Stability estimates for parabolic problems with Wentzell boundary conditions

被引:23
作者
Coclite, Giuseppe M. [1 ]
Goldstein, Gisele R. [2 ]
Goldstein, Jerome A. [2 ]
机构
[1] Univ Bari, Dept Math, I-70125 Bari, Italy
[2] Univ Memphis, Dept Math Sci, Memphis, TN 38152 USA
关键词
Heat equation; General Wentzell boundary condition; Stability;
D O I
10.1016/j.jde.2007.12.006
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Of concern is the uniformly parabolic problem u(t) = div(A del u), u(0,x)=f(x), u(1) + beta partial derivative(A)(v)u + gamma u - q beta Delta LBu = 0, for x epsilon ohm subset of R-N and t >= 0. Here A = {a(ij)(x)}(ij) is a real, hermitian, uniformly positive definite N x N matrix; beta, gamma epsilon C((ohm) over bar) with beta > 0; q epsilon (0, infinity) and partial derivative(A)(v)u is the conormal derivative of a with respect to A: and everything is sufficiently regular. The solution of this well-posed problem depends continuously on the ingredients of the problem, namely, A, beta, gamma, f. This is shown using semigroup methods in [G.M. Co-clite, A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian, Semigroup Forum, in press]. More precisely, if we have a sequence of such problems with solutions a, and if A(n) -> A, beta(n) (->) beta, etc. in a suitable sense, then u(n) -> u, the solution of the limiting problem. The abstract analysis associated with operator semigroup theory gives this conclusion, but no rate of convergence. Determining how fast the convergence of the solutions is requires detailed estimates. Such estimates are provided in this paper. (C) 2008 Elsevier Inc. All rights reserved.
引用
收藏
页码:2595 / 2626
页数:32
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