## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Dissipative property for non local evolution equations

Severino H. da Silva, Antonio R. G. Garcia, and Bruna E. P. Lucena

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.36045/bbms/1553047231

**Mathematical Reviews number (MathSciNet)**

MR3934083

**Zentralblatt MATH identifier**

07060318

**Subjects**

Primary: 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20] 45M05: Asymptotics 37B25: Lyapunov functions and stability; attractors, repellers

**Keywords**

Non local equation Well-posedness Smoothness orbit Global Attractor Lyapunov's functional

#### 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