scholarly journals On Fractional Diffusion Equation with Caputo-Fabrizio Derivative and Memory Term

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Binh Duy Ho ◽  
Van Kim Ho Thi ◽  
Long Le Dinh ◽  
Nguyen Hoang Luc ◽  
Phuong Nguyen
Keyword(s):  
Diffusion Equation ◽  
Local Existence ◽  
Main Idea ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Sobolev Embedding ◽  
Memory Term ◽  
New Techniques ◽  
Scale Theory ◽  
Local Existence And Uniqueness

In this paper, we examine a nonlinear fractional diffusion equation containing viscosity terms with derivative in the sense of Caputo-Fabrizio. First, we establish the local existence and uniqueness of lightweight solutions under some assumptions about the input data. Then, we get the global solution using some new techniques. Our main idea is to combine theories of Banach’s fixed point theorem, Hilbert scale theory of space, and some Sobolev embedding.

On Time Fractional Diffusion Equation with Memory Term

2021 ◽  
Vol 30 (1) ◽  
Author(s):  
Nguyen Duc Phuong ◽  
Nguyen Anh Tuan ◽  
Tran Thanh Binh ◽  
Tran Ngoc Thach
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Memory Term ◽  
Time Fractional Diffusion Equation ◽  
With Memory

scholarly journals Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abderrazak Nabti ◽  
Ahmed Alsaedi ◽  
Mokhtar Kirane ◽  
Bashir Ahmad
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Laplacian Operator ◽  
Nonlocal Source ◽  
Time Space ◽  
Nonexistence Of Solutions ◽  
Beta 2 ◽  
Fractional Laplacian Operator

Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s for $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ ( x , t ) ∈ R N × ( 0 , ∞ ) with initial data $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) , where $p,q,r>1$ p , q , r > 1 , $q(p+r)>q+r$ q ( p + r ) > q + r , $0<\gamma \leq 2 $ 0 < γ ≤ 2 , $0<\alpha <1$ 0 < α < 1 , $0<\beta \leq 2$ 0 < β ≤ 2 , $(-\Delta )^{\frac{\beta }{2}}$ ( − Δ ) β 2 stands for the fractional Laplacian operator of order β, the weight function $\nu (x)$ ν ( x ) is positive and singular at the origin, and $\Vert \cdot \Vert _{q}$ ∥ ⋅ ∥ q is the norm of $L^{q}$ L q space.

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