scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

The initial value problem for a nonlinear semi-infinite string

1978 ◽  
Vol 82 (1-2) ◽  
pp. 19-26 ◽  
Author(s):  
R. W. Dickey
Keyword(s):  
Boundary Value Problem ◽  
Classical Solution ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Galerkin Procedure ◽  
Initial Boundary Value ◽  
Infinite String

SynopsisThe existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

scholarly journals RELAXATION APPROXIMATION OF THE KERR MODEL FOR THE THREE-DIMENSIONAL INITIAL-BOUNDARY VALUE PROBLEM

2009 ◽  
Vol 06 (03) ◽  
pp. 577-614 ◽  
Author(s):  
GILLES CARBOU ◽  
BERNARD HANOUZET
Keyword(s):  
Boundary Value Problem ◽  
Response Time ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Debye Model ◽  
Impedance Boundary Condition ◽  
Impedance Boundary ◽  
Initial Boundary Value

The electromagnetic wave propagation in a nonlinear medium is described by the Kerr model in the case of an instantaneous response of the material, or by the Kerr–Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic and are endowed with a dissipative entropy. The initial-boundary value problem with a maximal-dissipative impedance boundary condition is considered here. When the response time is fixed, in both the one-dimensional and two-dimensional transverse electric cases, the global existence of smooth solutions for the Kerr–Debye system is established. When the response time tends to zero, the convergence of the Kerr–Debye model to the Kerr model is established in the general case, i.e. the Kerr model is the zero relaxation limit of the Kerr–Debye model.

scholarly journals On a third order initial boundary value problem in a plane domain

2012 ◽  
Vol 17 (3) ◽  
pp. 312-326
Author(s):  
Neringa Klovienė
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fluid Motion ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Plane Domain ◽  
Third Order ◽  
Initial Boundary Value

Third order initial boundary value problem is studied in a bounded plane domain σ with C4 smooth boundary ∂σ. The existence and uniqueness of the solution is proved using Galerkin approximations and a priory estimates. The problem under consideration appear as an auxiliary problem by studying a second grade fluid motion in an infinite three-dimensional pipe with noncircular cross-section.

An investigation of the Green–Lindsay three-dimensional model

2017 ◽  
Vol 23 (7) ◽  
pp. 987-1003 ◽  
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili ◽  
Wolfgang H Müller
Keyword(s):  
Boundary Value Problem ◽  
Variational Formulation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Relaxation Times ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Three Dimensional Model ◽  
Initial Boundary Value

In this paper we consider the Green and Lindsay nonclassical model for inhomogeneous anisotropic thermoelastic bodies with two relaxation times, which depend on space variables. We obtain a variational formulation for the initial-boundary value problem corresponding to the Green–Lindsay model. On the basis of the variational formulation we define the spaces of vector-valued distributions corresponding to the initial-boundary value problem and by applying suitable a priori estimates we prove the existence and uniqueness of the solution, an energy equality, and the continuous dependence of the solution on given data.

scholarly journals The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values

2000 ◽  
Vol 130 (3) ◽  
pp. 591-602 ◽  
Author(s):  
H. A. Levine ◽  
Q. S. Zhang
Keyword(s):  
Boundary Value Problem ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Values ◽  
Exterior Domains ◽  
Semilinear Heat Equation ◽  
Image Position ◽  
Initial Boundary Value

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problem we prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

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