Monotonicity in Time of Large Solutions to a Nonlinear Heat Equation
Victor A. Galaktionov, Andrew A. Lacey · Rocky Mountain Journal of Mathematics · 1998
We consider the Cauchy problem for a twodimensional semi-linear heat equation with radial symmetrywith smooth, bounded initial data u 0 (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 quasilinear equations are given.