Acta Mathematica Vietnamica

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The Backward Problem for Ginzburg-Landau-Type Equation
Dang Duc Trong, Bui Thanh Duy, Nguyen Dang Minh

 

Abstract

Let Ω be a bounded domain of Rn. In this paper, we consider a final value problem for the nonlinear parabolic equation

utuu(T)===Δu+aubu3+h(x,t),xΩ,t(0,T),0,xΩ,t(0,T),g,xΩ,

where g, h are given functions and the numbers a, b ( b > 0) are modeling parameters. The problem does not fulfill Hadamard’s postulates of well posedness: it might not have a solution in the strict sense; its solutions might not be unique or might not depend continuously on the data. Hence, its mathematical analysis is subtle. However, it has many applications in physics and other fields. For this reason, a regularization for the problem is proposed. In our problem, the function f(u) = a ub u 3 is not globally Lipschitz. So, we cannot apply directly recent methods that have been used in Trong et al. (Zeitschrift Analysis und ihre Anwendungen 26(2), 231–245, 2007), Trong et al. (Electron. J. Differ. Equ. 2009(109), 1–16, 2009), and Trong and Tuan (Electron. J. Differ. Equ. 2009(77), 1–13, 2009). We have to approximate the function f by a globally Lipschitz function and use an approximate equation to find the regularization solution of the problem. Error estimations between the exact solution and the approximate solution, established from noise data g ε , a δ , b δ , are given.

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