Large Time Behavior and Uniqueness of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

Umeda, Noriaki

doi:10.3836/tjm/1244208595

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…uniformly for x ∈ R N , and (1.2) holds. (See also [1,13,18,21] for related results.) In this paper we obtain the precise description of the large time behavior of the solution and show that the function U = U (x, t) defined by…”

Section: Introductionmentioning

confidence: 97%

Large time behavior of ODE type solutions to nonlinear diffusion equations

Eom¹,

Ishige²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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Consider the Cauchy problem for a nonlinear diffusion equationwhereThen the positive solution to problem (P) behaves like a positive solution to ODE ζ ′ = ζ α in (0, ∞) and it tends to +∞ as t → ∞. In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.

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Section: Introductionmentioning

confidence: 97%

Large time behavior of ODE type solutions to nonlinear diffusion equations

Eom¹,

Ishige²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

Ding

Dong

2022

DCDS-B

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The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), & \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &{\rm in} \ \overline{\Omega}. \end{array} \right. $\end{document} </tex-math></disp-formula>Here <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a spatial bounded region in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{n} \ (n\geq2) $\end{document}</tex-math></inline-formula> and the boundary <inline-formula><tex-math id="M3">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> of the spatial region <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is smooth. We give a sufficient condition to guarantee that the positive solution <inline-formula><tex-math id="M5">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> of the above problem must be a blow-up solution with a finite blow-up time <inline-formula><tex-math id="M6">\begin{document}$ t^* $\end{document}</tex-math></inline-formula>. In addition, an upper bound on <inline-formula><tex-math id="M7">\begin{document}$ t^* $\end{document}</tex-math></inline-formula> and an upper estimate of the blow-up rate on <inline-formula><tex-math id="M8">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> are obtained.

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Large time behavior of ODE type solutions to nonlinear diffusion equations

Eom¹,

Ishige²

2018

Preprint

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Large Time Behavior and Uniqueness of Solutions of a Weakly Coupled System of Reaction-Diffusion Equations

Cited by 3 publications

References 7 publications

Large time behavior of ODE type solutions to nonlinear diffusion equations

Large time behavior of ODE type solutions to nonlinear diffusion equations

Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

Large time behavior of ODE type solutions to nonlinear diffusion equations

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