Lagrangian approach to the study of level sets II: A quasilinear equation in climatology

Abstract

We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation u(t) - Delta p(u) + f is an element of aH(u - mu) in Q = Omega x (0, T], P is an element of (1, infinity), where Omega subset of R(n) is a ring-shaped domain, Delta(p)u is the p-Laplace operator, a and mu are given positive constants, and H(.) is the Heaviside maximal monotone graph: H(s) = 1 if s > 0, H(0) = [0, 1], H(s) = 0 if s < 0. The mathematical models of this type arise in climatology, the case p = 3 was proposed and justified by P. Stone in 1972. We establish the conditions on the initial data which guarantee that the level sets Gamma(mu)(t) = {x: u(x, t) = mu} are hypersurfaces, study the regularity of Gamma(mu)(t) and derive the differential equation that governs the dynamics of Gamma(mu)(t). The analysis is based on the introduction of a system of Lagrangian coordinates that transforms the moving surface Gamma(mu)(t) into a stationary one.