Some results of Gevrey and analytic regularity for semilinear weakly hyperbolic equations of Oleinik type (Q1915411)

We consider a real solution, \(u(t,x):[0,T) \times\mathbb{R}^n_x \to\mathbb{R}\), of the semilinear equation \(L(u)= f(t,x,u)\), where \(L(u)\) is the linear weakly hyperbolic operator defined by \[ L(u) \equiv u_{tt}-\sum^n_{i,j=1} \bigl(a_{ij} (t,x)u_{x_i} \bigr)_{x_j} +\sum^n_{j=1} b_j(t,x)u _{x_j} +b_0(t,x) u_t+ c(t,x)u. \] We suppose that \(a_{ij}\), \(b_j\), \(\partial_tb_0\), \(c\in C^0 ([0,T)\); \(\gamma^{(s)} (\mathbb{R}^n_x))\), \[ 0\leq\sum^n_{i,j=1} a_{ij} \xi_i\xi_j \leq\lambda |\xi |^2\quad (\forall \xi\in\mathbb{R}^n) \] and for positive constants \(A\), \(B>0\), \(\forall \xi\in \mathbb{R}^n\) \[ B\cdot \left(\sum^n_{j=1} b_j(t,x) \xi_j \right)^2\leq A\cdot \sum_{i,j} a_{i,j} (t,x)\xi_i \xi_j+ \sum_{i,j} \partial_ta_{i,j} (t,x)\xi_i \xi _j. \] while the nonlinear term, \(f(t,x,u): [0,T)\times \mathbb{R}^n_x \times\mathbb{R}_u \to\mathbb{R}\), satisfies \(f\in C^0 ([0,T)\); \(\gamma^{(s)} (\mathbb{R}^n_x \times \mathbb{R}_u))\), where \(\gamma^{(s)}\) is the space of Gevrey functions of order \(s\). Under these hypotheses, we prove the following result: A solution \(u(t,x): [0,T)\times \mathbb{R}^n_x \to\mathbb{R}\) of class \(C^2\), belongs to \(C^2([0,T)\); \(\gamma^{(s)} (\mathbb{R}^n_x))\) as soon as \(u(0,x)\), \(u_t(0,x) \in\gamma^{(s)} (\mathbb{R}^n_x)\).