一类奇异项依赖于梯度的奇异偏微分方程的研究

摘要/Abstract

摘要：

研究一类奇异偏微分方程, 其中奇异项依赖于梯度: ∂u/∂t-Δu=-μ|\nabla u|^l/μ^m+f(x, t), (x, t)∈Ω×(0, T], 并且u|_{∂Ω×(0, T]}=0, u(x, 0)=φ(x ∈Ω). 其中Ω 是R^N 的边界具有C² 光滑性的有界开区域, 0<T≤+∞, μ> 0, 1<m+1≤l<2 或 0<m<l=2. 称-Δu=-μ |\nabla u|^l/u^m+f(x) (x∈Ω), u|_∂Ω=0 为前面奇异问题的平衡解问题. 在对f 和φ某些条件下该文证明了这两个问题的正经典解的存在唯一性, 分别记为u, v. 其次, 提出了几个假定条件, 在这些条件下证明了lim_t→+∞u 是平衡解问题的正经典解, 即lim_t→+∞u =v.

关键词: 存在性, 惟一性, 奇异, 上下解, 一致收敛, 平衡解

Abstract:

In this paper, we consider a class of singular equations depending on quadratic gradient term in the form ∂u/∂t-Δu=-μ|\nabla u|^l/μ^m+f(x, t), (x, t)∈Ω×(0, T], with zero boundary and nonnegative initial conditions, where Ω is a bounded subset of R^N with ∂Ω of C² class, T>0 and μ>0, parameters 1<m+1≤l ≠ 2 or 0<m<l=2, the datum f and initial condition φ are both nonnegative functions satisfying some assumptions. We call problem -Δuμ |\nabla u|^l/u^m+f(x) in Ω, u|_∂Ω=0, is its stationary problem. First, we prove the existence and uniqueness of positive classical solutions of these two problems, denoted by u, v respectively. Secondly, under some assumptions, we prove lim_t→+∞u
=v is a positive classical solution of the stationary problems.

Key words: Existence, Uniquencess, Singular, Stationary, Uniform convergence

中图分类号:

35A05

运东方, 黄淑祥. 一类奇异项依赖于梯度的奇异偏微分方程的研究[J]. 数学物理学报, 2012, 32(5): 861-878.

YUN Dong-Fang, HUANG Shu-Xiang. On a Class of Singular Equations with Nonlinear Terms Depending on the Gradient[J]. Acta mathematica scientia,Series A, 2012, 32(5): 861-878.

参考文献

[1] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 2001

[2] Xia Li, Yao Zheng'an. Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation. J Math Anal Appl, 2009, 358: 182--188

[3] Zhou Wenshu, Lei Peidong. A one dimensional nonlinear heat equation with a singular term. J Math Anal Appl, 2010, 368: 711--726

[4] Gon\c{c}alves J V, Melo A L, Santos C A. Elliptic singular problems with a quadratic gradient term. Mat Contemp, 2009, 36: 107--129

[5] Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Walias Magdalena. Elliptic problems with nonlinear terms depending
on the gradient and singular on the boundary. Nonlinear Analysis TMA, 2011, 74: 1355--1371

[6] Zhang Zhijun. A remark on the existence of explosive solutions for a class of semilinear elliptic equations. Nonlinear Anal, 2000, 41: 143--148

[7] Tao Shuangping, Zhang Zhijun. On the existence of explosive solutions for semilinear elliptic problems. Nonlinear Anal, 2002, 48: 1040--1053

[8] Lazer A C, McKenna P J. On a problem of Bieberbach and Rademacher. Nonlinear Anal, 1993, 21: 327--335

[9] Lazer A C, McKenna P J. On a singular nonlinear elliptic boundary-value problem. Pro Am Math Soc, 1972, 111: 721--730

[10] Lazer A C. A singular elliptic boundary value problem. Appl Math Comput, 1994, 65: 183--194

