On the Cauchy Problem for Reaction-Diffusion Equations with White Noise

1988 ◽  
Vol 136 (1) ◽  
pp. 209-228 ◽  
Author(s):  
Ralf Manthey
Keyword(s):  
Cauchy Problem ◽  
White Noise ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
The Cauchy Problem

scholarly journals Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in R2

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Nai-Wei Liu
Keyword(s):  
Cauchy Problem ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
The Cauchy Problem ◽  
Subsolutions And Supersolutions

We consider the interaction of traveling curved fronts in bistable reaction-diffusion equations in two-dimensional spaces. We first characterize the growth of the traveling curved fronts at infinity; then by constructing appropriate subsolutions and supersolutions, we prove that the solution of the Cauchy problem converges to a pair of diverging traveling curved fronts in R2 under appropriate initial conditions.

Stability of travelling waves for degenerate reaction-diffusion equations of KPP-type

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Zsolt Biró
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behaviour ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
The Cauchy Problem ◽  
Nonlinear Degenerate

AbstractThe aim of this paper is to investigate the asymptotic behaviour as t → ∞ of the solutions to the Cauchy problem for the nonlinear degenerate KPP-type diffusion-reaction equation u

scholarly journals Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains

2018 ◽  
Vol 16 (1) ◽  
pp. 862-884
Author(s):  
Xiaoyao Jia ◽  
Xiaoquan Ding ◽  
Juanjuan Gao
Keyword(s):  
Dynamical Systems ◽  
White Noise ◽  
Reaction Diffusion ◽  
Random Parameter ◽  
Reaction Diffusion Equations ◽  
Random Dynamical Systems ◽  
Diffusion Equations ◽  
Uhlenbeck Process ◽  
Random Attractors ◽  
Multiplicative White Noise

AbstractIn this paper we investigate the stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn (n ≥ 2). We first transform the retarded reaction-diffusion equations into the deterministic reaction-diffusion equations with random parameter by Ornstein-Uhlenbeck process. Next, we show the original equations generate the random dynamical systems, and prove the existence of random attractors by conjugation relation between two random dynamical systems. In this process, we use the cut-off technique to obtain the pullback asymptotic compactness.

scholarly journals On Lr-regularity of global attractors generated by strong solutions of reaction-diffusion equations

2016 ◽  
Vol 1 (2) ◽  
pp. 375-390 ◽  
Author(s):  
José Valero
Keyword(s):  
Nonlinear Term ◽  
Mild Solutions ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Global Attractors ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
The Cauchy Problem ◽  
Weak Attractor

AbstractIn this paper we prove that the global attractor generated by strong solutions of a reaction-diffusion equation without uniqueness of the Cauchy problem is bounded in suitable Lr-spaces. In order to obtain this result we prove first that the concepts of weak and mild solutions are equivalent under an appropriate assumption.Also, when the nonlinear term of the equation satisfies a supercritical growth condition the existence of a weak attractor is established.

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