Metastable travelling-wave solutions of singularly-perturbed reaction-diffusion equations

This paper determines the asymptotic solution of certain initial-boundary value problems for singularly-perturbed reaction-diffusion equations, including the Allen–Cahn and Cahn–Hilliard equations, on bounded one-dimensional spatial domains for r[ges ]0. Attention is focused on the metastable evolution of a transition layer over an asymptotically exponentially-long time interval.

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The asymptotic behavior of solution for the singularly perturbed initial boundary value problems of the reaction diffusion equations in a part of domain

Applied Mathematics and Mechanics ◽

10.1007/bf02436455 ◽

2001 ◽

Vol 22 (10) ◽

pp. 1192-1197

Author(s):

Liu Qi-lin ◽

Mo Jia-qi

Keyword(s):

Asymptotic Behavior ◽

Boundary Value Problems ◽

Reaction Diffusion ◽

Initial Boundary ◽

Reaction Diffusion Equations ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Initial Boundary Value Problems ◽

Asymptotic Behavior Of Solution ◽

Initial Boundary Value

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A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation

Applied Mathematics-A Journal of Chinese Universities ◽

10.1007/s11766-007-0208-3 ◽

2007 ◽

Vol 22 (2) ◽

pp. 201-206 ◽

Cited By ~ 1

Author(s):

Jiaqi Mo

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Reaction Diffusion ◽

Initial Boundary ◽

Reaction Diffusion Equations ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Boundary Perturbation ◽

Initial Boundary Value Problems ◽

Initial Boundary Value

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The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. I. Continuous diffusivity

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1995.0019 ◽

1995 ◽

Vol 350 (1694) ◽

pp. 335-360 ◽

Cited By ~ 3

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Propagation Speed ◽

Reaction Diffusion ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Travelling Wave ◽

Reaction Diffusion Equations ◽

Classical Solutions ◽

Initial Boundary Value

We examine the effects of a concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics. The diffusivity is taken as a continuous monotone, a decreasing function of concentration that has compact support, of the form that arises in polymerization processes. We consider piecewise-classical solutions to an initial-boundary value problem. The existence of a family of permanent form travelling wave solutions is established, and the development of the solution of the initial-boundary value problem to the travelling wave of minimum propagation speed is considered. It is shown that an interface will always form in finite time, with its initial propagation speed being unbounded. The interface represents the surface of the expanding polymer matrix.

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The effects of variable diffusivity on the development of travelling waves in a class of reaction-diffusion equations

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1994.0090 ◽

1994 ◽

Vol 348 (1687) ◽

pp. 229-260 ◽

Cited By ~ 4

Keyword(s):

Boundary Value Problem ◽

Travelling Waves ◽

Boundary Value ◽

Reaction Diffusion ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Travelling Wave ◽

Reaction Diffusion Equations ◽

Classical Solutions ◽

Initial Boundary Value

In this paper we examine the effects of concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics and ecology. We consider piecewise classical solutions to an initial boundary-value problem. The existence of a family of permanent form travelling wave solutions is established and the development of the solution of the initial boundary-value problem to the travelling wave of minimum propagation speed is considered. For certain types of initial data, ‘waiting time’ phenomena are encountered.

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NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL

The ANZIAM Journal ◽

10.1017/s1446181113000175 ◽

2013 ◽

Vol 54 (3) ◽

pp. 153-170 ◽

Cited By ~ 3

Author(s):

RUNZHANG XU ◽

YANBING YANG ◽

SHAOHUA CHEN ◽

JIA SU ◽

JIHONG SHEN ◽

...

Keyword(s):

Boundary Value Problem ◽

Wave Equations ◽

Boundary Value ◽

Reaction Diffusion ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Nonlinear Wave Equations ◽

Initial Boundary Value

AbstractThis paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

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