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Philipp Sürig

Publications and source records attributed to Philipp Sürig.

11 recordsLinked to original sources
arXiv2026

Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds

We consider on a Riemannian manifold $M$ the Leibenson equation \begin{equation*}\label{eqabs}\partial _{t}u=Δ_{p}u^{q},\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\leq \exp\left(C r^{\frac{p}{p-1}}\right)$ for some $x_{0}\in M$ and all large enough $r>0$. When $q(p-1)< 1$, we prove this property assuming $V(x_0, r)\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)< 1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.

math.AP↗
arXiv2026

Leibenson's equation on graphs

In this paper we study on infinite graphs the Leibenson equation $$ \partial_t u = Δ_p u^q, $$ where $p>1$, $q>0$ and $Δ_p$ denotes the discrete $p$-Laplacian. We prove, for any integrable initial data $u_0$, the existence of a global solution, which is unique for a certain range of $p$ and $q$. Assuming a Faber--Krahn inequality, we obtain sharp $\ell^1$-$\ell^\infty$ smoothing estimates and quantitative bounds on the propagation of solutions with initially finite support. Under certain assumptions on $p$ and $q$, we also prove finite-time extinction results for solutions when the graph satisfies an \textit{isoperimetric inequality}. In particular, on Cayley graphs with polynomial volume growth, we establish the optimal large-time decay rate of the $\ell^\infty$-norm for nonnegative finite-mass solutions when $q(p-1)>1$, and demonstrate a sharp dichotomy regarding the finite-time extinction of exhaustion solutions.

math.AP↗
arXiv2026

Relative Faber--Krahn inequalities and Trudinger's equation on Riemannian Manifolds

We consider on Riemannian manifolds the Trudinger equation \begin{equation*}\partial _{t}u=Δ_{p}u^{\frac{1}{p-1}},\end{equation*} where $p>1$. We prove that a relative $p$--Faber--Krahn inequality is equivalent to the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions of Trudinger's equation. We also derive an improved long-time upper estimate under a uniform $p$--Faber--Krahn inequality.

math.AP↗
arXiv2026

Existence results for Leibenson's equation on Riemannian manifolds

We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=Δ_{p}u^{q}$, that is also known as a \textit{doubly nonlinear evolution equation}. We prove that if $p>1$ and $q>0$ then the Cauchy-problem \begin{equation*} \left\{ \begin{array}{ll} \partial _{t}u=Δ_{p}u^{q} &\text{in}~M\times (0, \infty), \\u(x, 0)=u_{0}(x)& \text{in}~M, \end{array}% \right. \end{equation*} has a unique weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$.

math.AP↗
arXiv2026

Parabolic Frequency for Doubly Nonlinear Equations on Manifolds

We establish monotonicity formulas for a parabolic frequency function associated with sign-changing solutions to a class of doubly nonlinear parabolic equations of the form $\partial_t u = \mathcal{L}_{p,φ} u^q$ on weighted complete Riemannian manifolds without any curvature assumption, where $\mathcal{L}_{p,φ}$ denotes the weighted $p$-Laplacian and $p>1$, $q>0$. As a consequence, we obtain results on backward uniqueness for $q(p-1)\geq 1$ and unique continuation at infinity for $q(p-1) > 1$. We further consider equations with a controlled nonlinear perturbation term and derive an almost-monotonicity formula for the parabolic frequency. By employing the parabolic frequency, we also establish some Liouville-type results for ancient solutions in the case $q(p-1)\geq 1$.

math.AP↗
arXiv2026

Sharp long distance upper bounds for solutions of Leibenson's equation on Riemannian manifolds

We consider on Riemannian manifolds the Leibenson equation $\partial _{t}u=Δ_{p}u^{q}$ that is also known as a doubly nonlinear evolution equation. We prove sharp upper estimates of weak subsolutions to this equation on Riemannian manifolds with non-negative Ricci curvature in the whole range of $p>1$ and $q>0$ satisfying $q(p-1)<1$. In this way, we improve the result of \cite{Grigoryan2024a} and prove Conjecture 1.2 from \cite{Grigoryan2024a}.

math.AP↗
arXiv2026

Finite extinction time for subsolutions of the weighted Leibenson equation on Riemannian manifolds

We consider on Riemannian manifolds the non-linear evolution equation $$ρ\partial _{t}u=Δ_{p}u^{q}.$$ Assuming that the manifold satisfies a \textit{(weighted) Sobolev inequality} and under certain assumptions on $p, q$ and function $ρ$, we prove that weak subsolutions to this equation have a finite extinction time. In particular, our main result holds in the case of a \textit{Cartan-Hadamard manifold}.

math.AP↗
arXiv2025

Gradient estimates for Leibenson's equation on Riemannian manifolds

We consider on Riemannian manifolds solutions of the Leibenson equation \begin{equation*} \partial _{t}u=Δ_{p}u^{q}. \end{equation*} This equation is also known as doubly nonlinear evolution equation. We prove gradient estimates for positive solutions $u$ under the condition that the Ricci curvature on $M$ is bounded from below by a non-positive constant. We distinguish between the case $q(p-1)>1$ (slow diffusion case) and the case $q(p-1)<1$ (fast diffusion case).

math.AP↗
arXiv2024

Sharp sub-Gaussian upper bounds for subsolutions of Trudinger's equation on Riemannian manifolds

We consider on Riemannian manifolds the nonlinear evolution equation \begin{equation*} \partial _{t}u=Δ_{p}(u^{1/(p-1)}), \end{equation*}% where $p>1$. This equation is also known as a doubly non-linear parabolic equation or Trudinger's equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including $\mathbb{R}^{n}$.

math.AP↗