On the Gevrey wellposedness of the Cauchy problem for some non-Kowalewskian equations
This paper is concerned with the wellposedness of the Cauchy problem for the equation in \(\mathbb{R}^d\): \[ u_{tt}+a(t)\Delta^2 u+p(t,\partial_t, \partial_x)u=0, \] where \(a(t)\) is a real-valued nonnegative function. In the case of \(a(t)\equiv 1\) the global wellposedness is well known in the usual Sobolev space. This paper studies the case that \(a(t)\) is not constant and sufficiently smooth. So, the wellposedness has to be considered in the Gevrey space. Using Fourier analysis, energy estimates and known mathematical results for weakly hyperbolic equations, the authors prove the \(\gamma^s\)-wellposedness for \(p(t,\partial_t,\partial_x)=0\), or \(=-b(t)\Delta\) or \(=-c(t)\Delta\partial_t\) provided some assumptions on \(s\) are satisfied (see also \textit{P. D'Ancona} [Publ. Res. Inst. Math. Sci. 24, No. 3, 433-449 (1988; Zbl 0706.35077)] on the \(\gamma^s\)-wellposedness).
scientific article; zbMATH DE number 1463873