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Long-time behavior of a class of nonlocal partial differential equations. (English) Zbl 1395.35199
Summary: 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 $u_t+(-\Delta)^{\sigma/2}u+f(u)=g.$ 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 $$(L_0^2(\Omega),L_0^2(\Omega))$$-absorbing sets and $$(L_0^2(\Omega),H_0^{\sigma/2}(\Omega))$$-absorbing sets for the solution semigroup $$\{S(t)\}_{t\geq q0}$$. Finally, we prove the existence of $$(L_0^2(\Omega),L_0^2(\Omega))$$-global attractor and $$(L_0^2(\Omega), H_0^{\sigma/2}(\Omega))$$-global attractor by a asymptotic compactness method.
##### MSC:
 35R11 Fractional partial differential equations 35D30 Weak solutions to PDEs 35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B41 Attractors
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