Exact Solutions and Critical Behaviour for a Linear Growth-Diffusion Equation on a Time-Dependent Domain

Abstract

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form, and derive the explicit expression in each case.Next we prove the precise behaviour near the boundary in a 'critical' case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line.Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with time-dependent radius.

Reaction-diffusion on a time-dependent interval: refining the notion of 'critical length'

“…We consider the linear equation and begin as in [1] by transforming onto a fixed spatial domain and making a change of variables. Let ψ(x, t) = u(ξ, t) where ξ = (x−A(t)) L(t) L 0 (for some L 0 > 0) to get…”

“…As such L(t) is either constant in time, or L(t) → 0 in a finite time, or L(t) → ∞ as t → ∞. The other result of [1] concerns the long-time behaviour near the boundaries, when the length of the interval tends to infinity with its endpoints moving at ±(2 Df ′ (0)t − α log(t + 1) − η(t)) with α > 0 and η(t) = O(1). The approach in [1] is based upon changes of variables (both of the spatial variable x and the independent variable ψ) which allows exact solutions, subsolutions and supersolutions to be constructed for the transformed equation.…”

“…The recent paper [1] by the current author considers equations ( 1), (2) on a time-dependent interval, and presents several exact solutions for the linear case. These involve domains where L(t) has the form L(t) = at 2 + 2bt + L 2 0 .…”

“…The other result of [1] concerns the long-time behaviour near the boundaries, when the length of the interval tends to infinity with its endpoints moving at ±(2 Df ′ (0)t − α log(t + 1) − η(t)) with α > 0 and η(t) = O(1). The approach in [1] is based upon changes of variables (both of the spatial variable x and the independent variable ψ) which allows exact solutions, subsolutions and supersolutions to be constructed for the transformed equation. This is the method which will also be employed in the current study, but here we use different upper and lower bounds on the transformed equation to derive a new sub-and supersolution pair.…”