Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 97, pp. 1-14.

Title: The heat equation and the shrinking

Authors: Masaki Kawagishi (Nihon University, Tokyo, Japan)
 Takesi Yamanaka (Nihon University, Tokyo, Japan)

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