On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

In this paper we analyze the porous medium equation

$ \begin{equation} u_t = \Delta u^m + a\int_\Omega u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad {\rm{in}}\quad \Omega \times I,\;\;\;\;\;\;(◇) \end{equation} $

where $ \Omega $ is a bounded and smooth domain of $ \mathbb R^N $, with $ N\geq 1 $, and $ I = [0,t^*) $ is the maximal interval of existence for $ u $. The constants $ a,b,c $ are positive, $ m,p,q $ proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of $ u $. Under some hypotheses on the data, including intrinsic relations between $ m,p $ and $ q $, and assuming that for some positive and sufficiently regular function $ u_0({\bf x}) $ the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution $ u = u({\bf x},t) $ on $ \Omega \times I $:

$ \triangleright $ when $ p>q $ and in 2- and 3-dimensional domains, we determine a lower bound of $ t^* $ for those $ u $ becoming unbounded in $ L^{m(p-1)}(\Omega) $ at such $ t^* $;

$ \triangleright $ when $ p<q $ and in $ N $-dimensional settings, we establish a global existence criterion for $ u $.