Room P3.10, Mathematics Building

Jorge Ferreira, Universidade Federal Rural de Pernambuco
On the asymptotic behaviour of nonlocal nonlinear problems

This lecture deals with nonlocal nonlinear problems. Our main results concern existence, uniqueness and asymptotic behavior of the weak solutions of a nonlinear parabolic equation of reaction-diffusion nonlocal type by an application of the Faedo-Galerkin approximation and Aubin-Lions compactness result. Moreover, we prove continuity with respect to the initial values, the joint continuity of the solution and a result on the existence of the global attractor for the problem \[ \begin{cases} u_{t}-a(l(u))\Delta u+| u|^{\rho }u=f(u) \phantom{.} \text{ in} \phantom{.} \Omega \times (0,T), \\ u(x,t)=0 \phantom{.} \text{ on }\phantom{.} \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) \phantom{.}\text{ in }\phantom{.} \Omega , \end{cases} \] when $0\lt \rho \le 2/(n-2)$ if $n\ge 3$ and $0\lt \rho \lt \infty $ if $n=1,2$, where $u=u(x,t)$ is a real valued function, $\Omega \subset \mathbb{R}^{n}$ is a bounded smooth domain, $n\ge 1$ with regular boundary $\Gamma =\partial \Omega $, $p\geq 2$. Moreover, $a$ and $f$ are continuous functions satisfying some appropriate conditions and $l: L^{2}(\Omega )\to\mathbb{R}$ is a continuous linear form.