## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Dissipative property for non local evolution equations

#### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 26, Number 1 (2019), 91-117.

Dates
First available in Project Euclid: 20 March 2019

https://projecteuclid.org/euclid.bbms/1553047231

Digital Object Identifier
doi:10.36045/bbms/1553047231

Mathematical Reviews number (MathSciNet)
MR3934083

Zentralblatt MATH identifier
07060318

#### Citation

da Silva, Severino H.; Garcia, Antonio R. G.; Lucena, Bruna E. P. Dissipative property for non local evolution equations. Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 1, 91--117. doi:10.36045/bbms/1553047231. https://projecteuclid.org/euclid.bbms/1553047231