The Global Existence of One Type of Nonlinear Kirchhoff String Equation

The Global Existence of One Type of Nonlinear Kirchhoff String Equation In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

The Global Existence of One Type of Nonlinear Kirchhoff String Equation

The Global Existence of One Type of Nonlinear Kirchhoff String Equation

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 477–484 The Global Existence of One Type of Nonlinear Kirchhoff String Equation Xiao-yi Zhang Beijing Graduate School of China Academy of Engineering Physics, Beijing 100088, China (E-mail: quanliuxyz@yahoo.com.cn) Abstract In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method. Keywords Kirchhoff equation, global well-posedness 2000 MR Subject Classification 35L70, 35B10, 37L10 1 Introduction Our purpose of this paper is to prove the existence and uniqueness of the solution to the mixed problem of the nonlinear Kirchhoff string equation 2 m−1 p−1 u − a |u| dx ∆ u + b |u | u = b |u| u, (x, t) ∈ Ω × (0,∞), (1.1) tt 1 t t 2 u(x, 0) = u (x),u (x, 0) = u (x),u(x, t)| =0. (1.2) 0 t 1 ∂Ω n ∂u Here Ω denotes a smooth bounded domain in R . u = , m> 1, p> 1, b ≥ 0, b ≥ 0, a(r) t 1 2 ∂t is a C function on [0,∞) such that a(r) ≥ a > 0, (1.3) m−1 for some a > 0. Roughly speaking, in equation (1.1), b |u | u is called the damping term, 0 1 t t its presence leads to the global existence of the solution. On the contrary, the source term p−1 b |u| u tends to generate blowing up. Thus, the global well posedness of (1.1)–(1.2) depends [2] on the interaction of the two terms. For instance, when b = b = 1, Georgiev and Todorova 1 2 proved the global existence and uniqueness when m ≥ p and the possibility of blowing up when m< p of the solution to the mixed problem of the equation m−1 p−1 u − ∆ u + |u | u = |u| u. (1.4) tt t t [3] Ikehata gave the decay estimate of the energy u (t) + ∇u(t) as t tends to infinity. t 2 2 [9] Following thesamevein, Ono extended the above results to the mixed problem of the equation 2 m−1 u − M |∇u|...
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2003 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0123-1
Publisher site
See Article on Publisher Site

Abstract

In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

References

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