Semilinear Caputo time-fractional pseudo-parabolic equations

In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefler function, we will give some results on the existence and the uniqueness of mild solution for the Problem \eqref{Main-Equation} below. When $ \nu=1 $, we will introduce some $ L^p-L^q $ estimates, and base on them we derive the global existence of a mild solution in the whole space $ \R^d. $

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“…Basic settings. (see [3]) We first consider the spectral problem for the negative Laplacian on a bounded domain D ⊂ R N as follows…”

Section: Preliminary Materialsmentioning

confidence: 99%

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

Phương

Tuấn²,

Kumar

et al. 2021

Math. Model. Nat. Phenom.

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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 equation [25], the time-space fractional Shrödinger equation with polynomial type nonlinearity [14], the time-fractional Navier-Stokes equations (FNS) [13], and also the time-fractional pseudo-parabolic equations [24]. Surprisingly in [13], it was shown that the order of time-fractional derivative influences the regularity not only in time variable but also in the spatial variable.…”

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confidence: 99%

“…Surprisingly in [13], it was shown that the order of time-fractional derivative influences the regularity not only in time variable but also in the spatial variable. In [24], a first attempt to explore the influence of the degree of nonlinearity on the dynamic behavior of the solution was conducted by considering the logarithmic nonlinearity and globally Lipschitz nonlinearity. All of the above achievements in this direction pushed us to consider not only the influence of the degree of the nonlinearities but also the order of the time-fractional derivative on the behavior of the solution to the time-fractional pseudo-parabolic equations with four distinguished types of nonlinearities, in which the degree of nonlinearities increase gradually.…”

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confidence: 99%

Global well-posedness for fractional Sobolev-Galpern type equations

Nguyen

Tuấn

Yang

2022

DCDS

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<p style='text-indent:20px;'>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 using an Orlicz space.</p>

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“…This derivative has interesting advantages, which can be seen in the Introduction of [38]. A good understanding for subdiffusion second parabolic equation with Caputo derivative can be found in [3,4,11,21,30,20] and references therein. Regarding the good works on the existence of solutions of nonlinear diffusion equation with Caputo-Fabrizio operator are typical articles such as [32,38].…”

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confidence: 99%

“…The main contribution of this paper is described in detail as follows The key tool evaluation relies on L p − L q estimates for solution operators, where we used a lemma about L p − L q for semigroup of biharmonic operator taken in the paper [17]. Part of this technique is learned from the interesting recent work of Tuan-Au-Xu [30]. To be successful in our evaluations we also need a good understanding of the embedding between Hilbert scales and L p spaces.…”

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confidence: 99%

Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

Tuan¹

2021

DCDS-S

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<p style='text-indent:20px;'>In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied <inline-formula><tex-math id="M1">\begin{document}$ L^p-L^q $\end{document}</tex-math></inline-formula> estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case <inline-formula><tex-math id="M2">\begin{document}$ 0<{\alpha}<1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ {\alpha} = 1 $\end{document}</tex-math></inline-formula>. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between <inline-formula><tex-math id="M4">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces and Hilbert scales spaces.</p>

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Semilinear Caputo time-fractional pseudo-parabolic equations

Cited by 47 publications

References 54 publications

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

Global well-posedness for fractional Sobolev-Galpern type equations

Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation

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