Long-time asymptotic estimate and a related inverse source problem for time-fractional wave equations

TL;DR

This paper studies time-fractional wave equations of order $\alpha\in(1,2)$ that interpolate between parabolic and hyperbolic dynamics and exhibit both decay and oscillation. It derives a long-time asymptotic formula for the homogeneous problem, $u(\cdot,t)=\sum_{j=0}^1 \frac{\mathcal{A}^{-1}u_j}{\Gamma(j+1-\alpha)} t^{j-\alpha}+O(t^{1-2\alpha})$ in $\mathcal{D}(\mathcal{A}^{\beta+1})$ as $t\to\infty$, showing the $t^{1-\alpha}$-term (driven by $u_1$) dominates the decay over $t^{-\lpha}$ (driven by $u_0$). From this, the authors obtain long-time sign information: the solution tends to the sign determined by $\mathcal{A}^{-1}u_1$ when nonzero, or by $\mathcal{A}^{-1}u_0$ in the distinguished case $\mathcal{A}^{-1}u_1=0$. Additionally, the paper proves a uniqueness result for an inverse source problem determining the temporal factor $\rho(t)$ from a single spatial observation, provided $\mathcal{A}^{-1}f(\mathbf x_0)\neq 0$ (and under related sign conditions on $f$), via the fractional Duhamel principle and spectral analysis.

Abstract

Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of order $α\in(1,2)$ in time inherit both decaying and oscillating properties. In this article, we establish a long-time asymptotic estimate for homogeneous time-fractional wave equations, which readily implies the strict positivity/negativity of the solution for $t\gg1$ under some sign conditions on initial values. As a direct application, we prove the uniqueness for a related inverse source problem on determining the temporal component.

Figures (1)

• Figure 1: Plots of Mittag-Leffler functions $E_{\alpha,1}(-t^\alpha)$ (a) and $t\,E_{\alpha,2}(-t^\alpha)$ (b) with several choices of $\alpha\in(1,2]$.