José M. Arrieta

Universidad Complutense de Madrid, Spain

Title: Nonautonomous degenerate logistic equations: boundedness vs unboundedness of solutions

Abstract:  In this presentation we consider a degenerate logistic equation of the type $u_t-\Delta u=\lambda u-n(x,t)u^p$ where $p>1$, $\lambda\in \mathbb{R}$ and $n(x,t)\geq 0$ which vanish in the set  $K(t)=\{ x\in \mathbb{R}^d: n(x,t)=0\}$, which acts as a refugee region from the biological interpretation. We analyze how the value of $\lambda$ and the geometry and motion of $K(t)$ affects the boundedness of the solutions of the equations. We will provide examples where for small velocity of the set $K(t)$ the solutions grow without bound as $t\to +\infty$ while for large velocity the solutions are stabilized and become bounded as $t\to +\infty$. This is a joint collaboration with Marcos Molina, Lucas A. Santos and Neus Cónsul. Finally, we will comment on an equivalent problem where the logistic term is placed at the boundary of the domain and for which similar questions as above may be posed. This problem was suggested by Sergio Oliva.