Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition

Lu Yang; Meihua Yang

doi:10.3934/dcdsb.2017102

Article Contents

2017, Volume 22, Issue 7: 2627-2650. Doi: 10.3934/dcdsb.2017102

This issue Previous Article Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model Next Article Extinction in stochastic predator-prey population model with Allee effect on prey

Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition

Lu Yang^1, and
Meihua Yang^2, ,

1.
School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
2.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author
* Corresponding author

Received: June 30, 2016

Revised: August 31, 2016

Published: October 2017

The first author was supported by the NSFC grants(11361053,11471148), the Fundamental Research Funds for the Central Universities Grant (lzujbky-2016-98) and the State Scholarship Fund (201506185006) of China Scholarship Council. The second author was supported by the NSFC grant(11571125) and the NCET-12-0204.

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Abstract

In this paper, we study the dynamic behavior of a stochastic reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms $f$ and $h$ satisfy the polynomial growth condition of arbitrary order. Some higher-order integrability of the difference of the solutions near the initial time, and the continuous dependence result with respect to initial data in $H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$ were established. As a direct application, we can obtain the existence of pullback random attractor $A$ in the spaces $L^{p}(\mathcal{O})× L^{p}(Γ)$ and $H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$ immediately.

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Mathematics Subject Classification: Primary:37L05, 35B40;Secondary:35B41.

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