一类奇异项依赖于梯度的奇异偏微分方程的研究

Abstract

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

CLC Number:

35A05

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.

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On a Class of Singular Equations with Nonlinear Terms Depending on the Gradient

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