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

Abstract

Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of order $\alpha \in (1,2)$ 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 $t\gg 1$ 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.

arXiv comment15 pages, 1 figure

References in corpus (9)

• Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients
• Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
• Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
• Strong Maximum Principle for Multi-Term Time-Fractional Diffusion Equations and its Application to an Inverse Source Problem
• Reconstruction of the Temporal Component in the Source Term of a (Time-Fractional) Diffusion Equation
• Initial-boundary value problems for coupled systems of time-fractional diffusion equations
• Inverse moving source problem for time-fractional evolution equations: Determination of profiles
• Inverse moving source problem for fractional diffusion(-wave) equations: Determination of orbits
• Well-posedness of initial-boundary value problem for time-fractional diffusion-wave equation with time-dependent coefficients

Cited by in corpus (2)

• Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems
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