# 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

, Volume 19 (3) – Mar 3, 2017

## 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|...

/lp/springer_journal/the-global-existence-of-one-type-of-nonlinear-kirchhoff-string-yIvom0zCUx
Publisher
Springer Journals
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

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