Global solution of the Cauchy problem for a class of abstract nonlinear hyperbolic equations
The author treats the Cauchy problem in a general situation including the following equation: \[ u_{tt}-(\int_{\Omega}u^ 2_ xdx dy)^ pu_{xx}-(\int_{\Omega}u^ 2_ ydx dy)^ qu_{yy}=0 \] \[ u(x,y,0)=u_ 0(x,y);\quad u_ t(x,y,0)=u_ 1(x,y) \] where \(\Omega\) is a given rectangle in \(R^ 2\), and p,q\(\geq 1\). Consider the following nonlinear equation: \[ u_{tt}-\phi_ 1(u)A_ 1u+\phi_ 2(u)A_ 2u=0;\quad u(0)=u_ 0;\quad u'(0)=u_ 1, \] where \(A_ 1,A_ 2\) are nonnegative symmetric operators on a Hilbert space, while \(\phi_ 1(u),\phi_ 2(u)\) are nonnegative functions. This is a generalization of preceding research for the form \[ \phi_ 1(u)=f_ 1(<Pu,u>),\quad and\quad \phi_ 2(u)\equiv 0, \] by e.g. \textit{R. W. Dickey} [J. Differ. Equations 8, 16-26 (1970; Zbl 0218.34016)]. The main result is stated as follows: Let \(A_ i(i=1,2)\) be bounded symmetric nonnegative linear operator in a suitably defined Hilbert space to its dual, and \(\phi_ 1(\cdot)\), \(\phi_ 2(\cdot)\) be Lipschitz continuous. In this case the global well posedness can be derived for this Cauchy problem.
scientific article; zbMATH DE number 4118818
scientific article