Global existence, asymptotic behavior and self-similar solutions for a class of semilinear parabolic systems

Snoussi, Seifeddine; Tayachi, Slim

doi:10.1016/s0362-546x(00)00170-x

Cited by 26 publications

(15 citation statements)

References 19 publications

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“…Since the functions f, g of interest usually define vector fields with invariant regions, as defined in [34] (think of the FitzHugh-Nagumo equations), global solutions (τ = +∞) are expected from a maximum principle on A. It is interesting to note that global existence results for systems of reaction-diffusion equations, similar to our bidomain model, are only available when the elliptic operators in the system are a Laplacian or more generally a second order elliptic operator reading the form ∇ · (σ ∇·) [35][36][37]33,38,34,39,40]. In these cases, essential properties of elliptic second order differential operators are required to derive comparison theorems or maximum principles.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

Bourgault

Coudière

Pierre

2009

Nonlinear Analysis: Real World Applications

125

163

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We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch models.

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Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

Bourgault

Coudière

Pierre

2009

Nonlinear Analysis: Real World Applications

125

163

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“…See details in [38]. For other studies, see [4,37,42]. It is natural to consider problem (1.1) with general nonlinearity satisfying (1.2)-(1.3).…”

Section: Introductionmentioning

confidence: 99%

“…They obtained that there exists a unique solution ( -regular mild solution) of (1.10), and applied their abstract results to the heat equation and Navier-Stokes equation. For other related studies, we refer readers to [7,8,23,24,37,42].…”

Section: Introductionmentioning

confidence: 99%

Attractors for non-autonomous parabolic problems with singular initial data

Ruan

2011

Journal of Differential Equations

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In this paper, we study the asymptotic behavior of solutions of non-autonomous parabolic problems with singular initial data. We first establish the well-posedness of the equation when the initial data belongs to L r (Ω) (1 < r < ∞) and W 1,r (Ω) (1 < r < N), respectively. When the initial data belongs to L r (Ω), we establish the existence of uniform attractors in L r (Ω) for the family of processes with external forces being translation bounded but not translation compact in L p loc (R; L r (Ω)). When we consider the existence of uniform attractors in H 1 0 (Ω), the solution of equation lacks the higher regularity, so we introduce a new type of solution and prove the existence result. For the long time behavior of solutions of the equation in W 1,r (Ω), we only obtain the uniform attracting property in the weak topology.

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“…The only previous result we know concerning solutions of (1.2) with a = 1 which are asymptotic to self-similar solutions whose profiles decay slowly are due to Cazenave and the third author [3]. Since then, the ideas in these papers have been further developed and applied to several different semilinear problems: to the Navier-Stokes system [8,21], to nonlinear heat equations [3,22,27], to the nonlinear Schrödinger equation [3][4][5]7,18,20,23], to the nonlinear wave equation [19,24] and to semilinear parabolic systems [25,26]. See Theorem 6.2 in [3].…”

Section: Introductionmentioning

confidence: 99%

Asymptotically self-similar global solutions of a general semilinear heat equation

2001

Self Cite

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We consider the general nonlinear heat equation ∂ t u = ∆u+a|u| p 1 −1 u+g(u), u(0) = ϕ, on (0, ∞) × IR n , where a ∈ IR, p 1 > 1 + (2/n) and g satisfies certain growth conditions. We prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation. We also prove that some of those global solutions are asymptotic for large time to self-similar solutions of the single power nonlinear heat equation, i.e. with g ≡ 0.We emphasize that although equation (1.1) itself has no self-similar structure, it does have solutions which are asymptotically self-similar. Also, we are primarilywhere f is called the profile of the solution. The standard approach to studying self-similar solutions starts with the equation for the profile f obtained by substituting the self-similar ansatz into (1.2). Basic tools for studying the asymptotic behavior of solutions include the maximum and comparison principles, variational methods, and dynamical systems.In the case a = −1, where all solutions of (1.2) are global, we refer the reader to Gmira and Veron [11], Kamin and Peletier [13], Escobedo, Kavian and Matano [6], Galaktionov, Kurdyumov, and Samarskii [9] and Herraiz [12], as well as the references cited in these papers. While these papers contain a great wealth of information, we recall here only those results which have direct bearing on the present article. Assume that n(p 1 − 1)/2 > 1. Let ϕ ≥ 0 be such that 0 < lim |x|→∞ |x| α ϕ(x) = A < ∞ for some some α > 0, with

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Global existence, asymptotic behavior and self-similar solutions for a class of semilinear parabolic systems

Cited by 26 publications

References 19 publications

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

Attractors for non-autonomous parabolic problems with singular initial data

Asymptotically self-similar global solutions of a general semilinear heat equation

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