Abstract differential equations and nonlinear dispersive systems (Q1308508)

Let \(M\) be a closed operator \(D(M)\subset X\to X\), where \(X\) is a Banach space. We shall assume that \(M\) is invertible with a bounded inverse \(M^{-1}\). Let \(Y\) be another Banach space such that \[ D(M)\subset Y\subset X\qquad\text{with continuous injections}.\tag{1.1} \] Let \(f\) be an operator \(f:[0,\infty)\times Y\to X\), where \(f(t,u)\) is continuous in \(t\geq 0\) and locally Lipschitz continuous in the variable \(u\), with the Lipschitz constant uniform in \(t\) for \(t\in[0,T]\), any \(T<\infty\). We study questions related to local existence, a priori bounds (global existence) and regularity of the solutions to the following Cauchy problem \[ \begin{cases} Mu_ t(t)=f(t,u(t)),\quad t\geq 0, \\ u(0)=u_ 0\in D(M). \end{cases} \tag{1.2} \] There are several equations of physical interest which can be recasted in the form (1.2), for example, the Benjamin-Bona-Mahony equation and its extensions. We quote in a particular way the very recent paper by \textit{J. A. Goldstein}, \textit{R. Kajikiya} and \textit{S. Oharu}, Differ. Integral Equ. 3, No. 4, 617-632 (1990; Zbl 0735.35103) because in fact our abstract results give many of their assertions when applied to that dispersive equation.