scholarly journals Maximum principles for time-fractional Cauchy problems with spatially non-local components

2018 ◽  
Vol 21 (5) ◽  
pp. 1335-1359 ◽  
Author(s):  
Anup Biswas ◽  
József Lőrinczi
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Existence And Uniqueness ◽  
Strong Maximum Principle ◽  
Maximum Principles ◽  
Inverse Source Problem ◽  
Cauchy Problems ◽  
Probabilistic Representation ◽  
Local Operators ◽  
Non Local

Abstract We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate for the weak solutions of a Cauchy problem featuring Caputo time-derivatives and non-local operators in space variables given in terms of Bernstein functions of the Laplacian. To achieve this, first we propose a suitable meaning of a weak solution, show their existence and uniqueness, and establish a probabilistic representation in terms of time-changed Brownian motion. As an application, we also discuss an inverse source problem.

scholarly journals The complement value problem for non-local operators

2020 ◽  
Vol 20 (03) ◽  
pp. 2050043
Author(s):  
Wei Sun
Keyword(s):  
Weak Solution ◽  
Lipschitz Domain ◽  
Dirichlet Forms ◽  
Probabilistic Representation ◽  
Kernel Estimates ◽  
Mild Conditions ◽  
Heat Kernel Estimates ◽  
Bounded Lipschitz Domain ◽  
Local Operators ◽  
Non Local

Let [Formula: see text] be a bounded Lipschitz domain of [Formula: see text]. We consider the complement value problem [Formula: see text] Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in our approach.

scholarly journals A non-linear hyperbolic equation

1980 ◽  
Vol 3 (3) ◽  
pp. 505-520 ◽  
Author(s):  
Eliana Henriques de Brito
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Real Number ◽  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Linear Hyperbolic Equation ◽  
Non Linear

In this paper the following Cauchy problem, in a Hilbert spaceH, is considered:(I+λA)u″+A2u+[α+M(|A12u|2)]Au=fu(0)=u0u′(0)=u1Mandfare given functions,Aan operator inH, satisfying convenient hypothesis,λ≥0andαis a real number.Foru0in the domain ofAandu1in the domain ofA12, ifλ>0, andu1inH, whenλ=0, a theorem of existence and uniqueness of weak solution is proved.

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

scholarly journals Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

2021 ◽  
Vol 7 (1) ◽  
pp. 260-275
Author(s):  
Zihan Cai ◽  
◽  
Yan Liu ◽  
Baiping Ouyang ◽  
Keyword(s):  
Cauchy Problem ◽  
Phase Space ◽  
Coupled Systems ◽  
Decay Properties ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Thermoelastic Plate ◽  
Local Operators ◽  
Non Local

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

scholarly journals Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

2021 ◽  
Author(s):  
Yan Liu ◽  
Zihan Cai ◽  
Shuanghu Zhang
Keyword(s):  
Cauchy Problem ◽  
Phase Space ◽  
Coupled Systems ◽  
Decay Properties ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Thermoelastic Plate ◽  
Local Operators ◽  
Non Local

In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $L^p-L^q$ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

scholarly journals Transforms on white noise functionals with their applications to Cauchy problems

1997 ◽  
Vol 147 ◽  
pp. 1-23 ◽  
Author(s):  
Dong Myung Chung ◽  
Un Cig Ji
Keyword(s):  
Cauchy Problem ◽  
White Noise ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Characterization Theorem ◽  
Cauchy Problems ◽  
Continuous Linear ◽  
The Cauchy Problem ◽  
White Noise Functionals ◽  
Generalized Laplacian

AbstractA generalized Laplacian ΔG(K) is defined as a continuous linear operator acting on the space of test white noise functionals. Operator-parameter - and -transforms on white noise functionals are introduced and then prove a characterization theorem for and -transforms in terms of the coordinate differential operator and the coordinate multiplication. As an application, we investigate the existence and uniqueness of solution of the Cauchy problem for the heat equation associated with ΔG(K)

scholarly journals Existence and uniqueness of weak solutions of the Cauchy problem for parabolic delay-differential equations

1980 ◽  
Vol 21 (1) ◽  
pp. 65-80 ◽  
Author(s):  
S. Nababan ◽  
K.L. Teo
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Divergence Form ◽  
Partial Delay ◽  
The Cauchy Problem ◽  
Delay Differential

In this paper, a class of systems governed by second order linear parabolic partial delay-differential equations in “divergence form” with Cauchy conditions is considered. Existence and uniqueness of a weak solution is proved and its a priori estimate is established.

scholarly journals THE NON-LOCAL CAUCHY PROBLEM FOR SEMILINEAR INTEGRODIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT

2001 ◽  
Vol 44 (1) ◽  
pp. 63-70 ◽  
Author(s):  
K. Balachandran ◽  
M. Chandrasekaran
Keyword(s):  
Cauchy Problem ◽  
Integrodifferential Equation ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Subject Classification ◽  
Classical Solutions ◽  
Deviating Argument ◽  
Primary 34K05 ◽  
Non Local

AbstractThe aim of this paper is to prove the existence and uniqueness of mild and classical solutions of the non-local Cauchy problem for a semilinear integrodifferential equation with deviating argument. The results are established by using the method of semigroups and the contraction mapping principle. The paper generalizes certain results of Lin and Liu.AMS 2000 Mathematics subject classification: Primary 34K05; 34K30

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