Existence and continuity of global attractors and nonhomogeneous equilibria for a class of evolution equations with non local terms

Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

“…Repeating the same argument, starting from inequality u ( τ , u τ )≥−2 k , for almost every x ∈Ω, we obtain and thus for each t > τ , and the claim follows by continuity. □ Remark In the particular case where the function a (in ) is constant, the functional energy that we obtained earlier coincides with the functional Lyapunov found in ; in this sense, our result generalize Theorem 5.4 in .…”

“…For any u. , / in the ball centered at the origin and radius k in L 1 .OE , T /, from (H4), it follows that In the particular case where the function a (in (1.1)) is constant, the functional energy that we obtained earlier coincides with the functional Lyapunov found in [12]; in this sense, our result generalize Theorem 5.4 in [12].…”

“…The long time behavior of the solution of nonlocal diffusion equations like (1.1) for autonomous case has attracted the attention of many researchers in the last years, see for instance, [3][4][5][6][7][8][9][10]21] and [11]. In a recent paper, [12] generalized the results contained in some of these papers, proving the existence and continuity of global attractors and non-homogeneous equilibria for a large class of nonlocal autonomous evolution equations. On the other hand, as far as we know, there are no results on pullback attractor for the nonlocal nonautonomous diffusion equations.…”

“…In Section 3, under hypotheses ( H 1 ) and ( H 4 ), we show the existence of global pullback attractor for . In Section 4, motivated by functionals from , , and , we exhibit a functional to evolution process generated by this problem that decreases along of solutions. And finally, in Section 5, we will show that the origin is locally pullback asymptotically stable.…”

Lower Semicontinuity of Global Attractors for a Class of Evolution Equations of Neural Fields Type in a Bounded Domain

“…In "Lower Semicontinuity of the Attractors" section, using the same techniques of [23], we prove the property of lower semicontinuity of the attractors. To the extent of our knowledge, with the exception of [23], the proofs of this property available in the literature assume that the equilibrium points are all hyperbolic and therefore isolated (see for example [2,5,19,20]). However, this property cannot hold true in our case, due to the symmetries present in the equation.…”