On a class of fully nonlinear parabolic equations

Advances in Nonlinear Analysis

article

Stanislav Antontsev¹  Sergey Shmarev³ 

[1] Lavrentyev Institute of Hydrodynamics of SB RAS;University of Lisbon;Universidad de Oviedo

关键词: Fully nonlinear parabolic equation;    strong solution;    asymptotic behavior;    extinction in a finite time;   

DOI  :  10.1515/anona-2016-0055

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We study the homogeneous Dirichlet problem for the fully nonlinear equation u t = | Δ ⁢ u | m - 2 ⁢ Δ ⁢ u - d ⁢ | u | σ - 2 ⁢ u + f   in  Q T = Ω × ( 0 , T ) , u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters m > 1 {m>1} , σ > 1 {\sigma>1} and d ≥ 0 {d\geq 0} . At the points where Δ ⁢ u = 0 {\Delta u=0} , the equation degenerates if m > 2 {m>2} , or becomes singular if m ∈ ( 1 , 2 ) {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as t → ∞ {t\to\infty} . Sufficient conditions for exponential or power decay of ∥ ∇ ⁡ u ⁢ ( t ) ∥ 2 , Ω {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

【 授权许可】

CC BY