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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
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