–Orlicz spaces in the class of renormalized solutions

Chlebicka

Gwiazda

Zatorska–Goldstein

2018

Journal of Differential Equations

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We prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consideron a Lipschitz bounded domain in R N . The growth of the weakly monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N -function M . The approach does not require any particular type of growth condition of M or its conjugate M * (neither ∆ 2 , nor ∇ 2 ). The condition we impose on M is continuity of log-Hölder-type, which results in good approximation properties of the space. However, the requirement of regularity can be skipped in the case of reflexive spaces. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments. Uniqueness results from the comparison principle.

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“…The equivalence with the notion of entropy solutions is shown in [16]. For such a soft measure, an extension to equations of type (b(u)) t −∆ p u = µ is given in [6]; another formulation is used in [23] for solving a perturbed problem from (1.1) by an absorption term.…”

Section: Introductionmentioning

confidence: 99%