Monotonicity in time of large solutions to a nonlinear heat equation

We consider the Cauchy problem for a two-dimensional semi-linear heat equation with radial symmetry u_t := u_rr+u_r/r + e^u in R⁺ × (0, T); with smooth, bounded initial data u₀(r). We prove that the solution u(r, t) becomes strictly monotone in time, u_t > 0, at any point where u is large enough. The proof is based on intersection comparison of u(r, t) with the set {w(·)} of stationary solutions satisfying w? + w' /r + e^w = 0 for r > 0. The above monotonicity result is shown to depend essentially on the global structure of the set {w}. The same result is found to hold for positive solutions u to the equation with power nonlinearity u_t = ?u + u^p, 1 < p < (N + 2)/(N - 2)+. Several generalizations to boundary value problems and quasi-linear equations are given.