In this work we consider the nonlocal evolution equation $$\displaystyle \frac{\partial u(w,t)}{\partial t}=-u(w,t)+ \int _{S^{1}}J(wz^{-1})f(u(z,t))\mathrm{d}z+ h, \quad h > 0,$$∂u(w,t)∂t=-u(w,t)+∫S1J(wz-1)f(u(z,t))dz+h,h>0,which arises in models of neuronal activity, in $$L^{2}(S^{1})$$L2(S1), where $$S^{1}$$S1… Click to show full abstract
In this work we consider the nonlocal evolution equation $$\displaystyle \frac{\partial u(w,t)}{\partial t}=-u(w,t)+ \int _{S^{1}}J(wz^{-1})f(u(z,t))\mathrm{d}z+ h, \quad h > 0,$$∂u(w,t)∂t=-u(w,t)+∫S1J(wz-1)f(u(z,t))dz+h,h>0,which arises in models of neuronal activity, in $$L^{2}(S^{1})$$L2(S1), where $$S^{1}$$S1 denotes the unit sphere. We obtain more interesting results on existence of global attractors and the associate Lypaunov functional than the already existing in the literature. Furthermore, we prove the result, not yet known in the literature, of lower semicontinuity of global attractors with respect to connectivity function J.
Journal Title: Differential Equations and Dynamical Systems
Year Published: 2018
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