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 u_t = Δu^m + a ∫_Ω u^p − bu^q − c|∇√u|² in Ω × I, (◇) where Ω is a bounded and smooth domain of R^N, with N ≥ 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 u₀(x) the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution u = u(x,t) on Ω × I:▹ when p>q and in 2- and 3-dimensional domains, we determine a lower bound oft∗ for those u becoming unbounded in L^m(p−1)(Ω) at such t∗;▹ when p < q and in N-dimensional settings, we establish a global existence criterion for u. [ABSTRACT FROM AUTHOR]