On the Cauchy problem for non linear PDEs in the Gevrey class with shrinkings.

*(English)*Zbl 1032.35059Authors’ summary: The Cauchy problems (1) and (2) below are studied:
\[
\partial_t u(x,t)= f\biggl(u(x,t), \partial^p_x u\bigl(x, \alpha (t) t\bigr), x,t\biggr), \quad u(x,0)=0;\tag{1}
\]

\[ \partial_t u(x,t)=f \Bigl( u(x, t), \partial_\xi^p u(\xi,t)|_{\xi=\alpha (x,t)x},x,t\Bigr),\quad u(x, 0)=0. \tag{2} \] In (1) and (2) \(u(x,t)\) is a real valued unknown function of real variables \(x\) and \(t\). \(f(v,w,x,t)\) is a continuous function of \((v,w,x,t)\) with \(|v|<R\), \(|w|<R\), \(|x|<r_0\), \(|t|<T_0\). It is assumed that \(f(v,w,x,t)\) is a Gevrey function of \((v,w,x)\). \(p\) is an arbitrary positive integer. \(\alpha(t)\) and \(\alpha(x,t)\) are continuous functions satisfying \(\sup|\alpha(t) |<1\) and \(\sup|\alpha(x,t) |<1\), respectively. For this reason they are called shrinking with respect to \(t\) and \(x\), respectively. It is further assumed that \(\alpha(x,t)\) is analytic in \(x\). Under these conditions it is proved by means of the contraction principle that there is a positive number \(T_1\leq T_0\) with the property that each of the problems (1) and (2) has a unique solution \(u:(-r_0, r_0)\times (-T_1,T_1) \to\mathbb{R}\) such that \(u(x,t)\) is Gevrey in \(x\). The Cauchy problem (1) and (2) above are of the same type as those treated in Kawagishi’s earlier paper [Proc. Japan Acad. Ser. A, 75, 184-187 (1999; Zbl 0969.35007)]. In [loc. cit.], however, \(f(v,w,x,t)\) was assumed to be analytic in \((v,w,x)\), while in the present paper it is assumed to be Gevrey in \((v,w,x)\). The word shrinking was used in [loc.cit.] for the first time.

\[ \partial_t u(x,t)=f \Bigl( u(x, t), \partial_\xi^p u(\xi,t)|_{\xi=\alpha (x,t)x},x,t\Bigr),\quad u(x, 0)=0. \tag{2} \] In (1) and (2) \(u(x,t)\) is a real valued unknown function of real variables \(x\) and \(t\). \(f(v,w,x,t)\) is a continuous function of \((v,w,x,t)\) with \(|v|<R\), \(|w|<R\), \(|x|<r_0\), \(|t|<T_0\). It is assumed that \(f(v,w,x,t)\) is a Gevrey function of \((v,w,x)\). \(p\) is an arbitrary positive integer. \(\alpha(t)\) and \(\alpha(x,t)\) are continuous functions satisfying \(\sup|\alpha(t) |<1\) and \(\sup|\alpha(x,t) |<1\), respectively. For this reason they are called shrinking with respect to \(t\) and \(x\), respectively. It is further assumed that \(\alpha(x,t)\) is analytic in \(x\). Under these conditions it is proved by means of the contraction principle that there is a positive number \(T_1\leq T_0\) with the property that each of the problems (1) and (2) has a unique solution \(u:(-r_0, r_0)\times (-T_1,T_1) \to\mathbb{R}\) such that \(u(x,t)\) is Gevrey in \(x\). The Cauchy problem (1) and (2) above are of the same type as those treated in Kawagishi’s earlier paper [Proc. Japan Acad. Ser. A, 75, 184-187 (1999; Zbl 0969.35007)]. In [loc. cit.], however, \(f(v,w,x,t)\) was assumed to be analytic in \((v,w,x)\), while in the present paper it is assumed to be Gevrey in \((v,w,x)\). The word shrinking was used in [loc.cit.] for the first time.