Finite Time Blow-up for a Non-linear Parabolic Equation with a Gradient Term and Applications

Abstract

We give new finite time blow-up results for the non-linear parabolic equations ut−Δu = up and ut−Δu+μ∣∇u∣q = up. We first establish an a priori bound in Lp+1 for the positive non-decreasing global solutions. As a consequence, we prove in particular that for the second equation on ℝN, with q = 2p/(p+1) and small μ>0, blow-up can occur for any N≥1, p>1, (N−2)p<N+2 and without energy restriction on the initial data. Incidentally, we present a simple model in population dynamics involving this equation.