Gevrey regularity in time of solutions to nonlinear partial differential equations (Q2902451)

The author considers nonlinear partial differential equations of the form NEWLINE\[NEWLINEt^k D^m_tu= G(t,x,D^j_t D^\alpha_x u)_{j< m,|\alpha|\leq L},NEWLINE\]NEWLINE with \(k\geq 0\). The analytic and Gevrey regularity of the solutions \(u(t,x)\) is studied, under suitable assumptions on the nonlinearity \(G\). Namely, the problem under consideration is the following: Let \(u(t,x)\) be a solution of the equation, with \(C^\infty\) regularity with respect to the time variable \(t\), Gevrey with respect to the space variables \(x\). Can we conclude Gevrey regularity also with respect to the variable \(t\)?NEWLINENEWLINE Precise results are given, improving the results of the author [J. Fac. Sci., Univ. Tokyo, Sect. I A 39, No. 3, 555--582 (1992; Zbl 0774.35044)] for singular hyperbolic equations. As interesting example of application, we also mention the generalized KdV equation NEWLINE\[NEWLINE\partial_t u=\partial^M_x u+ u^p\partial^L_x u,\quad u(0,x)= u_0(x).NEWLINE\]