Existence and Uniqueness of a Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems

Blanchard, Dominique; Murat, François; Redwane, Hicham

doi:10.1006/jdeq.2000.4013

Cited by 146 publications

(128 citation statements)

References 14 publications

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“…To handle this difficulty we apply a time-regularization procedure, introduced first by Landes [17] and thereafter employed by many authors to solve nonlinear parabolic equations with L 1 or measure data (see [11,4,23,3]). …”

Section: )) We Argue Again As In the Proof Of Lemma 52 To Obtain H(mentioning

confidence: 99%

Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue

Bendahmane¹,

Karlsen²

2006

Networks &Amp; Heterogeneous Media

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Abstract. We prove well-posedness (existence and uniqueness) results for a class of degenerate reaction-diffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the Faedo-Galerkin method, and the compactness method.1. Introduction. Our point of departure is a widely accepted model, the so-called bidomain model, for describing the cardiac electric activity in a physical domain Ω ⊂ R 3 (the cardiac muscle) over a time span (0, T ), T > 0. In this model the cardiac muscle is viewed as two superimposed (anisotropic) continuous media, referred to as the intracellular (i) and extracellular (e), which occupy the same volume and are separated from each other by the cell membrane.To state the model, we let u i = u i (t, x) and u e = u e (t, x) represent the spatial cellular at time t ∈ (0, T ) and location x ∈ Ω of the intracellular and extracellular electric potentials, respectively. The difference v = v(t, x) = u i − u e is known as the transmembrane potential. The anisotropic properties of the two media are modeled by conductivity tensors M i (t, x) and M e (t, x). The surface capacitance of the membrane is represented by a constant c m > 0. The transmembrane ionic current is represented by a nonlinear (cubic polynomial) function h (t, x, v) depending on time t, location x, and the value of the potential v. The stimulation currents applied to the intra-and extracellular space are represented by a function I app = I app (t, x).A prototype system that governs the cardiac electric activity is the following degenerate reaction-diffusion system (known as the bidomain equations)

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Section: )) We Argue Again As In the Proof Of Lemma 52 To Obtain H(mentioning

confidence: 99%

Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue

Bendahmane¹,

Karlsen²

2006

Networks &Amp; Heterogeneous Media

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“…Later, they have also been considered in other situations; let us mention in particular the contributions of Blanchard, Boccardo, Murat and their co-workers in the framework of second-order elliptic and parabolic PDEs; see [6,7,4,5] and the references therein; see also [19] and [8] for more related results.…”

Section: D(u)mentioning

confidence: 99%

Weak-renormalized solutions for a system that models non-isothermal solidification

Fernández-Cara

Vaz

2012

SeMA

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The aim of this paper is to prove the existence of weak-renormalized solutions to a system of the Navier-Stokes-Boussinesq kind. This system may be regarded as a modified version of the non-isothermal solidification problem with melt convection. This task will be accomplished satisfactorily in the two-dimensional case. Some nontrivial and deep difficulties will be found, however, in three dimensions in space.

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“…Among all of them we refer to [19] where the author proved the existence of an entropy solution for all data in (f, u 0 ) ∈ L 1 (Ω T ) × L 1 (Ω). The case of general measure data was studied in [6], [7], where the existence of renormalized solution is obtained.…”

Section: Introductionmentioning

confidence: 99%

On fractional p-Laplacian parabolic problem with general data

Abdellaoui

Attar

Bentifour

et al. 2017

Annali di Matematica

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Abstract. In this article the problem to be studied is the followingwhere Ω is a bounded domain, and (−∆ s p ) is the fractional p-Laplacian operator defined by|x − y| N+ps dy with 1 < p < N , s ∈ (0, 1) and f, u 0 are measurable functions. The main goal of this work is to prove, problem (P ) has a weak solution with suitable regularity. In addition, if f 0 , u 0 are nonnegative, we show that the problem above has a nonnegative entropy solution.In the case of nonnegative data, we give also some quantitative and qualitative properties of the solution according the values of p.

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Existence and Uniqueness of a Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems

Cited by 146 publications

References 14 publications

Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue

Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue

Weak-renormalized solutions for a system that models non-isothermal solidification

On fractional p-Laplacian parabolic problem with general data

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