The heat equation and the shrinking

Abstract

<p>This article concerns the Cauchy problem for the partial differential equation</p> <pre>∂₁u(t, x) - α∂²₂u(t, x) = ƒ(t, x, ∂^p₂ u(μ(t) t, x), ∂^q₂ u(t, v(t)x)).</pre> <p>Here t and x are real variables, p and q are positive integers greater than 1, and the <i>shrinking factors</i> μ(t), v(t) are positive-valued functions such that their suprema are less than 1.</p>