A sixth order degenerate equation with the higher order p-laplacian operator

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Changchun Liu
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sixth Order ◽  
Laplacian Operator ◽  
Degenerate Equation ◽  
Discrete Method ◽  
Finite Speed Of Propagation ◽  
Order Degenerate ◽  
Speed Of Propagation ◽  
Initial Boundary Value

AbstractWe consider a initial-boundary value problem for a sixth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the time-discrete method. The uniqueness, asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed.

Initial-boundary value problem for a fractional type degenerate heat equation

2018 ◽  
Vol 28 (06) ◽  
pp. 1199-1231
Author(s):  
Gerardo Huaroto ◽  
Wladimir Neves
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Diffusion ◽  
Laplacian Operator ◽  
Initial Boundary Value ◽  
Fractional Laplacian Operator ◽  
Fractional Type

In this paper, we study a fractional type degenerate heat equation posed in bounded domains. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition. The nonlocal diffusion effect relies on an inverse of the [Formula: see text]-fractional Laplacian operator, and the solvability is proved for any [Formula: see text].

High order approximations in space and time of a sixth order Cahn–Hilliard equation

2015 ◽  
Vol 30 (6) ◽  
Author(s):  
Oleg Boyarkin ◽  
Ronald H. W. Hoppe ◽  
Christopher Linsenmann
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Sixth Order ◽  
Fourth Order Equation ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractWe consider an initial-boundary value problem for a sixth order Cahn-Hilliard equation describing the formation of microemulsions. Based on a Ciarlet-Raviart type mixed formulation as a system consisting of a second order and a fourth order equation, the spatial discretization is done by a C

scholarly journals Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 735 ◽  
Author(s):  
Dumitru Baleanu ◽  
Vladimir E. Fedorov ◽  
Dmitriy M. Gordievskikh ◽  
Kenan Taş
Keyword(s):  
Control System ◽  
Fractional Order ◽  
Approximate Controllability ◽  
Homogeneous Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Degenerate Equation ◽  
Fractional Order Control ◽  
Infinite Dimensional ◽  
Initial Boundary Value

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.

scholarly journals Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Changming Song ◽  
Jina Li ◽  
Ran Gao
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Type Equation ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Bond Number ◽  
Singularly Perturbed ◽  
Sixth Order ◽  
Initial Boundary Value

We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.

scholarly journals Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Zhao Wang ◽  
Changchun Liu
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sixth Order ◽  
Classical Solutions ◽  
Oil Water ◽  
Initial Boundary Value

We study the initial boundary value problem for a sixth-order Cahn-Hilliard-type equation which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. We show that the solutions might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time. We also discuss the existence of global attractor.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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