scholarly journals On the Sirs Epidemic Model with Free Boundaries

2021 ◽  
Vol 66 (1) ◽  
pp. 21-47
Author(s):  
J. O. Takhirov ◽  
◽  
Z. K. Djumanazarova ◽  
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Global Solvability ◽  
Reproductive Number ◽  
Free Boundaries ◽  
Sirs Epidemic Model ◽  
Priori Estimates ◽  
Initial Area

We investigate an epidemic non-linear reaction-diffusion system with two free boundaries. A free boundary is introduced to describe the expanding front of the infectious environment. A priori estimates of the required functions are established, which are necessary for the correctness and global solvability of the problem. We get sufficient conditions for the spread or disappearance of the disease. It has been proven that with a base reproductive number the disease disappears in the long term if the initial values and the initial area are sufficiently small.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

A reaction-diffusion-advection competition model with a free boundary

2021 ◽  
Vol 65 (3) ◽  
pp. 25-37
Keyword(s):  
Free Boundary ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Competition Model ◽  
Baseline Data ◽  
The Other ◽  
Quasilinear System ◽  
Priori Estimates ◽  
Competitive Diffusion

In this paper, we study a competitive diffusion quasilinear system with a free boundary. First, we investigate the mathematical questions of the problem. A priori estimates of Schauder type are established, which are necessary for the solvability of the problem. One of two competing species is an invader, which initially exists on a certain sub-interval. The other is initially distributed throughout the area under consideration. Examining the influence of baseline data on the success or failure of the first invasion. We conclude that there is a dichotomy of spread and extinction and give precise criteria for spread and extinction in this model.

scholarly journals Solvability Analysis of a Mixed Boundary Value Problem for Stationary Magnetohydrodynamic Equations of a Viscous Incompressible Fluid

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2088
Author(s):  
Gennadii Alekseev ◽  
Roman V. Brizitskii
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Mixed Boundary ◽  
Boundary Value ◽  
A Priori ◽  
Sufficient Conditions ◽  
Mixed Boundary Conditions ◽  
Global Solvability ◽  
Tangential Component ◽  
Mhd Equations

We investigate the boundary value problem for steady-state magnetohydrodynamic (MHD) equations with inhomogeneous mixed boundary conditions for a velocity vector, given the tangential component of a magnetic field. The problem represents the flow of electrically conducting viscous fluid in a 3D-bounded domain, which has the boundary comprising several parts with different physical properties. The global solvability of the boundary value problem is proved, a priori estimates of the solutions are obtained, and the sufficient conditions on data, which guarantee a solution’s local uniqueness, are determined.

On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type

1973 ◽  
Vol 16 (1) ◽  
pp. 137-141
Author(s):  
K. A. Zischka
Keyword(s):  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Equations ◽  
Priori Estimates ◽  
The Given ◽  
Uniqueness Of A Solution

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].

scholarly journals A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 75
Author(s):  
Hristo Kiskinov ◽  
Magdalena Veselinova ◽  
Ekaterina Madamlieva ◽  
Andrey Zahariev
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Integral Representations ◽  
Distributed Delays ◽  
Finite Time Stability ◽  
Gronwall’S Inequality ◽  
Priori Estimates ◽  
The Stability ◽  
Traditional Approaches

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

Existence and Uniqueness of Weak Solution for a Class of Elliptic Reaction-Diffusion Systems

2008 ◽  
Vol 15 (4) ◽  
pp. 619-625
Author(s):  
Abdelfatah Bouziani ◽  
Ilham Mounir
Keyword(s):  
Weak Solution ◽  
Existence And Uniqueness ◽  
Iterative Process ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Priori Estimates ◽  
Quasilinear Elliptic

Abstract We present a simple proof of the existence and uniqueness of a weak solution for a class of quasilinear elliptic reaction-diffusion systems. The proof is based on an iterative process and on some a priori estimates.

scholarly journals Method of Monotone Solutions for Reaction-Diffusion Equations

2017 ◽  
Vol 63 (3) ◽  
pp. 437-454
Author(s):  
V Volpert ◽  
V Vougalter
Keyword(s):  
Existence Of Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Monotone Solutions

Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reactiondiffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

scholarly journals On the solvability of a class of reaction-diffusion systems

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Abdelfatah Bouziani ◽  
Ilham Mounir
Keyword(s):  
Iterative Process ◽  
Continuous Dependence ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Priori Estimates ◽  
Quasilinear Systems ◽  
Linearized Problem

We deal with a class of parabolic reaction-diffusion systems. We use an iterative process based on results obtained for a linearized problem, then we derive some a priori estimates to establish the existence, uniqueness, and continuous dependence of the weak solution for a class of quasilinear systems.

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