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On the Cauchy problem for non linear PDEs in the Gevrey class with shrinkings. (English) Zbl 1032.35059
Authors’ 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.

##### MSC:
 35G25 Initial value problems for nonlinear higher-order PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35A10 Cauchy-Kovalevskaya theorems 35R10 Partial functional-differential equations
##### Keywords:
delay; Gevrey function; contraction principle
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