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Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods

Vicente Vergara, Rico Zacher

arXiv:1310.0209v1math.AP

TL;DR

The paper addresses sharp long-time decay for non-local subdiffusion equations on bounded domains, including time-fractional and ultraslow diffusion. It uses energy estimates and a new inequality for convolutional integro-differential operators to obtain optimal rates in weak divergence-form settings and selected quasilinear problems. The resulting decay can be algebraic or logarithmic and differs markedly from classical parabolic behavior.

  • Problem

    Sharp decay behavior for broad non-local subdiffusion equations, especially time-fractional and ultraslow models, requires analysis beyond classical parabolic diffusion.

  • Method

    The paper combines energy estimates with a new Lp-norm inequality for operators of the form ∂t(k ∗·), under bounded measurable uniformly parabolic coefficients.

  • Results

    The L2(Ω)-norm decays like ct^−α for time-fractional diffusion and c(log t)^−1 for ultraslow diffusion, with optimal estimates extending to selected quasilinear equations.

  • Takeaways & Limitations

    Non-local diffusion produces decay patterns that reflect its slower dynamics and differ markedly from the exponential decay of classical diffusion on bounded domains.

  • Takeaways & Limitations

    For the time-fractional p-Laplace problem, existence and uniqueness of weak solutions are not known in the literature, though construction is believed possible at least for p ≥ 2N/(N+2).

Abstract

from arXiv · show

We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the time-fractional and ultraslow diffusion equation, which have seen much interest during the last years, mostly due to their applications in the modeling of anomalous diffusion. We study the case where the equation is in divergence form with bounded measurable coefficients. Our proofs rely on energy estimates and make use of a new and powerful inequality for integro-differential operators of the form $\partial_t (k\ast \cdot)$. The results can be generalized to certain quasilinear equations. We illustrate this by looking at the time-fractional $p$-Laplace and porous medium equation. Here it turns out that the decay behaviour is markedly different from that in the classical parabolic case.

1 Introduction and main results

The paper develops sharp long-time decay estimates for non-local subdiffusion equations on bounded domains, including time-fractional and ultraslow cases. Energy methods and a new Lp-norm inequality extend the analysis to weak divergence-form and selected quasilinear problems.

  • Problem setting: The paper studies long-time behavior for non-local in time diffusion equations on bounded domains with homogeneous Dirichlet boundary conditions.The coefficients are assumed measurable, bounded, and uniformly parabolic, while the time kernel satisfies condition (PC).
  • Examples: For the standard fractional kernel, the equation becomes a time-fractional diffusion equation of order α ∈ (0, 1), while another kernel yields distributed-order ultraslow diffusion.The fractional operator is the Riemann–Liouville derivative in the special case A = νI.
  • Motivation: The motivation comes from anomalous diffusion and material-memory models, where non-local equations describe slower spreading and memory-dependent dynamics.Mean square displacement is linear in the classical case, behaves like ct^α for time-fractional diffusion, and like c log t for ultraslow diffusion.
  • Main decay results: The main results give optimal decay estimates: the L2(Ω)-norm behaves like ct^−α for the fractional case and c(log t)^−1 for the ultraslow case.These rates contrast with exponential decay for classical diffusion on bounded domains and reflect different degrees of diffusion slowness.
  • Quasilinear extensions: The estimates extend to certain quasilinear equations, improving a previously known t^−α/2 bound to the optimal t^−α rate for a time-fractional p-Laplace problem.For time-fractional p-Laplace and porous medium equations, the decay differs markedly from the classical parabolic case; finite-time extinction no longer occurs in the stated p-range.
  • Methods: The proofs use energy estimates, a new Lp-norm inequality for ∂t(k ∗·), and a subsolution inequality for positive parts.The inequality is supported by the fundamental identity for operators of the form ∂t(k ∗·), with suitable time regularization needed in the weak setting.

2 Preliminaries

This section develops regularization, comparison, and Tauberian tools for Volterra equations with kernels in (k,l)∈PC. These tools support weak formulations and asymptotic decay analysis.

