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 equationut - Delta u(m) + a integral(Omega) u(p) - bu(q) - c vertical bar del root u vertical bar(2) in Omega x I, (lozenge)where Omega 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 u(0)(x) the Initial Boundary Value Problem (IBVP) associated to (lozenge) possesses a positive classical solution u = u(x, t) on Omega x I:when p > q and in 2- and 3-dimensional domains, we determine a lower bound of t* for those u becoming unbounded in Lm(p-1) (Omega) at such t*;when p < q and in N-dimensional settings, we establish a global existence criterion for u.