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Dissipative property for a class of non local evolution equations (1705.09702v1)

Published 26 May 2017 in math.DS

Abstract: In this work we consider the non local evolution problem [ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h), ~x \in\Omega, ~t\in[0,\infty[;\ u(x,t)=0, ~x\in\mathbb{R}N\setminus\Omega, ~t\in[0,\infty[;\ u(x,0)=u_0(x),~x\in\mathbb{R}N, \end{cases} ] where $\Omega$ is a smooth bounded domain in $\mathbb{R}N, ~g,f: \mathbb{R}\to\mathbb{R}$ satisfying certain growing condition and $K$ is an integral operator with symmetric kernel, $ Kv(x)=\int_{\mathbb{R}{N}}J(x,y)v(y)dy.$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has a gradient property.

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