scholarly journals Global Well-Posedness for a Class of Kirchhoff-Type Wave System

2017 ◽  
Vol 2017 ◽  
pp. 1-18
Author(s):  
Xiaoli Jiang ◽  
Xiaofeng Wang
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Wave System ◽  
Well Posedness ◽  
Initial Value ◽  
Kirchhoff Type ◽  
Type Wave ◽  
Initial Boundary Value

In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars’ attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.

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.

Blow up of Solution for the Generalized Boussinesq Equation with Damping Term

2006 ◽  
Vol 61 (5-6) ◽  
pp. 235-238
Author(s):  
Necat Polat ◽  
Doğan Kaya
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Damping Term ◽  
Initial Boundary Value ◽  
Generalized Boussinesq Equation

We consider the blow up of solution to the initial boundary value problem for the generalized Boussinesq equation with damping term. Under some assumptions we prove that the solution with negative initial energy blows up in finite time

scholarly journals Extinction and Positivity of the Solutions for a -Laplacian Equation with Absorption on Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Qiao Xin ◽  
Chunlai Mu ◽  
Dengming Liu
Keyword(s):  
Boundary Value Problem ◽  
Numerical Experiment ◽  
Finite Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Simple Graph ◽  
Weighted Graphs ◽  
Laplacian Equation ◽  
Initial Boundary Value ◽  
Standard Weight

We deal with the extinction of the solutions of the initial-boundary value problem of the discretep-Laplacian equation with absorption withp> 1,q> 0, which is said to be the discretep-Laplacian equation on weighted graphs. For 0 <q< 1, we show that the nontrivial solution becomes extinction in finite time while it remains strictly positive for , and . Finally, a numerical experiment on a simple graph with standard weight is given.

scholarly journals Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky–Hunter equation

2015 ◽  
Vol 12 (02) ◽  
pp. 221-248 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Boundary Value Problem ◽  
Nonlinear Evolution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Depth ◽  
Well Posedness ◽  
Rotating Fluid ◽  
Bounded Solutions ◽  
Initial Boundary Value

The Ostrovsky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. Here the well-posedness of bounded solutions for a non-homogeneous initial-boundary value problem associated with this equation is studied.

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