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March/April 2016 Global well-posedness and singularity propagation for the BBM-BBM system on a quarter plane
Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia
Adv. Differential Equations 21(3/4): 301-332 (March/April 2016).

Abstract

Nonlinear, dispersive wave equations arise as models of various physical phenomena. A major preoccupation on the mathematical side of the study of such equations has been to settle the fundamental issues of local and global well-posedness in Hadamard's classical sense. The development so far has been mostly for the initial-value problem for single equations.

However, systems of such equations have also received consideration, and there is now theory for pure initial-value problems where data are given on the entire space or on the torus. Here, consideration is given to non-homogeneous initial-boundary-value problems for a class of BBM-type systems having the form \begin{equation*} \begin{aligned} u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \\ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \end{aligned} \end{equation*} where $P$ and $Q$ are homogeneous, quadratic polynomials, $u$ and $v$ are real-valued functions of a spatial variable $x$ and a temporal variable $t$, and subscripts connote partial differentiation. Local in time well-posedness is established in the quarter plane $\{(x,t): x \geq 0, \, t \geq 0 \}$. Under certain restrictions on the coefficients of the nonlinearities $P$ and $Q$, global well posedness is also shown to obtain.

Citation

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Jerry L. Bona. Hongqiu Chen. Chun-Hsiung Hsia. "Global well-posedness and singularity propagation for the BBM-BBM system on a quarter plane." Adv. Differential Equations 21 (3/4) 301 - 332, March/April 2016.

Information

Published: March/April 2016
First available in Project Euclid: 18 February 2016

zbMATH: 1382.35207
MathSciNet: MR3461296

Subjects:
Primary: 35M33 , 35Q35 , 35Q51 , 35Q53 , 45G15 , 76B03 , 76B15 , 76B25 , 76B55

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 3/4 • March/April 2016
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