On the well‐posedness of a Volterra equation with applications in the Navier‐Stokes problem

In this paper, we study a backward problem for a fractional diffusion equation with nonlinear source in a bounded domain. By applying the properties of Mittag‐Leffler functions and Banach fixed point theorem, we establish some results above the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill‐posedness of our problem in the sense of Hadamard. The regularized solution is given, and the convergence rate between the regularized solution and the exact solution is also obtained.

“…where we have applied Hölder inequality and noting that p < 1 . Combining (9), (18), and (22), we obtain that…”

Section: The Existence and The Uniqueness Of The Mild Solutionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

On a final value problem for fractional reaction‐diffusion equation with Riemann‐Liouville fractional derivative

Tran

Zhou

et al. 2019

“…With the integrodifferential equation

{\partial}_t\mathbf{u}-\Delta \mathbf{u}-{\int}_0&amp;amp;amp;amp;#x0005E;{\infty }f(s)\Delta \mathbf{u}\left(t-s\right)\mathrm{d}s&amp;amp;amp;amp;#x0002B;g\left(\mathbf{u}\right)&amp;amp;amp;amp;#x0003D;h,

derived from the Coleman‐Gurtin Principle of Heat Conduction in, 24 the authors showed the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semi‐group of solutions. For local existence, regularity, and continuity dependence upon the initial data of

\epsilon

‐regular mild solutions for the abstract integrodifferential equation we refer the reader to 25 ). Several kinds of plate equations with memory and the Navier‐Stokes equation were studied in 26–29 and see the reference therein.…”

Section: Introductionmentioning

confidence: 99%

“…derived from the Coleman-Gurtin Principle of Heat Conduction in, 24 the authors showed the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related semi-group of solutions. For local existence, regularity, and continuity dependence upon the initial data of 𝜖-regular mild solutions for the abstract integrodifferential equation we refer the reader to 25 ). Several kinds of plate equations with memory and the Navier-Stokes equation were studied in [26][27][28][29] and see the reference therein.…”

Section: Introductionmentioning

confidence: 99%

The dependence on fractional orders of mild solutions to the fractional diffusion equation with memory

Akdemir

Binh²,

O’Regan

et al. 2022

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time-fractional derivative is taken in the Caputo type sense.We give a representation of solutions under Fourier series and analyze initial value problems for the semi-linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag-Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings.

“…To the best of our knowledge, there are some works on the initial value problem for , for example, de Andrade et al . However, there are not any results on inverse initial value problem for , for example, to .…”

Section: Introductionmentioning

confidence: 99%

On a backward problem for fractional diffusion equation with Riemann‐Liouville derivative

Tuan

Tuấn

Băleanu

et al. 2019

In the present paper, we study the initial inverse problem (backward problem) for a two‐dimensional fractional differential equation with Riemann‐Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well‐posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L2 and Hτ norms is considered and illustrated by numerical example.