[11] Phillips D. Existence of solutions of quenching problems. Appl Anal, 1987, 24: 253--264

[12] Arcoya David, Carmona José, Leonori Tommaso, Mart\'{\i}nez-Aparicio Pedro J, Orsina Luigi, Petitta Francesco. Existence and nonexistence of solutions for singular quadratic quasilinear equations. J Differential Equations, 2009, 246: 4006--4042

[13] Arcoya David, Barile Sara, Mart\'{\i}nez-Aparicio Pedro J. Singular quasilinear equations with quadratic growth in the gradient
without sign condition. J Math Anal Appl, 2009, 350: 401--408

[14] Boccardo Lucio. Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM Control Optim Calc Var, 2008, 14(3): 411--426

[15] Arcoya David, Boccardo Lucio, Leonori Tommaso, Porretta Alessio. Some elliptic problems with singular natural growth lower order terms. J Differential Equations, 2010, 249: 2771--2795

[16] Mart\'{\i}nez-Aparicio P J, Petitta Francesco. Parabolic equations with nonlinear singularities. Nonlinear Analysis TMA, 2011, 74: 114--131

[17] Arcoya D, Segura de Le\'{o}n S. Uniqueness of solutions for some elliptic equations with a quadratic gradient term. ESAIM Control Optim. Calc Var, 2010, 16(2): 327--336

[18] Wang Ying, Wang Mingxin. Solutions to nonlinear elliptic equations involving a singular lower order term with respect to the gradient.
Nonlinear Anal, 2011, doi:10.1016/j.na.2011.01.34

[19] Zhang Zhijun. Nonlinear elliptic equations with singular boundary conditions. J Math Anal Appl, 1997, 216: 390--397

[20] Leonori Tommaso, Petitta Francesco. Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data. Asymptot Anal, 2006, 48(3): 219--233

[21] Xia Li, Yao Zheng'an. Positive solutions for a class of singular boundary-value problems. Electronic J Diff Equa, 2007, 43: 1--5

[22] Giachetti Daniela, Murat Fran\c{c}ois. An elliptic problem with a lower order term having singular behaviour. Boll Un Mat Ital, 2009, 2(2): 349--370

[23] Arcoya David, Carmona Jos\'{e}, Mart\'{\i}nez-Aparicio Pedro J. Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms. Adv Nonlinear Stud, 2007, 7(2): 299--317

[24] Arcoya David, Mart\'{\i}nez-Aparicio Pedro J. Quasilinear equations with natural growth. Rev Mat Iberoam, 2008, 24(2): 597--616

[25] Arcoya David, Carmona Jos\'{e}, Mart\'{\i}nez-Aparicio Pedro J. Bifurcation for quasilinear elliptic singular BVP. Comm Partial Differential Equations, 2011, 36(4): 670--692

[26] Mart\'{\i}nez-Aparicio Pedro J. Singular Dirichlet problems with quadratic gradient. Boll Unione Mat Ital, 2009, 2(3): 559--574

[27] Aouaoui Sami. Existence of a solution for some singular quasilinear problem with variable exponent and containing gradient term.
Commun Math Anal, 2011, 11(2): 46--69

[28] Friedman A. Partial Differential Equations of Parabolic Type. Englewood Cliffs, N J: Prentice-Hall, INC, 1964

[29] Ladyzenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasi-linear Equations of Parabolic Type. Providence, RI: Amer Math Soc, 1968

[30] 辜联昆. 二阶抛物型偏微分方程. 厦门: 厦门大学出版社, 2002

[31] Wu Zhuoqun, Yin Jingxue, Wang Chunpeng. Elliptic & Parabolic Equations. Singapore: World Scientific, 2006

[32] Lieberman G M. Second Order Parabolic Differential Equation. Singapore: World Scientific, 1996

[33] Ladyzenskaja O A, Ural\'{c}eva N N. Linear and Quasilinear Elliptic Equations. New York: Academic Press, 1968