The Backward Problem for Ginzurg-Landau-Type Equation

Abstract

Let be a bounded domain of R n . In this paper, we consider a final value problem for the nonlinear parabolic equationwhere 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) = au − bu 3 is not globally Lipschitz. So, we cannot apply directly recent methods that have been used in ). 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.

Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source

“…There have been several papers devoted to backward nonlinear parabolic equations: (1) backward uniqueness [12,21]; (2) regularization methods [24-26, 36, 37, 39-41]; (3) stability estimates [11,[37][38][39][40][41]. However, the stability results for the case with locally Lipschitz source are very few [36,37,39]. Since our work is devoted to the case with the source function satisfying a local Lipschitz condition close to that of Trong et al [37,39], let us summarize their stability results.…”

On a backward problem for multidimensional Ginzburg–Landau equation with random data

“…But she did not give a regularization result. In 2015, Trong et al [27] have given a regularization method for (1.2) when Λ(t) = 1. Their idea is to approximate the function f (u) = u − u 3 with a globally Lipschitz function and using this function to find the regularized solution of the problem.…”

“…In [26], the authors deal with a discrete random model in 1-D case for Hemholtz equation. In [27], the authors considered a discrete random model in 2D case. In both papers, the spectral methods together with trigonometric least squares method in nonparametric regression has been applied.…”

“…Although spectral method is effective for linear random models, it is not possible to approximate the solution of problem (1.2) using the spectral method.Until now, to the best of our knowledge, there do not exist any results for approximating the solution of the problem (1.2) with the random model (1.6) and (1.7). We want to emphasize that our random model here is multidimensional case which is a generalization of results in [26,27].…”