一类渗流方程解在边界上的不稳定性和Blow-up

基金项目: 国家自然科学基金资助项目(60274008;10171084)

Asymptotic Non-Stability and Blow-up at the Boundary for the Solutions of a Filtration Equation

1.
Department of Mathematics, Xiamen University, Xiamen, Fujian 361005, P. R. China;

摘要: 对一类具有非线性第二、第三边值条件的非线性渗流方程,证明了解的先验的界可以用初值和解在区域边界上的积分来估计和控制．这一先验估计是通过迭代技巧来建立的．根据这个估计,解可能在边界上爆破(Blow-up)从而解有渐近不稳定性．
- 渗流方程 /
- 解的先验估计 /
- 渐近不稳定性 /
- 解在边界上的Blow-up
Abstract: For a class of nonlinear Filtration equation with nonlinear second-third boundary value condition,it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region.The priori estimate of the solutions was established by iterative method.By using this estimate the solutions may blow-upon the boundary of the region and thus it may have a symptotic non-stability.
- filtr ation equation /
- prior iestimate for the solution /
- asymptotic no n-stability /
- blow-up on the boundary

[1]	Rothe F.Uniform bounds from bounded L-functionals in reaction-diffusion equations[J].J Differential Equations,1982,45(2):207—233. doi: 10.1016/0022-0396(82)90067-5
[2]	Friedman A,Lacey A A.Blow up of solutions of semilinear parabolic equations[J].J Math Anal Appl,1988,132(1):171—186. doi: 10.1016/0022-247X(88)90052-2
[3]	Friedman A Mcleod B.Blow-up of positive solutions of semilinear heat equations[J].Indian Univ Math J,1985,34(2):425—447. doi: 10.1512/iumj.1985.34.34025
[4]	Gomez Lope J,Marquez V,Wolanski N.Blow-up results and localization of blow up points for the heat equation with a nonlinear boundary condition[J].J Differential Equations,1991,92(2):384—401. doi: 10.1016/0022-0396(91)90056-F
[5]	Alikakos N D.An application of the invariance principle to reaction-diffusion equations[J].J Differential Equations,1979,33(2):201—225. doi: 10.1016/0022-0396(79)90088-3
[6]	CAO Zhen-chao,GU Lian-kun.Initial-boundary value problem for a degenerate quasilinear parabolic equation of order 2m[J].J Partial Differential Equations,1990,3(1):13—20.
[7]	Ladyzenskaja O A,Solonnikov V A,Uralceva N N.Linear and Quasilinear Equations of Parabolic Type[M].AMS Translations of Mathematical Monographs,Vol 23,Rhode Island:AMS,1968.
[8]	Levin H A.Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time[J].J Differential Equations,1974,16(2):319—334. doi: 10.1016/0022-0396(74)90018-7