  • Kernel regularization: The resolvent kernel h_µ associated with µl satisfies h_µ + µ(h_µ∗l) = µl and equals −ṡ_µ.It is nonnegative, and regularized kernels k_µ = µs_µ are nonnegative, nonincreasing, and locally H1.
  • Weak formulation: Approximating the singular kernel by a more regular kernel gives an equivalent weak formulation for solutions, subsolutions, and supersolutions.The approximation uses kernels h_n and k_n and applies to data in the stated weak-solution spaces.
  • Fundamental identity: A fundamental identity for ∂t(k∗·) serves as an analogue of the chain rule and remains valid for regularized operators.For singular kernels such as g1−α, the identity applies when the function is sufficiently smooth.
  • Comparison principles: Comparison arguments show that ordered data and nondecreasing nonlinearities preserve ordering for Volterra equations, including weak sub- and supersolutions.The result yields u≤v or v≤w almost everywhere under the corresponding assumptions.
  • Tauberian analysis: The Karamata-Feller Tauberian theorem connects small-z Laplace-transform asymptotics with large-time asymptotics of monotone functions.If the transform behaves like z^-βL(1/z), then the function behaves like gβ(t)L(t), and conversely.

3 The Lp-norm inequality

This section introduces a new Lp-norm inequality for integro-differential operators ∂t(k∗·). The inequality extends across function and Hilbert-space settings and also accommodates singular kernels under smoothness conditions.

  • Main inequality: The new inequality is the key tool for sharp Lp-norm decay estimates in linear and nonlinear integro-differential equations.It is formulated for nonnegative, nonincreasing kernels and extends the energy method beyond the classical time derivative.
  • Lp setting: For 1<p<∞, the inequality applies to functions in Lp([0,T];Lp(Ω)) on arbitrary measurable subsets Ω of RN.The proof uses the fundamental identity, Fubini’s theorem, and Hölder’s inequality.
  • Singular kernels: The inequality remains valid for sufficiently smooth functions with singular kernels, including k=g1−α for α∈(0,1).This produces an Lp-norm inequality for the fractional derivative ∂α_t.
  • Further extensions: The framework also extends to positive measures on RN and to functions valued in the sequence space lp(N).The sequence-space formulation uses the norm |x|lp for 1<p<∞.
  • Hilbert-space setting: The result has a Hilbert-space version for u∈L2([0,T];H), extending the p=2 case beyond scalar spatial functions.The extension follows directly from the Hilbert-space fundamental identity.

4 On the positive part of a subsolution

This section derives a subsolution inequality for the positive part of a weak subsolution. The proof regularizes the positive-part function, exploits convexity, and passes to the limit.

  • Positive-part regularization: A convex regularization Hε of the positive part is used as a test-function device for weak subsolutions.Hε converges to y+ as ε→0, while its convexity controls the integro-differential term.
  • Testing argument: Testing with the derivative of Hε and using the fundamental identity yields an inequality for the regularized positive part.The construction also uses u0≤[u0]+ to control the initial datum.
  • Limit passage: The approximation kernels are sent to the singular limit, and the nonnegative gradient contribution can be discarded.The limit uses the approximation property of h_n and convergence Hε(y)→y+.
  • Final subsolution inequality: The resulting inequality is first established for bounded test functions and then extended to all nonnegative η∈Ȟ1,2(ΩT) by truncation and approximation.The supersolution case follows by applying the subsolution result to −u.

5 Proof of Theorem 1.1 and Corollary 1.1

This section proves the main decay estimate by reducing the positive part to a scalar Volterra comparison problem, then combines positive and negative parts for solutions. It also records extensions to other norms.

  • Proof of Theorem 1.1: The positive part is bounded by a scalar comparison solution obtained after regularization and parameter limits.The scalar problem contains the first Dirichlet eigenvalue λ1 and the ellipticity parameter ν.
  • Proof of Theorem 1.1: The comparison solution is V(t)=sνλ1(t)|[u0]+|L2(Ω), yielding the desired estimate for subsolutions.The supersolution case follows by applying the argument to −u.
  • Proof of Corollary 1.1: For a global weak solution, separate estimates for the positive and negative parts combine through their L2-orthogonality and the Pythagorean theorem.Squaring and adding the two estimates gives the corollary for u.
  • Lp extensions: Testing with |u|p−2u extends the decay framework to suitably defined solutions with initial data in Lp(Ω), 1<p<∞.The argument assumes v:=|u|(p−2)/2u belongs to V(T) for every T>0.
  • Lp extensions: Bounded initial data satisfy the maximum estimate |u(t,·)|∞≤|u0|∞ for almost every t>0.This follows by taking p→∞, since ρ(p)→0 and s0≡1.

6 Decay behaviour for some specific examples

The relaxation function determines solution decay, and the admissible kernel pairs produce exponential, algebraic, logarithmic, and even non-vanishing behaviours. Several examples show that the resulting estimates are optimal or identify which kernel component controls the asymptotics.

