[Paper Review] Decay estimates for evolutionary equations with fractional time-diffusion

Serena Dipierro, Enrico Valdinoci+1 more|arXiv (Cornell University)|Jul 25, 2017

Fractional Differential Equations SolutionsMathematics71 references48 citations

TL;DR

This paper establishes general decay estimates for solutions to evolutionary equations with fractional time-diffusion (order α ∈ (0,1)) in bounded domains, using energy methods and a structural condition on the nonlocal operator. The key result is a power-law decay rate of the L^s-norm of solutions as ‖u‖_{L^s(Ω)}(t) ≤ C⋆ / (1 + t^{α/γ}), which generalizes classical exponential decay and captures the slower, polynomial decay typical of fractional time dynamics.

ABSTRACT

We consider an evolution equation whose time-diffusion is of fractional type, and we provide
 decay estimates in time for the L^s-norm of the solutions in a bounded domain. The spatial operator that we
 take into account is very general and comprises classical local and nonlocal diffusion equations.

Motivation & Objective

• To derive decay estimates for solutions of time-fractional evolutionary equations with general space-diffusion operators in bounded domains.
• To establish a unified framework for decay rates applicable to both local and nonlocal, linear and nonlinear diffusion operators.
• To generalize existing results on classical and fractional diffusion by incorporating nonlinear, nonlocal, and geometric operators.
• To provide a structural condition (1.4) under which power-law decay in time is guaranteed for the L^s-norm of solutions.
• To recover and extend known decay results for specific equations such as the fractional heat, p-Laplacian, porous medium, and fractional mean curvature equations.

Proposed method

• Formulate the time-fractional evolution equation using the Caputo derivative of order α ∈ (0,1), modeling anomalous diffusion in time.
• Introduce a general structural condition (1.4) involving the L^s-norm and the action of the spatial operator N[u], linking the solution's energy to its dissipation.
• Apply energy methods and Volterra integral representations to derive a differential inequality involving the Caputo derivative of the L^s-norm.
• Use the comparison principle and inversion of the Caputo derivative via Volterra kernels to derive the decay estimate (1.6).
• Verify the structural condition (1.4) for specific operators (e.g., Laplacian, p-Laplacian, fractional Laplacian) using fractional Sobolev embeddings and pointwise inequalities.
• Employ Hölder’s inequality and interpolation to extend decay estimates from higher to lower Lebesgue norms when necessary.

Experimental results

Research questions

• RQ1 What is the general decay rate of solutions to time-fractional evolutionary equations with general space-diffusion operators?
• RQ2 How does the decay rate depend on the fractional order α and the nonlinearity of the spatial operator?
• RQ3 Can the classical exponential decay of classical parabolic equations be recovered or generalized in the fractional time-diffusion setting?
• RQ4 What conditions on the spatial operator ensure that the L^s-norm of the solution decays as a power law in time?
• RQ5 How do the decay estimates extend to nonlocal, nonlinear, and geometric operators such as the fractional p-Laplacian or mean curvature operators?

Key findings

• The solution's L^s-norm decays as ‖u‖_{L^s(Ω)}(t) ≤ C⋆ / (1 + t^{α/γ}), where γ > 0 is a parameter from the structural condition (1.4), establishing a power-law decay distinct from classical exponential decay.
• The decay rate t^{-α/γ} is sharp and consistent with the Mittag-Leffler function behavior observed in fractional heat equations, as confirmed by the eigenfunction solution in a ball.
• For the p-Laplacian with m=1, the decay rate is t^{-α/(m(p-1))} = t^{-α/(p-1)}, recovering known results and extending them to fractional time.
• For the porous medium equation (p=2, m>0), the decay rate is t^{-α/m}, which matches known asymptotic behavior in the classical case.
• For the fractional p-Laplacian and sum of fractional operators, the decay rate is t^{-α/(p_max - 1)} where p_max is the largest exponent, showing the dominant role of the strongest nonlinearity.
• For the fractional mean curvature equation, the decay rate is t^{-α/m} under the condition s ≥ m−1, extending the framework to geometric nonlocal operators.