On existence and regularity of a terminal value problem for the time fractional diffusion equation

Tuan, Nguyen Huy; Ngoc, Tran Bao; Zhou, Yong; O’Regan, Donal

doi:10.1088/1361-6420/ab730d

Cited by 18 publications

(6 citation statements)

References 63 publications

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“…Because the fractional derivative will help us to capture the viscoelastic properties of the flow, the time-fractional pseudo-parabolic equations are useful for describing the behavior of some non-Newtonian fluids. In mathematical aspect, it is a hot stream to consider the time-fractional version of the classical mathematical models including the parabolic type equations or diffusion models [20,37,38,49], the time-space fractional Shrödinger equation with polynomial type nonlinearity [22], the time-fractional Navier-Stokes equations (FNS) [12,45], and also the time-fractional pseudo-parabolic equations [36]. Surprisingly in [12], it was shown that the order of time-fractional derivative influences the regularity not only in time variable but also in the spatial variable.…”

Section: Background Of the Problemmentioning

confidence: 99%

Global well-posedness for fractional Sobolev-Galpern type equations

Nguyen¹,

Tuấn²,

Liu³

2021

Preprint

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This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally use of Orlicz space.

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Section: Background Of the Problemmentioning

confidence: 99%

Global well-posedness for fractional Sobolev-Galpern type equations

Nguyen¹,

Tuấn²,

Liu³

2021

Preprint

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“…The uniqueness and some stability estimate can be found in the pioneer work [32]. We also refer [22,35,[37][38][39] for different regularization methods, and [41] for error analysis of fully discrete schemes. The analysis heavily relies on the asymptotic behaviors of Mittag-Leffler functions, or equivalently the smoothing properties of solution operators.…”

Section: Introductionmentioning

confidence: 99%

Stability and numerical analysis of backward problem for subdiffusion with time-dependent coefficients

ZHANG¹

2023

Inverse Problems

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Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiﬀusion with time dependent coeﬃcients. First of all, by using the smoothing property of solution operators and a perturbation argument of freezing the diﬀusion coeﬃcients, we show a stability estimate in Sobolev spaces, under some smallness/largeness condition on the terminal time. Moreover, in case of noisy observation, we apply a quasi-boundary value method to regularize the problem and then show the convergence of the regularization scheme. Finally, to numerically reconstruct the initial data, we propose a completely discrete scheme by applying the ﬁnite element method in space and backward Euler convolution quadrature in time. An a priori error estimate is then established. The proof is heavily built on a perturbation argument dealing with time dependent coeﬃcients and some nonstandard error estimates for the direct problem. The error estimate gives a useful guide for balancing discretization parameters, regularization parameter and noise level. Some numerical experiments are presented to illustrate our theoretical results.

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“…The reason they are of interest comes from the model sticking to memory eects. We temporarily list some articles about Caputo or Riemann-Liouville [1,2,3,18,10,12,14,9,11,15,20,13,19] and some other Email address: tiennv55@fe.edu.vn (Van Tien Nguyen) derivatives, see [21, 22? , 23, 24, 25].…”

Section: Introductionmentioning

confidence: 99%

On Caputo fractional elliptic equation with nonlocal condition

NGUYEN

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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This paper is first study for considering nonlocal elliptic equation with Caputo derivative. We obtain the upper bound of the mild solution. The second contribution is to provide the lower bound of the solution at terminal time. We prove the non-correction of the problem in the sense of Hadamard. The main tool is the use of upper and lower bounds of the Mittag-Lefler function, combined with analysis in Hilbert scales space.

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On existence and regularity of a terminal value problem for the time fractional diffusion equation

Cited by 18 publications

References 63 publications

Global well-posedness for fractional Sobolev-Galpern type equations

Global well-posedness for fractional Sobolev-Galpern type equations

Stability and numerical analysis of backward problem for subdiffusion with time-dependent coefficients

On Caputo fractional elliptic equation with nonlocal condition

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