Global existence for quasi-linear dissipative wave equations with large data and small parameter
We consider the hyperbolic quasi-linear initial value problem for \[ u=u(x,t),\quad v=v(x,t)\in {\mathbb{R}}^ 1,\quad x\in {\mathbb{R}},\quad t\in {\mathbb{R}}^+,\quad \epsilon >0, \] \[ \epsilon u_ t+u-\delta (v)_ x=0;\quad v_ t-u_ x=0, \] \[ u(x,0)=u_ 0(x);\quad v(x,0)=v_ 0(x), \] where \(\delta\) : \({\mathbb{R}}\to {\mathbb{R}}\) is smooth and satisfies \(\delta '(0)>0\), \(\delta\) ''(0)\(\neq 0\). We show that the smooth solution of such problem exists globally in time, provided that the \(L^{\infty}\) norm of \(u_ 0\), \(v_ 0\) and the parameter \(\epsilon\) be sufficiently small. In particular, the \(L^{\infty}\) norm of \(u_{0x}\) and \(v_{0x}\) is not required to be small, but the smallness of \(\epsilon\) will depend on the size of such norm.
scientific article; zbMATH DE number 4058065