scholarly journals A viscosity solution of the Hamilton–Jacobi equation with exponential dependence of Hamiltonian on the momentum

2021 ◽  
pp. 273-276
Author(s):  
Lyubov Shagalova
Keyword(s):  
Viscosity Solution ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
The State ◽  
Exponential Dependence ◽  
Time Interval ◽  
Momentum Variable ◽  
Initial Boundary Value

The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.

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.

scholarly journals Parabolic problems in generalized Sobolev spaces

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Valerii Los ◽  
Vladimir Mikhailets ◽  
Aleksandr Murach
Keyword(s):  
Sobolev Spaces ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Local Regularity ◽  
Parabolic Problems ◽  
Function Parameter ◽  
Generalized Derivatives ◽  
Initial Boundary Value

<p style='text-indent:20px;'>We consider a general inhomogeneous parabolic initial-boundary value problem for a <inline-formula><tex-math id="M1">\begin{document}$ 2b $\end{document}</tex-math></inline-formula>-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers <inline-formula><tex-math id="M2">\begin{document}$ s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ s/(2b) $\end{document}</tex-math></inline-formula> and with a function <inline-formula><tex-math id="M4">\begin{document}$ \varphi:[1,\infty)\to(0,\infty) $\end{document}</tex-math></inline-formula> that varies slowly at infinity. The function parameter <inline-formula><tex-math id="M5">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set.</p>

scholarly journals On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

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 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 On Spectral and Evolutional Problems Generated by a Sesquilinear Form

2020 ◽  
Vol 66 (2) ◽  
pp. 335-371
Author(s):  
A. R. Yakubova
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Superposition Principle ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Time Interval ◽  
Sesquilinear Form ◽  
Two Parameters ◽  
Initial Boundary Value ◽  
Necessary And Sufficient

On the base of boundary-value, spectral and initial-boundary value problems studied earlier for the case of single domain, we consider corresponding problems generated by sesquilinear form for two domains. Arising operator pencils with corresponding operator coefficients acting in a Hilbert space and depending on two parameters are studied in detail. In the perturbed and unperturbed cases, we consider two situations when one of the parameters is spectral and the other is fixed. In this paper, we use the superposition principle that allow us to present the solution of the original problem as a sum of solutions of auxiliary boundary-value problems containing inhomogeneity either in the equation or in one of the boundary conditions. The necessary and sufficient conditions for the correct solvability of boundary-value problems on given time interval are obtained. The theorems on properties of the spectrum and on the completeness and basicity of the system of root elements are proved.

scholarly journals INITIAL-BOUNDARY VALUE PROBLEM FOR ONE-DIMENSION HYPERBOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

2017 ◽  
Vol 17 (8) ◽  
pp. 95-101
Author(s):  
M.V. Strigun
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Integral Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Integral Boundary ◽  
Nonlocal Integral Condition ◽  
Initial Boundary Value

In this paper, we study an initial-boundary value problem with nonlocal integral condition for a hyperbolic equation. The existence and uniqueness of a generalized solution of the problem is proved.

Sign in / Sign up

Export Citation Format

Share Document