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The heat equation and the shrinking. (English) Zbl 1039.35048
Consider the differential equation $\partial_1u(t,x)-a\partial_2^2u(t,x)= f(t,x,\partial_2^pu(\mu(t)t,x),\partial_2^qu (t,\nu(t)x)) \tag{1}$ with the initial value $u(0,x)=0,\tag{2}$ which is equivalent to an integral equation $u(t,x)=\int^t_0d\tau\int_{-\infty}^{\infty}G(t-\tau,x-s)f(\tau,s,\partial_2^pu(\mu(\tau)\tau,s), \partial_2^qu(\tau,\nu(\tau)s))\,ds,\tag{3}$ where $G(t,x)=\frac{1}{2\sqrt{\pi at}}e^{-\frac{x^2}{4at}} \quad (t>0,-\infty<x<\infty).$ Assume that $$a$$ is a positive constant; $$\mu(t)$$ and $$\nu(t)$$ are positive-valued functions satisfying $$\sup_t\mu(t)<1$$ and $$\sup_t\nu(t)<1$$, which are called the time shrinking factor and the space shrinking factor, respectively; $$p$$ and $$q$$ denote integers greater than 1. The function $$f(t,x,v,w)$$ is assumed to be continuous and a Gevrey function. The result of this article is a theorem on the existence of solution for (1)-(2) or (3) which is a Gevrey function, too.

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 35K55 Nonlinear parabolic equations
##### Keywords:
shrinking factor; Cauchy problem; Gevrey function
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