Backward heat equations with locally lipschitz source

Trong, Dang Duc; Duy, Bui Thanh; Minh, Mach Nguyet

doi:10.1080/00036811.2014.963063

Cited by 11 publications

(10 citation statements)

References 10 publications

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

2018

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Let H be a Hilbert space with the inner product •, • and the norm • , A(t) a positive self-adjoint unbounded time-dependent operator on H and ε > 0. We establish stability estimates of Hölder type and propose a regularization method with error estimates of Hölder type for the ill-posed backward semilinear parabolic equationwith the source function f satisfying a local Lipschitz condition.

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Section: Introductionmentioning

confidence: 99%

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

2018

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“…Nonhomogeneous and nonlinear backward parabolic problems were considered in a lot of papers. [8][9][10][11][12][13] Recently, a lot of fractional ill-posed problem were studied (see, eg, other studies 1, [14][15][16][17][18] ). However, in the present paper, we consider the ill-posedness depended on the parameters.…”

Section: Introductionmentioning

confidence: 99%

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

Dien

Duc

2019

Math Methods in App Sciences

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In this paper, we study the forward and the backward in time problems for a class of nonlinear diffusion equations with respect to the pseudo‐differential operator. Herein, we investigate the stability of the solution of the forward problem in relationship with parameters of the pseudo‐differential operator and initial data. Besides, as known, the backward in time problem is instability. Hence, we give a method to regularize the solution of the backward problem in the case of the parameters are perturbed.

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“…Later in , M. Denche and K. Bessila applied quasi‐boundary value method approximated with

\begin{matrix} u_{t}^{ε} (t) - A u^{ε} (t) & = & 0 0 < t < T \\ - ε u_{t}^{ε} (x, 0) + u^{ε} (x, T) & = & g (x) \end{matrix}\}

There are many papers for the linear case of final value problems, but very less literature for nonlinear case. In , authors use non‐homogeneous nonlinear cases of backward heat equation by using quasi‐boundary value method, and in , authors have discussed about nonlinear backward heat equation using different methods.…”

Section: Introductionmentioning

confidence: 99%

“…There are many papers for the linear case of final value problems, but very less literature for nonlinear case. In [1,[8][9][10], authors use non-homogeneous nonlinear cases of backward heat equation by using quasi-boundary value method, and in [11][12][13][14][15][16], authors have discussed about nonlinear backward heat equation using different methods.…”

Section: Introductionmentioning

confidence: 99%

Backward heat equation with time dependent variable coefficient

Hapuarachchi

2016

Math Methods in App Sciences

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Backward heat equation with time dependent variable coefficient is severely ill‐posed in the sense of Hadamard, so we need regularization. In this paper, we consider Backward heat equation with time dependent variable coefficient, and by small perturbing, we obtain an approximation problem. We show this approximation problem is well‐posed with small parameter. Also, we show this approximation system converges to the original problem when parameter goes to zero. Here, we use modified‐quasi boundary value method to regularize this problem. Copyright © 2016 John Wiley & Sons, Ltd.

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Backward heat equations with locally lipschitz source

Cited by 11 publications

References 10 publications

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

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

Stability of solutions of a class of nonlinear fractional diffusion equations with respect to a pseudo‐differential operator

Backward heat equation with time dependent variable coefficient

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