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march 2019 Dissipative property for non local evolution equations
Severino H. da Silva, Antonio R. G. Garcia, Bruna E. P. Lucena
Bull. Belg. Math. Soc. Simon Stevin 26(1): 91-117 (march 2019). DOI: 10.36045/bbms/1553047231

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\linebreak 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. Furthermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has the gradient property.

Citation

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Severino H. da Silva. Antonio R. G. Garcia. Bruna E. P. Lucena. "Dissipative property for non local evolution equations." Bull. Belg. Math. Soc. Simon Stevin 26 (1) 91 - 117, march 2019. https://doi.org/10.36045/bbms/1553047231

Information

Published: march 2019
First available in Project Euclid: 20 March 2019

zbMATH: 07060318
MathSciNet: MR3934083
Digital Object Identifier: 10.36045/bbms/1553047231

Subjects:
Primary: 37B25, 45J05, 45M05

Rights: Copyright © 2019 The Belgian Mathematical Society

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Vol.26 • No. 1 • march 2019
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