Quenching for a parabolic system with general singular terms

Pei, Haijie; Li, Zhongping

doi:10.22436/jnsa.009.08.14

Cited by 5 publications

(3 citation statements)

References 15 publications

(18 reference statements)

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“…The specific equations and parameters would depend on the particular system and the underlying mechanisms involved. We find many details, real models, and techniques used to study such problems in Barka et al [2], Berestycki et al [3], de Bonis [7], Ji et al [10], Kawarada [12], Mesbahi [15,16], Murray [17,18], Pei and Li [19], Salin [22], Wang [24], Zhou et al [26], Zhu et al [27], as well as in the sources mentioned there.…”

Section: Introductionmentioning

confidence: 99%

Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences

Mesbahi,

Djemai,

Saffidine

2024

ETJ

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This research paper revolves around investigating the phenomenon of quenching in reaction-diffusion systems and highlighting its significance. The primary focus is on analyzing a specific type of parabolic singular reaction-diffusion model that incorporates positive Dirichlet boundary conditions. The objective is to establish the sufficiency of certain conditions for quenching to occur within a finite time frame and to demonstrate the global existence of solutions. The novelty of this paper lies in the simplicity of the conditions imposed on the nonlinearity. This simplicity allows us to choose it from a wide range of possibilities, thus facilitating the application of the model to numerous singular reaction-diffusion phenomena. To bolster our findings, we will present various real-world applications in the fields of bioengineering and life sciences, showcasing the practical relevance of quenching phenomena. Finally, the paper ends with a conclusion and some potential future perspectives for further research in this area.

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Section: Introductionmentioning

confidence: 99%

Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences

Mesbahi,

Djemai,

Saffidine

2024

ETJ

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“…Nonlinear parabolic systems like (1.1)-(1.4) come from chemical reactions, heat transfer, etc, where u and v represent the temperatures of two different materials during heat propagation. The quenching phenomenon of parabolic problems has been the issue of intensive study (see for example [3,4,[8][9][10] and the references cited therein), particulary the study of heat equations system with nonlinear boundary conditions has been the subject of investigation of several authors in recent years (see [6,7,14,15,17] and the references cited therein). In [7] the authors study this problem, they prove that the solution (u, v) quenches in finite time T and the quenching occurs only at the boundary x = 0 for 0 < u 0 , v 0 ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

Numerical Quenching for Heat Equations with Coupled Nonlinear Boundary Flux

Edja

Koffi

Koua

et al. 2019

ijaa

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In this paper, we study a numerical approximation of the following problem ut = uxx, vt = vxx, 0 < x < 1, 0 < t < T ; ux(0, t) = u −m (0, t) + v −p (0, t), vx(0, t) = u −q (0, t) + v −n (0, t) and ux(1, t) = vx(1, t) = 0, 0 < t < T, where m, p, q and n are parameters. We prove that the solution of a semidiscrete form of above problem quenches in a finite time only at first node of the mesh. We show that the time derivative of the solution blows up at quenching node. Some conditions under which the non-simultaneous or simultaneous quenching occurs for the solution of the semidiscrete problem are obtained. We establish the convergence of the quenching time. Finally, some numerical results to illustrate our analysis are given.

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“…, and a, b are positive constants such that a ≤ b. The definition of quenching was different from References [2,3]. In Reference [4], the solution u or v is said to quench if there exists a finite time Γ such that…”

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confidence: 99%

Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations

2020

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This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.

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Quenching for a parabolic system with general singular terms

Cited by 5 publications

References 15 publications

Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences

Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences

Numerical Quenching for Heat Equations with Coupled Nonlinear Boundary Flux

Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations

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