ANY LARGE SOLUTION OF A NONLINEAR HEAT-CONDUCTION EQUATION BECOMES MONOTONIC IN TIME

In this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u(t) = (phi(u))xx + Q(u) in omega-T = (0, T] X R1 with bounded sufficiently smooth initial function u(0, x) = u0(x) greater-than-or-equal-to 0 in R1. We assume that phi(u) and Q(u) are smooth functions in [0, + infinity) and phi'(u) > 0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)phi'(u) at infinity, it is proved that if u(t0, x0) becomes sufficiently large at some point (t0, x0) is-an-element-of-omega-T, then u(t)(t, x0) greater-than-or-equal-to 0 for all t is-an-element-of[t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.