On the Blow-up of Solutions to Some Semilinear and Quasilinear Reaction-Diffusion Systems

Hollis, Selwyn; Morgan, Jeff

doi:10.1216/rmjm/1181072348

Cited by 14 publications

(19 citation statements)

References 8 publications

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“…A further related result by Hollis and Morgan [20] showed that if blow-up (here that is a concentration phenomena since the total mass is conserved) occurs in one concentration a i (t, x) at some time t and position x, then at least one more concentration has to blow-up (i.e. concentrate) at the same time and position.…”

Section: Reaction-diffusion Systems For Reversible Chemistrymentioning

confidence: 94%

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Exponential Convergence to Equilibrium for Nonlinear Reaction-Diffusion Systems Arising in Reversible Chemistry

Desvillettes

Fellner

2014

IFIP Advances in Information and Communication Technology

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Abstract. We consider a prototypical nonlinear reaction-diffusion system arising in reversible chemistry. Based on recent existence results of global weak and classical solutions derived from entropy-decay related apriori estimates and duality methods, we prove exponential convergence of these solutions towards equilibrium with explicit rates in all space dimensions. The key step of the proof establishes an entropy entropy-dissipation estimate, which relies only on natural a-priori estimates provided by massconservation laws and the decay of an entropy functional.

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Section: Reaction-diffusion Systems For Reversible Chemistrymentioning

confidence: 94%

“…[20,21,12] and the references therein), the parabolic problem (11) satisfies for all T > 0 and Ω T = (0, T ) × Ω and for all space dimensions N ≥ 1 the following a-priori estimate…”

Section: Entropy Structure and Duality Methodsmentioning

confidence: 99%

Exponential Convergence to Equilibrium for Nonlinear Reaction-Diffusion Systems Arising in Reversible Chemistry

Desvillettes

Fellner

2014

IFIP Advances in Information and Communication Technology

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“…in the subregions Q 1 where c > 0 and Q 2 where c < 0. Indeed, on Q 1 , the first equation is "good", and u is obviously locally bounded: then, so is v by application of a local version of the proof of Theorem 3.1 (see [64,58,32,33]). The same remark is valid for Q 2 if we exchange the roles of u and v. Therefore, blow up may occur only around the region where the sign of c(t, x) changes!…”

Section: Open Problems a Main Extra Assumption In Theorem 31 Besidementioning

confidence: 99%

“…The quadratic case is also handled for some systems in dimension N ≤ 2 (see [26,20,33]). Let us finally mention the case of coupled or cross-diffusions where a few techniques have been developed to prove global existence of classical solutions (see [15,16,40]).…”

Section: More Remarks On Global Classical Solutions Use Of Lyapunov Fmentioning

confidence: 99%

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Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

2010

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International audienceThe goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or - in mathematical terms-on the L(1)-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well

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Duality and entropy methods for reversible reaction‐diffusion equations with degenerate diffusion

Desvillettes

Fellner

2015

Math Methods in App Sciences

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A combination of entropy and duality methods has been successfully used in the past to study the existence and smoothness of reaction-diffusion equations arising in reversible chemistry in many situations, but in general under the requirement that the diffusion is non-degenerate (or at least that all diffusion coefficients are strictly positive a.e.). Here, we wish to describe how these methods also enable to treat reaction-diffusion systems which are strongly degenerate in the sense that one or more of the species do not diffuse at all.

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On the Blow-up of Solutions to Some Semilinear and Quasilinear Reaction-Diffusion Systems

Cited by 14 publications

References 8 publications

Exponential Convergence to Equilibrium for Nonlinear Reaction-Diffusion Systems Arising in Reversible Chemistry

Exponential Convergence to Equilibrium for Nonlinear Reaction-Diffusion Systems Arising in Reversible Chemistry

Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

Duality and entropy methods for reversible reaction‐diffusion equations with degenerate diffusion

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