Long-time behavior of a class of nonlocal partial differential equations

Chang Zhang; Fang Li; Jinqiao Duan

doi:10.3934/dcdsb.2018041

Article Contents

2018, Volume 23, Issue 2: 749-763. Doi: 10.3934/dcdsb.2018041

This issue Previous Article Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction Next Article Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion

Long-time behavior of a class of nonlocal partial differential equations

1.
School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China
2.
School of Mathematics and Statistics, Xidian University, Xi'an 710126, China
3.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

* Corresponding author
* Corresponding author

Received: September 30, 2016

Revised: September 30, 2017

Early access: November 2017

Published: June 2018

Zhang was supported by NSFC Grant (11701230)

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Abstract

This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain

$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$

Firstly, due to the lack of an upper growth restriction of the nonlinearity $f$, we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of $(L^2_0(Ω), L^2_0(Ω))$-absorbing sets and $(L^2_0(Ω), H^{σ/2}_0(Ω))$-absorbing sets for the solution semigroup $\{S(t)\}_{t≥q 0}$. Finally, we prove the existence of $(L^2_0(Ω), L^2_0(Ω))$-global attractor and $(L^2_0(Ω), H^{σ/2}_0(Ω))$-global attractor by a asymptotic compactness method.

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Mathematics Subject Classification: Primary: 35R11, 35D30, 35K61; Secondary: 35A01, 35B41.

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