scholarly journals Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shihe Xu ◽  
Minhai Huang
Keyword(s):  
Time Delays ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Unique Global Solution ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
A Free Boundary Problem ◽  
Initial Boundary Value

We study a mathematical model for the growth of necrotic tumors with time delays in proliferation. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval[0,T].

scholarly journals On the Evolutionary Form of the Constraints in Electrodynamics

2018 ◽  
Author(s):  
István Rácz
Keyword(s):  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Of Solutions ◽  
Maxwell Theory ◽  
Constraint Equations ◽  
First Order ◽  
Global Existence And Uniqueness ◽  
Hyperbolic Form ◽  
Initial Boundary Value

The constraint equations in Maxwell theory are investigated. In analogy with some recent results on the constraints of general relativity it is shown, regardless of the signature and dimension of the ambient space, that the "divergence of a vector field" type constraints can always be put into linear first order hyperbolic form for which global existence and uniqueness of solutions to an initial-boundary value problem is guaranteed.

scholarly journals On the Evolutionary Form of the Constraints in Electrodynamics

Symmetry ◽  
2018 ◽  
Vol 11 (1) ◽  
pp. 10 ◽  
Author(s):  
István Rácz
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Of Solutions ◽  
Maxwell Theory ◽  
Constraint Equations ◽  
First Order ◽  
Global Existence And Uniqueness ◽  
Hyperbolic Form ◽  
Initial Boundary Value ◽  
Type Constraint

The constraint equations in Maxwell theory are investigated. In analogy with some recent results on the constraints of general relativity, it is shown, regardless of the signature and dimension of the ambient space, that the “divergence of a vector field”-type constraint can always be put into linear first order hyperbolic form for which the global existence and uniqueness of solutions to an initial-boundary value problem are guaranteed.

DYNAMICS OF MULTISECTION SEMICONDUCTOR LASERS

2005 ◽  
Vol 9 (1) ◽  
pp. 51-66 ◽  
Author(s):  
J. Sieber ◽  
M. Radžiūnas ◽  
K. R. Schneider
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Semiconductor Lasers ◽  
Invariant Manifolds ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Low Dimension ◽  
Global Existence And Uniqueness ◽  
Numerical Bifurcation ◽  
Initial Boundary Value

We investigate the longitudinal dynamics of multisection semiconductor lasers based on a model, where a hyperbolic system of partial differential equations is nonlinearly coupled with a system of ordinary differential equations. We present analytic results for that system: global existence and uniqueness of the initial‐boundary value problem, and existence of attracting invariant manifolds of low dimension. The flow on these manifolds is approximately described by the so‐called mode approximations which are systems of ordinary differential equations. Finally, we present a detailed numerical bifurcation analysis of the two-mode approximation system and compare it with the simulated dynamics of the full PDE model.

Stability for a nonlinear coupled system of elasticity and thermoelasticity with second sound

2014 ◽  
Vol 13 (01) ◽  
pp. 45-75 ◽  
Author(s):  
Yi-Ping Meng ◽  
Ya-Guang Wang
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Middle Part ◽  
Second Sound ◽  
Qualitative Properties ◽  
Thermal Change ◽  
Nonlinear Coupled System ◽  
Initial Boundary Value

In this paper, we study the qualitative properties of solutions to a nonlinear system describing the motion of a bar in which the middle part is sensitive to the thermal change, while the outer parts are insensible. By the energy method, we show that the initial boundary value problem for this coupled system of wave equations and thermoelastic equations with second sound in one space variable is well-posed globally in time, and it is also stable exponentially as the time goes to infinity when the wave speed of the outer parts is properly large, under certain restrictions on the initial data and the growth rate of the nonlinear terms.

scholarly journals An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
T. S. Jang
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Boussinesq Equation ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Successive Approximations ◽  
Initial Boundary Value

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initial-boundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.

On existence and uniqueness of solutions to a pantograph type equation

2021 ◽  
Vol 62 ◽  
pp. 489-512
Author(s):  
Muhammad Mohsin ◽  
Ali Ashher Zaidi
Keyword(s):  
Existence And Uniqueness ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cell Distribution ◽  
Uniqueness Of Solutions ◽  
Initial Cell ◽  
Daughter Cells ◽  
Initial Boundary Value ◽  
Nonlocal Terms

We show existence and uniqueness of solutions to an initial boundary value problem that entails a pantograph type functional partial differential equation with two advanced nonlocal terms. The problem models cell growth and division into two daughter cells of different sizes. There is a paucity of information about the solution to the problem for an arbitrary initial cell distribution. doi:10.1017/S144618112100002X

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