The Cauchy problem for a two-component generalized \(\theta \)-equations
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.
scientific article; zbMATH DE number 5736453
scientific article