Unified transform method for nonautonomous linear evolution equations: rigorous analysis and qualitative theory

Abstract

Established herein is the rigorous extension of the celebrated Fokas unified transform method for linear evolution PDE with time-dependent coefficients. As illustrative examples, considered below and rigorously analysed in detail are non-homogeneous initial-boundary-value problems, formulated on the half-line, for the heat equation and the modified heat (also widely known as Barenblatt's pseudo-parabolic) equation with reaction terms and temporally varying coefficients. Solution formulae are derived formally via suitable implementation of the UTM, and rigorously verified a posteriori, in the classical sense. The latter task relies on complex-analytic tools and careful interpretation of the obtained integral representations. The explicit solutions established here are then used for investigating qualitative properties. These include regularity, boundary behavior near the axes of the domain as well as long-range and eventual periodicity asymptotics. Notably, smoothness of the solution up to the boundaries depends on compatibility conditions between initial, boundary and forcing data, into which interjected now are initial values of the coefficients. Further results pertaining to solution uniqueness, analyticity and controllability of the addressed problems are thence deduced in light of the preceding qualitative theory.