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{document}$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$ \end{document} 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 \begin{document}$(L^2_0(Ω), L^2_0(Ω))$\end{document} -absorbing sets and \begin{document}$(L^2_0(Ω), H^{σ/2}_0(Ω))$\end{document} -absorbing sets for the solution semigroup \begin{document}$\{S(t)\}_{t≥q 0}$\end{document} . Finally, we prove the existence of \begin{document}$(L^2_0(Ω), L^2_0(Ω))$\end{document} -global attractor and \begin{document}$(L^2_0(Ω), H^{σ/2}_0(Ω))$\end{document} -global attractor by a asymptotic compactness method.
Highlights
We consider the existence of global attractors for the following nonlocal nonlinear reaction-diffusion equations:
The main purpose of the present paper is to study the long-time dynamical behavior of solutions for the fractional reaction-diffusion equations (1)
We examine the existence of absorbing sets in L20(Ω) and H0σ/2(Ω) for the semigroup {S(t)}t≥0 corresponding to the fractional reactiondiffusion equations (1) and the existence of a global attractor in L20(Ω)
Summary
We consider the existence of global attractors for the following nonlocal nonlinear reaction-diffusion equations: ut. Fractional Laplacian, long-time behavior, weak solution, nonlocal partial differential equation, bounded domain. Utilizing the norm-to-weak continuous semigroup method in [32], we prove the existence of a global attractor in H0σ/2(Ω)). Combining Theorem 1.1 with Lemma 2.6 ( section), we can define a solution semigroup {S(t)}t≥0 in L20(Ω) as. (See [32]) Let X, Y be two Banach spaces satisfy the assumptions just above, {S(t)}t≥0 be a semigroup on X and Y , respectively, and assume that {S(t)}t≥0 is continuous or weak continuous on Y. (See [32]) Let {S(t)}t≥0 is a norm-to-weak continuous semigroup on (Y, X).
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