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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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