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

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.