A nonlinear diffusion equation with reaction localized in the half-line

We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line u(t) = (u(m))(xx) + a(x)u(p), m, p > 0 and a(x) = 1 for x > 0, a(x) = 0 for x < 0. We first characterize the global existence exponent p(0) = 1 and the Fujita exponent p(c) = m + 2. Then we pass to study the grow-up rate in the case p <= 1 and the blow-up rate for p > 1. In particular we show that the grow-up rate is different as for global reaction if p > m or p not equal 1, m.

Automated summary

We study the behaviour of solutions to a quasilinear heat equation with a reaction restricted to a half-line. We characterize the global existence exponent and Fujita exponent, and then examine the grow-up rate and blow-up rate under different conditions. We show that the grow-up rate differs from the case of global reaction if certain conditions are met.