Electron. J. Diff. Eqns., Vol. 2003(2003), No. 97, pp. 1-14.

The heat equation and the shrinking

Masaki Kawagishi & Takesi Yamanaka

Abstract:
This article concerns the Cauchy problem for the partial differential equation
$$ \partial_1 u(t,x)-a\partial_2^2 u(t,x) =f(t,x,\partial_2^p u(\mu(t)t,x),\partial_2^q u(t,\nu(t)x))\,. $$
Here $t$ and $x$ are real variables, $p$ and $q$ are positive integers greater than 1, and the shrinking factors $\mu(t)$, $\nu(t)$ are positive-valued functions such that their suprema are less than 1.

Submitted April 1, 2003. Published September 17, 2003.
Math Subject Classifications: 35K05, 35K55, 35R10, 49K25.
Key Words: Partial differential equation, heat equation, shrinking, delay, Gevrey

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Masaki Kawagishi
College of Science and Technology, Nihon University
Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
email: masaki@suruga.ge.cst.nihon-u.ac.jp
Takesi Yamanaka
College of Science and Technology, Nihon University
Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
email: yamanaka@math.cst.nihon-u.ac.jp

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