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 equation u t − Δ p u + f ∈ a H ( u − μ ) in Q = Ω × ( 0 , T ] , p ∈ ( 1 , ∞ ) , where Ω ⊂ R n is a ring-shaped domain, Δ p u is the p-Laplace operator, a and μ 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 Γ μ ( t ) = { x : u ( x , t ) = μ } are hypersurfaces, study the regularity of Γ μ ( t ) and derive the differential equation that governs the dynamics of Γ μ ( t ) . The analysis is based on the introduction of a system of Lagrangian coordinates that transforms the moving surface Γ μ ( t ) into a stationary one.