scholarly journals On Initial Boundary Value Problem for Parabolic Differential Operator with Non-coercive Boundary Conditions

2020 ◽  
pp. 547-558
Author(s):  
Alexander N. Polkovnikov
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Type Spaces ◽  
Bochner Space ◽  
Initial Boundary Value

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in Rn with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space

ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE

2006 ◽  
Vol 08 (02) ◽  
pp. 189-217 ◽  
Author(s):  
ROSA E. CARDIEL ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Pseudo Differential Operator ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ > 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

EXISTENCE OF A WEAK SOLUTION IN AN INFINITE VISCOELASTIC STRIP WITH A SEMI-INFINITE CRACK

2004 ◽  
Vol 14 (07) ◽  
pp. 975-986 ◽  
Author(s):  
HIROMICHI ITOU ◽  
ATUSI TANI
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Viscoelastic Strip ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Initial Boundary Value

We study an initial-boundary value problem in an infinite viscoelastic strip with a semi-infinite fixed crack. For this problem we prove the existence and uniqueness of a weak solution which is prescribed on each side of the extended crack in Sobolev-type spaces.

scholarly journals ON A 3D INITIAL-BOUNDARY VALUE PROBLEM FOR DETERMINING THE DYNAMICS OF IMPURITIES CONCENTRATION IN A HORIZONTAL LAYERED FINE-PORE MEDIUM

2019 ◽  
Vol 3 ◽  
pp. 60
Author(s):  
Sharif E. Guseynov ◽  
Ruslans Aleksejevs ◽  
Jekaterina V. Aleksejeva
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Analytical Approach ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Unsteady State ◽  
State Boundary ◽  
Initial Boundary Value

In the present paper, we propose an analytical approach for solving the 3D unsteady-state boundary-value problem for the second-order parabolic equation with the second and third types boundary conditions in two-layer rectangular parallelepipedic domain.

Definitions of solutions to the IBVP for multi-dimensional scalar balance laws

2018 ◽  
Vol 15 (02) ◽  
pp. 349-374 ◽  
Author(s):  
Elena Rossi
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
General Framework ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Balance Laws ◽  
Initial Boundary Value

We consider four definitions of solution to the initial-boundary value problem (IBVP) for a scalar balance laws in several space dimensions. These definitions are extended to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.

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.

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