Regularity for degenerate evolution equations with strong absorption

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:264

Article

Vitor da Silva, Joao¹  Ochoa, Pablo^2,3  Silva, Analia^4,5 

[1] Univ Buenos Aires, Dept Matemat, FCEyN, Ciudad Univ,Pabellon 1, RA-1428 Buenos Aires, DF, Argentina

[2] Univ Nacl Cuyo, RA-5500 Mendoza, Argentina

[3] Consejo Nacl Invest Cient & Tecn, RA-5500 Mendoza, Argentina

[4] Univ Nacl San Luis, Inst Matemat Aplicada San Luis, Ejercito Andes 950,D5700HHW San Luis, San Luis, Argentina

[5] Consejo Nacl Invest Cient & Tecn, Ejercito Andes 950,D5700HHW San Luis, San Luis, Argentina

关键词: p-Laplacian type operators;    Dead-core problems;    Sharp and improved intrinsic regularity;    Liouville type results;   

DOI  :  10.1016/j.jde.2018.02.013

来源: Elsevier

【 摘 要 】

In this manuscript, we study geometric regularity estimates for degenerate parabolic equations of p-Laplacian type (2 <= p < infinity) under a strong absorption condition: Delta(p)u-partial derivative u/partial derivative t =lambda(0)u(+)(q) in Omega(T) := Omega x (0, T), where 0 <= q < 1. This model is interesting because it yields the formation of dead-core sets, i.e., regions where non-negative solutions vanish identically. We shall prove sharp and improved parabolic C-alpha regularity estimates along the set F-0(u, Omega T) = partial derivative{u > 0} boolean AND Omega(T) (the free boundary), where alpha = p/p-1-q >= 1 + 1/p-1. Some weak geometric and measure theoretical properties as non-degeneracy, positive density, porosity and finite speed of propagation are proved. As an application, we prove a Liouville-type result for entire solutions. A specific analysis for Blow-up type solutions will be done as well. The results are new even for dead-core problems driven by the heat operator. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free