[Paper Review] Asymptotic estimates of solutions to time-fractional diffusion equations with space-dependent variable order

Yavar Kian, Diomba Sambou+1 more|arXiv (Cornell University)|Jan 9, 2019

Fractional Differential Equations SolutionsMathematics20 references3 citations

TL;DR

This paper establishes asymptotic estimates for solutions to time-fractional diffusion equations with space-dependent variable order (VO) in ℝᵈ. Using complex analysis and Fourier multiplier techniques, it derives sharp time-decay bounds: for long times (t > 1), the L²-norm decays as t^{-(αₘ - α* (1 - d/s))}, and for short times (t ≤ 1), it decays as t^{αₘ - αₘₐₓ}, revealing how spatial heterogeneity in the fractional order affects diffusion dynamics.

ABSTRACT

We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.

Motivation & Objective

• To extend existence and uniqueness results for time-fractional diffusion equations with variable order (VO) from bounded domains to the whole space ℝᵈ.
• To analyze the short-time (t → 0⁺) and long-time (t → ∞) asymptotic behavior of solutions to VO time-fractional diffusion equations.
• To quantify how compactly supported spatial inhomogeneity in the fractional order α(x) affects the decay rate of the solution in L²(ℝᵈ).
• To establish sharp L²-norm estimates for the solution under minimal regularity assumptions on the initial data u₀.

Proposed method

• The analysis employs the Fourier transform and complex analysis techniques, particularly the use of a sectorial contour in the complex plane to represent the solution via inverse Laplace transform.
• The solution is decomposed into three parts: u⁰(t,·) corresponding to the real axis and u±(t,·) corresponding to the two rays of the sector, enabling separate estimation.
• Estimates are derived using the boundedness of Fourier multipliers associated with the resolvent operator (−Δ + z^α)^{-1} for z in the sector, with bounds depending on the variable order α(x) ∈ [αₘ, αₘₐₓ].
• The decay rates are obtained by estimating oscillatory integrals involving e^{tr cos θ} and power-law weights r^{α(x) - 1}, exploiting the homogeneity of the fractional Laplacian.
• The Marcinkiewicz interpolation theorem is applied to extend L² estimates to initial data in L^{2s′/(2s′−1)}(ℝᵈ), ensuring broader applicability.
• A density argument is used to extend the results from L¹ ∩ L² to the full function space L² ∩ L^{2s′/(2s′−1)}.

Experimental results

Research questions

• RQ1 How does the time-asymptotic behavior of solutions to time-fractional diffusion equations change when the fractional order α(x) is space-dependent rather than constant?
• RQ2 What is the precise decay rate of the L²-norm of the solution as t → ∞ for variable-order equations in ℝᵈ?
• RQ3 How does the short-time behavior (t → 0⁺) of the solution depend on the minimum and maximum values of the variable order α(x)?
• RQ4 In what way does a compactly supported perturbation of the fractional order (i.e., α(x) ≠ α* in a bounded set K) affect the long-time decay rate of the solution?
• RQ5 Can sharp L²-estimates be derived for solutions with initial data in L² ∩ L^{2s′/(2s′−1)}(ℝᵈ) under variable-order dynamics?

Key findings

• For long times (t > 1), the L²-norm of the solution decays as t^{-(αₘ - α* (1 - d/s))}, where αₘ and αₘₐₓ are the essential infimum and supremum of α(x), and s > max(d, 4).
• For short times (t ≤ 1), the L²-norm decays as t^{αₘ - αₘₐₓ}, which is consistent with known results for constant-order equations when αₘ = αₘₐₓ.
• The long-time decay rate depends on the interplay between the minimal order αₘ, the spatial dimension d, and the integrability parameter s, with the exponent αₘ - α* (1 - d/s) being the key determinant.
• The decay rate is sharper than the constant-order case when αₘ < αₘₐₓ, indicating that spatial heterogeneity in α(x) can significantly alter long-time diffusion behavior.
• The results are valid for initial data in L²(ℝᵈ) ∩ L^{2s′/(2s′−1)}(ℝᵈ), with s′ satisfying 1/s + 1/s′ = 1, and the constants in the estimates depend only on d, s, K, αₘ, αₘₐₓ, and α*.
• The analysis confirms that the variable-order model captures more complex diffusion dynamics than constant-order models, especially in the presence of spatial inhomogeneities.