  • The relaxation function sµ determines the solution’s decay, while admissible kernel pairs allow exponential, algebraic, and logarithmic decay.The section also includes an example where the relaxation function does not tend to zero.
  • The classical time-fractional case: In the classical time-fractional case, the Mittag-Leffler relaxation function has algebraic decay matching the kernel, yielding a corresponding algebraic solution estimate.The estimate follows from the Mittag-Leffler bound and Corollary 1.1.
  • Weighted time-fractional kernels: Exponential weighting produces exponential relaxation and solution decay with rate ω, which is optimal because the Laplace transform has a singularity at −ω.Here ω is the unique solution of ω = µ(γ −ω)^(1−α).
  • Weighted time-fractional kernels: When l is integrable, sµ need not vanish at infinity; switching the weighted kernels therefore yields non-decaying relaxation.The conclusion uses l ∈ L1(R+) and the associated estimate involving (1 ∗l)(t).
  • A sum of two fractional derivatives: For a sum of fractional derivatives, the lowest fractional order determines the relaxation decay and leads to an algebraic solution estimate governed by α.The asymptotic relation is derived using the Karamata-Feller Tauberian theorem.
  • Ultraslow and switched kernels: In the distributed-order ultraslow example, the relaxation and solution estimates decay logarithmically, while a switched-kernel example improves an initial log(t)/t bound to an optimal t^-1 rate.The switched example explicitly states that the logarithmic correction is not optimal.

7 On a basic nonlinear fractional differential equation

The section compares the time-fractional nonlinear scalar equation with its classical counterpart by constructing sub- and supersolutions. For α < 1, the solution remains positive and decays algebraically across the considered parameter range, unlike the classical extinction regime.

  • Comparison construction: Sub- and supersolutions bound the nonlinear fractional solution from below and above, producing two-sided decay estimates.The comparison gives v(t) ≤ u(t) ≤ w(t) for all t ≥ 0.
  • Comparison with the classical case: For α < 1, the fractional solution’s decay behaviour differs markedly from the classical α = 1 cases of algebraic decay, exponential decay, or finite-time extinction.The classical alternatives depend on whether γ is greater than, equal to, or less than 1.

8 On the time-fractional p-Laplace equation

The time-fractional p-Laplace section derives L2-based decay estimates on bounded Lipschitz domains using energy inequalities, Sobolev embeddings, and scalar fractional comparison equations. The rates are optimal in part of the parameter range and differ from the classical limit.

  • Problem and theorem: The theorem considers weak solutions of the time-fractional p-Laplace problem on bounded Lipschitz domains with homogeneous Dirichlet data.The stated setting has α ∈ (0, 1), 1 < p < ∞, and initial data in L2(Ω).
  • Proof strategy: Energy testing, Sobolev embeddings, and scalar fractional inequalities reduce the PDE estimate to a weak subsolution problem.The reduction is performed separately across the p-regimes.
  • Decay regimes: The decay estimates cover both 2N/(N+2) ≤ p < ∞ and 1 < p < 2N/(N+2), with the latter handled through a different exponent choice.The proof invokes the scalar comparison result in both regimes.
  • Optimality: The rates are optimal at least for p > 2N/(N+2), as shown using a separable solution based on a first p-Laplacian eigenfunction.The construction reduces the PDE to a scalar fractional equation.
  • Optimality: For α < 1 and p > 2N/(N+2), the constructed solution stays positive, so finite-time extinction does not occur in that example.The paper contrasts this with the classical α = 1 decay rate.

9 On the time-fractional porous medium equation

The time-fractional porous medium section establishes decay estimates for nonnegative weak solutions through energy and Sobolev arguments. It emphasizes optimality in part of the range and contrasts fractional behaviour with classical finite-time extinction and decay rates.

  • Problem setting: The section studies nonnegative weak solutions of the time-fractional porous medium problem on bounded Lipschitz domains with homogeneous Dirichlet data.The assumptions include α ∈ (0, 1), m > 0, and N > 2.
  • Proof strategy: Testing the equation with u^m and applying Hölder’s inequality, Sobolev embedding, and the energy inequality yields a fractional differential inequality.This scalar inequality implies the asserted decay estimate.
  • Comparison with the classical case: For the fractional problem, finite-time extinction no longer occurs in the stated range m > N/(N+2), unlike the classical case with 0 < m < 1.The paper also notes that the fractional and classical algebraic rates differ when m > 1.
  • Optimality: The rates are optimal at least when m > 2N/(N+2), as demonstrated by a positive separable solution.The temporal factor solves the scalar equation used in the p-Laplace analysis.
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