Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions
José M. Arrieta, Anı́bal Rodriguez-Bernal · Communications in Partial Differential Equations · 2004
In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u t − Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ∂u/∂n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = −βu p , β ≥ 0, and g(x, u) = u q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β 0 and p > 1, we show that blow-up occurs only on the boundary.