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

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