Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39698

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

Andreas Chatziafratis·Spyridon Kamvissis·Alain Miranville·Tohru Ozawa
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.APmath-phmath.CVmath.DSmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andreas Chatziafratis, Spyridon Kamvissis, Alain Miranville, Tohru Ozawa. 2026-10-29. Unified transform method for nonautonomous linear evolution equations: rigorous analysis and qualitative theory. https://arxiv.org/abs/2610.39698

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain $\mathbb{T}\times \mathbb{R}$. We begin by directly investigating the Couette shear flow, where we characterize the linear growth of the compressible part of the fluid while proving time decay for the incompressible part (inviscid damping with slower rates). Then we extend the analysis to monotone shear flows near Couette, where we are able to give an upper bound, superlinear in time, for the compressible part of the fluid. The incompressible part enjoys an inviscid damping property, analogous to the Couette case. In the pure Couette case, we exploit the presence of an additional conservation law (which connects the vorticity and the density on the moving frame) in order to reduce the number of degrees of freedom of the system. The result then follows by using weighted energy estimates. In the general case, unfortunately, this conservation law no longer holds. Therefore we define a suitable weighted energy functional for the whole system, which can be used to estimate the irrotational component of the velocity but does not provide sharp bounds on the solenoidal component. However, even in the absence of the aforementioned additional conservation law, we are still able to show the existence of a functional relation which allows us to recover somehow the vorticity from the density, on the moving frame. By combining the weighted energy estimates with the functional relation we also recover the inviscid damping for the solenoidal component of the velocity.

math.AP↗
arXiv2026

The evolution equation and the eigenvalue problem for the Laplacian in a regular tree

In this paper, our main goal is to study the evolution problem associated with the Laplacian operator with Dirichlet boundary conditions on a regular tree. To this end, we place special emphasis on the associated first eigenvalue problem, which provides the fundamental tool for describing the long-time dynamics. First, we prove existence and uniqueness of solutions when the initial condition is compatible with the boundary condition. Next, we address the asymptotic behavior of the solutions and show that they decay to zero exponentially fast. This decay rate is determined by the associated first eigenvalue, which we also analyze in detail.

math.AP↗
arXiv2026

Invariant Densities on Cylinders with an Interface: Existence, Uniqueness, and Exponential Asymptotics

We study the stationary double-divergence equation on the infinite cylinder $\mathbb R\times\mathbb T^{d-1}$, whose coefficients coincide with two possibly different periodic media outside a bounded interface layer. Under the centering conditions for the two periodic media, we construct a positive invariant density that converges exponentially in $C^1$ at each end to a positive multiple of the corresponding periodic invariant density. Every bounded distributional solution is a scalar multiple of this density. The two asymptotic amplitudes are determined by the full interface problem, up to a common normalization. Motivated by the invariant-measure analysis of Hairer and Manson [Ann. Probab. 39(2011) 648-682], our proof uses only PDE methods: a transformation based on the periodic invariant densities, adjoint Neumann problems on finite cylinders, and energy estimates.

math.AP↗