The heat equation and the shrinking

Abstract

This article concerns the Cauchy problem for the partial differential equation ∂1u(t, x) - α∂2 2u(t, x) = ƒ (t, x, ∂p 2 u (μ(t) t, x), ∂q 2 u (t, v(t)x)). Here t and x are real variables, p and q are positive integers greater than 1, and the shrinking factors μ(t), v(t) are positive-valued functions such that their suprema are less than 1.