The Cauchy problem for a two-component generalized \(\theta \)-equations
type: Item
instance of: scholarly article
title: The Cauchy problem for a two-component generalized \(\theta \)-equations
zbMATH Open document ID: 1195.37052
author: Lidiao Ni
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-07-13T00:00:00Z
review text: The paper is concerned with the third-order nonlinear system of PDEs, named two-component generalized \(\theta\)-equations \[ \begin{cases} u_t-u_{xxt}+uu_x-(1-\theta)u_xu_{xx}-\theta uu_{xxx}+\sigma \rho\rho_x=0,\,\,\,x\in\mathbb{R},\,\,\,t>0,\\ \rho_t+\theta \rho_x u+(1-2\theta)\rho u_x=0,\,\,\,x\in\mathbb{R},\,\,\,t>0,\,\,\,\sigma\in\{1,-1\},\end{cases} \] with the initial data \(u(x,0)=u_0(x),\,\,\,\rho(x,0)=\rho_0(x),\,\,\,x\in\mathbb{R}\), which arises in shallow water theory. The author gives sufficient conditions on the initial data \(u_0,\,\rho_0\) to guarantee wave breaking for the above system. He also analyzes some persistence properties of the strong solutions, and then he presents a detailed description of the infinite propagation speed on the solution in its lifespan. More precisely, the author proves that the solution \(u\) does not have compact spatial support for \(t>0\) though \(u_0\in C_0^{\infty}(\mathbb{R})\), while the other one \(\rho\) is always compactly supported under the assumption of its initial compact support condition.
reviewed by: Rodica Luca
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 26A12
Mathematics Subject Classification ID: 58E35
zbMATH DE Number: 5736453
zbMATH Keywords: two-component \(\theta\)-equations
zbMATH Keywords: blow-up
zbMATH Keywords: persistence properties
zbMATH Keywords: propagation speed
MaRDI profile type: Publication
full work available at URL: j.na.2010.04.064
OpenAlex ID: W1976562005
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DOI: 10.1016/J.NA.2010.04.064
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description: scientific article; zbMATH DE number 5736453