Long time upper bounds for solutions of Leibenson's equation on Riemannian manifolds

Abstract: We consider on Riemannian manifolds the Leibenson equation $$\partial {t}u=Δ{p}u^{q}.$$ We prove that a certain upper bound for weak solutions of this equation is equivalent to a euclidean-type Sobolev inequality.

• The paper establishes an equivalence between Sobolev inequalities and long-time decay bounds for weak solutions.
• It uses refined analysis with log-Sobolev inequalities, energy estimates, and weak solution theory to derive optimal decay rates.
• The study extends classical results by covering a full range of parameters on geodesically complete Riemannian manifolds.

Long-time Upper Bounds for Solutions of Leibenson’s Equation on Riemannian Manifolds

Introduction and Motivation

The paper "Long time upper bounds for solutions of Leibenson's equation on Riemannian manifolds" (2604.23227) investigates the asymptotic behavior of bounded weak solutions to the doubly nonlinear parabolic PDE

$\partial_t u = \Delta_p u^q$

on geodesically complete Riemannian manifolds $M$. Here, $p > 1$, $q > 0$, $u \geq 0$, and $\Delta_p$ is the $p$-Laplacian operator. This equation encompasses numerous classical models: for $p=2$, it reduces to the porous medium equation, and for $q=1, p=2$ it becomes the linear heat equation. The physical significance and mathematical structure of (\ref{evoeq}) have been discussed in prior literature (e.g., [grigor2024finite], [leibenzon1945general], [leibenson1945turbulent]).

The main objective is to prove a sharp equivalence: an upper decay bound on $L^\infty$-norms of solutions is equivalent to the validity of a Euclidean-type Sobolev inequality on $M$0. This generalizes classical results known for linear and porous medium cases to the full parametric range of $M$1 and $M$2 for which certain structural constraints are met. The analysis encompasses global inequalities and time-decay rates, offering insight into the interplay between geometric analysis and nonlinear diffusions.

Main Results

Let $M$3 denote a structural parameter. The core theorem asserts that for $M$4 and $M$5, the Sobolev inequality \begin{equation*} \left(\int_M |v|^{\frac{pn}{n-p}}\,d\right)^{\frac{n-p}{n}} \leq C \int_M |v|^p\,d \end{equation*} is equivalent to the following decay estimate for all bounded, non-negative solutions $M$6 with initial data $M$7: \begin{equation*} |u(t)|{L^\infty(M)} \leq C|u_0|{L^1(M)}^{\frac{p}{p-nD}} t^{-\frac{n}{p-nD}} \end{equation*} where $M$8 depends on $M$9. This decay exponent and scaling match the explicit Barenblatt-type self-similar solutions constructed in Euclidean space, confirming optimality in this context ([barenblatt1952self], [bonforte2006super]).

The equivalence is established via two directions:

• The Sobolev inequality implies the $p > 1$0 decay estimate.
• The $p > 1$1 decay estimate implies the Sobolev inequality.

The proofs are based on a refined analysis using log-Sobolev inequalities, functional interpolation, weak solution theory, and Caccioppoli-type energy estimates.

This result subsumes earlier findings for the parabolic $p > 1$2-Laplace equation ($p > 1$3; [bonforte2007singular]) and the porous medium case ($p > 1$4; [bonforte2008fast]). The novelty is the extension to the general parameter regime with structural restrictions.

Strong claims include:

• The decay estimate holds for all Cartan-Hadamard manifolds (simply connected, non-positive curvature), since Sobolev inequalities are known in this setting ([hoffman1974sobolev]).
• The decay exponent matches the optimal Barenblatt profile when the structural parameter $p > 1$5 meets the criterion $p > 1$6.

Theoretical Analysis

The analysis hinges on the weak solution framework for (\ref{dtv}) and interpolates between $p > 1$7 and $p > 1$8 norms using log-Sobolev inequalities. Functional estimates and ODE techniques are deployed to extract time decay rates.

A key lemma establishes monotonicity of $p > 1$9 norms for weak solutions, and a Caccioppoli-type inequality bounds the $q > 0$0 norm of derivatives. These technical results facilitate the derivation of explicit decay rates.

It is shown that an $q > 0$1 decay estimate for solutions is sufficient to recover a Sobolev inequality of the relevant exponent. Conversely, the existence of a Sobolev inequality yields the desired time decay. This completes the equivalence in both directions, consolidating the link between geometric analysis (Sobolev inequalities) and nonlinear PDE dynamics.

Practical and Theoretical Implications

The findings have direct implications in geometric analysis, nonlinear diffusion theory, and PDE modeling:

• Practical: Benchmarking diffusion rates for various geometric backgrounds, relevant in physical models governed by variable curvature.
• Theoretical: The equivalence gives a characterization of Sobolev inequalities via evolution equations, suggesting a route to explore the geometry/PDE interface.
• Extension potential: The results pave the way for further studies of nonlinear parabolic equations on singular spaces or under weaker curvature assumptions, as well as possible probabilistic generalizations ([barbu2025leibenson]). There is also scope for quantitative refinements and stability under perturbation.

In the broader context of nonlinear analysis and geometric PDEs, this equivalence strengthens the conceptual unity between functional inequalities and asymptotic solution behavior. The argument and sharp exponents are expected to influence further research into sharp bounds for doubly nonlinear evolution on geometric manifolds, including stochastic variants.

Conclusion

This paper establishes a sharp equivalence between Sobolev-type inequalities and long-time upper bounds for solutions of the doubly nonlinear Leibenson equation on Riemannian manifolds. The argument generalizes classical results to a wider class of parameters and backgrounds, demonstrating optimality via comparison to explicit Barenblatt solutions. The analysis integrates energy methods, functional inequalities, and geometric insights to reveal deep structural links between Riemannian geometry and nonlinear parabolic PDEs. Future research directions include extension to stochastic frameworks, singular spaces, and quantitative stability estimates.