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Díaz Díaz, Jesús Ildefonso and Shmarev, Sergey
(2009)
*Lagrangian approach to the study of level sets II: A quasilinear equation in climatology.*
Journal of Mathematical Analysis and Applications, 352
(1).
pp. 475-495.
ISSN 0022-247X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X08009426

## 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.

Item Type: | Article |
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Uncontrolled Keywords: | regularity; parabolic p-laplacian; lagrangian coordinates; climatic energy balance models; free boundary problem |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15117 |

Deposited On: | 07 May 2012 08:35 |

Last Modified: | 12 Dec 2018 15:07 |

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