Conditional stability for an inverse coefficient problem of a weakly coupled time-fractional diffusion system with half order by Carleman estimate

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Caixuan Ren ◽  
Xinchi Huang ◽  
Masahiro Yamamoto
Keyword(s):  
A Priori ◽  
Coefficient Matrix ◽  
Stability Estimate ◽  
Fractional Diffusion ◽  
Carleman Estimate ◽  
Diffusion System ◽  
Conditional Stability ◽  
Inverse Coefficient Problem ◽  
Weakly Coupled ◽  
Spatially Varying

Abstract Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate.

Conditional stability in a backward Cahn–Hilliard equation via a Carleman estimate

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yunxia Shang ◽  
Shumin Li
Keyword(s):  
A Priori ◽  
Stability Estimate ◽  
Carleman Estimate ◽  
Time Interval ◽  
Conditional Stability ◽  
Hilliard Equation ◽  
Intermediate Data ◽  
Backward Problem ◽  
Final Data ◽  
Cahn Hilliard Equation

AbstractWe consider a Cahn–Hilliard equation in a bounded domain Ω in {\mathbb{R}^{n}} over a time interval {(0,T)} and discuss the backward problem in time of determining intermediate data {u(x,\theta)}, {\theta\in(0,T)}, {x\in\Omega} from the measurement of the final data {u(x,T)}, {x\in\Omega}. Under suitable a priori boundness assumptions on the solutions {u(x,t)}, we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{L^{2}(\Omega)}\leq C\lVert u(\,\cdot\,,T)% \rVert_{H^{2}(\Omega)}^{\kappa_{0}},and a conditional stability estimate for the linear Cahn–Hilliard equation\lVert u(\,\cdot\,,\theta)\rVert_{H^{\beta}(\Omega)}\leq C\lVert u(\,\cdot\,,T% )\rVert_{H^{2}(\Omega)}^{\kappa_{1}},where {\theta\in(0,T)}, {\beta\in(0,4)} and {\kappa_{0},\kappa_{1}\in(0,1)}. The proof is based on a Carleman estimate with the weight function {\mathrm{e}^{2s\mathrm{e}^{\lambda t}}} with large parameters {s,\lambda\in\mathbb{R}^{+}}.

scholarly journals Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Yavar Kian ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Boundary Data ◽  
Stability Estimate ◽  
Carleman Estimate ◽  
Parabolic Problems ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Lateral Boundary ◽  
Spatial Component

Abstract For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

scholarly journals Inverse problem for fractional diffusion equation

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Vu Tuan
Keyword(s):  
Inverse Problem ◽  
Diffusion Coefficient ◽  
Lower Bound ◽  
Diffusion Equation ◽  
Spectral Data ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
One Dimensional ◽  
Initial Distributions

AbstractWe prove that by taking suitable initial distributions only finitely many measurements on the boundary are required to recover uniquely the diffusion coefficient of a one dimensional fractional diffusion equation. If a lower bound on the diffusion coefficient is known a priori then even only two measurements are sufficient. The technique is based on possibility of extracting the full boundary spectral data from special lateral measurements.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

2021 ◽  
pp. 1-25
Author(s):  
HONG-MING YIN
Keyword(s):  
Diffusion Coefficients ◽  
A Priori ◽  
Main Tool ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sobolev Embedding ◽  
System Modelling ◽  
Strongly Coupled ◽  
Elliptic And Parabolic Equations

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

scholarly journals Integrated Time-Fractional Diffusion Processes for Fractional-Order Chaos-Based Image Encryption

Sensors ◽  
2021 ◽  
Vol 21 (20) ◽  
pp. 6838
Author(s):  
Fudong Ge ◽  
Zufa Qin ◽  
YangQuan Chen
Keyword(s):  
Boundary Conditions ◽  
Image Encryption ◽  
Fractional Order ◽  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Spatiotemporal Chaos ◽  
Diffusion System ◽  
Encryption Algorithm ◽  
Secret Key ◽  
Image Encryption Algorithm

The purpose of this paper is to explore a novel image encryption algorithm that is developed by combining the fractional-order Chua’s system and the 1D time-fractional diffusion system of order α∈(0,1]. To this end, we first discuss basic properties of the fractional-order Chua’s system and the 1D time-fractional diffusion system. After these, a new spatiotemporal chaos-based cryptosystem is proposed by designing the chaotic sequence of the fractional-order Chua’s system as the initial condition and the boundary conditions of the studied time-fractional diffusion system. It is shown that the proposed image encryption algorithm can gain excellent encryption performance with the properties of larger secret key space, higher sensitivity to initial-boundary conditions, better random-like sequence and faster encryption speed. Efficiency and reliability of the given encryption algorithm are finally illustrated by a computer experiment with detailed security analysis.

Coexistence states in a cross-diffusion system of a predator–prey model with predator satiation term

2018 ◽  
Vol 28 (11) ◽  
pp. 2131-2159 ◽  
Author(s):  
Willian Cintra ◽  
Cristian Morales-Rodrigo ◽  
Antonio Suárez
Keyword(s):  
A Priori ◽  
Diffusion System ◽  
Predator Satiation ◽  
Linear Diffusion ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Coexistence States ◽  
Predator Model

In this paper, we study the existence and non-existence of coexistence states for a cross-diffusion system arising from a prey–predator model with a predator satiation term. We use mainly bifurcation methods and a priori bounds to obtain our results. This leads us to study the coexistence region and compare our results with the classical linear diffusion predator–prey model. Our results suggest that when there is no abundance of prey, the predator needs to be a good hunter to survive.

The spreading property for a prey-predator reaction-diffusion system with fractional diffusion

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Hongmei Cheng ◽  
Rong Yuan
Keyword(s):  
Fractional Laplacian ◽  
Stable State ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Diffusion Term ◽  
Stable Lévy Process ◽  
Front Position ◽  
Spreading Property

AbstractThis paper is devoted to the study of some spreading properties of a prey-predator reaction-diffusion system where the diffusion term is replaced by the fractional Laplacian. We focus on the invasion of the introduced predator in some environment which is initially well-populated of prey. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Lévy process, the front position is exponential in time.

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