scholarly journals SOLVABILITY OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE QUASILINEAR EQUATION OF HEAT CONDUCTIVITY IN DOMAINS THAT CAN BE TRANSFORMED INTO RECTANGLES

2020 ◽  
Vol 70 (2) ◽  
pp. 36-46
Author(s):  
S.E. Aytzhanov ◽  
◽  
S.Z. Saidalimov ◽  
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Quasilinear Equation ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Existence Of A Solution ◽  
Initial Boundary Value

In this paper, we study the initial-boundary-value problem for the quasilinear heat equation in regions that are reduced to rectangular. Mathematical modeling of many processes taking place in the real world leads to the study of the problems of equations of mathematical physics, when the areas are not rectangular. The theory of nonlinear problems is an actively developing section of the theory of modern differential equations. In the theory of nonlinear equations, a special place is occupied by the study of unbounded solutions or, in other words, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally unsolvable: solutions grow unlimitedly over a finite period of time. In this paper, the initial-boundary-value problem for the quasilinear heat equation in regions that can be reduced to rectangular ones, the existence of a solution is proved by the Galerkin method. The uniqueness of the solution was proved by the obtained a priori estimates. Sufficient conditions for the destruction of the solution in a finite time in a bounded domain are obtained. The exponential decay of the solution with an infinite increase in time is proved. In the final time, it was proved that the solution is localized, i.e. disappears (nullifies).

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.

GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO THE MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

2009 ◽  
Vol 19 (07) ◽  
pp. 1099-1138 ◽  
Author(s):  
ZHI-QIANG SHAO
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Local Existence ◽  
Discontinuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quasilinear Hyperbolic Systems ◽  
Initial Boundary Value

In this paper, we consider the mixed initial–boundary value problem for first-order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0}. Based on the fundamental local existence results and global-in-time a priori estimates, we prove the global existence of a unique weakly discontinuous solution u = u(t, x) with small and decaying initial data, provided that each characteristic with positive velocity is weakly linearly degenerate. Some applications to quasilinear hyperbolic systems arising in physics and other disciplines, particularly to the system describing the motion of the relativistic closed string in the Minkowski space R1+n, are also given.

scholarly journals Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

2021 ◽  
Vol 65 (3) ◽  
pp. 17-24
Author(s):  
Ravshan Ashurov ◽  
◽  
Oqila Muhiddinova
Keyword(s):  
Boundary Value Problem ◽  
Initial Function ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Subdiffusion Equation ◽  
Initial Boundary Value ◽  
The Right

An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo.

scholarly journals INITIAL-BOUNDARY VALUE PROBLEM FOR HYPERBOLIC EQUATIONS WITH AN ARBITRARY ORDER ELLIPTIC OPERATOR

2020 ◽  
pp. 8-19
Author(s):  
Р.Р. Ашуров ◽  
А.Т. Мухиддинова
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Order Elliptic Operator ◽  
Initial Boundary Value

В настоящей работе исследуется начально-краевые задачи для гиперболических уравнений, эллиптическая часть которых имеет наиболее общий вид и определена в произвольной многомерной области (с достаточно гладкой границей). Установливаются требования на правую часть уравнения и начальные функции, при которых к рассматрываемую задачу применим классический метод Фурье. Другими словами, доказывается методом Фурье существование и единственность решения смешанной задачи и показана устойчивость найденного решения от данных задачи: от начальных функций и правой части уравнения. Введено понятие обобщенного решения и доказана теорема о его существования. Аналогичные результаты справедливы и для параболических уравнений. An initial-boundary value problem for a hyperbolic equation with the most general elliptic differential operator, defined on an arbitrary bounded domain, is considered. Uniqueness, existence and stability of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. The notion of a generalized solution is introduced and existence theorem is proved. Similar results are formulated for parabolic equations too.

Sign in / Sign up

Export Citation Format

Share Document