Existence results for Leibenson's equation on Riemannian manifolds

TL;DR

This work establishes global existence of bounded weak solutions for the doubly nonlinear Leibenson equation $\partial_t u = \Delta_p u^q$ on geodesically complete Riemannian manifolds under $p>1$, $q>0$, $pq\ge 1$, for initial data $u_0 \in L^1(M) \cap L^\infty(M)$. The authors develop a robust approximation framework via auxiliary problems on bounded domains and a Galerkin-type construction, derive universal energy estimates (Caccioppoli inequalities) and a nonlinear mean value inequality, and pass to the limit to obtain a global solution on $M$. They also prove finite propagation speed for a broad class of parameters by leveraging a mean-value inequality and De Giorgi iterations, and relate propagation rates to geometric conditions such as the relative Faber-Krahn inequality, yielding sharp rates in model settings. Overall, the results extend existence and qualitative properties of Leibenson-type diffusion beyond Euclidean spaces with minimal geometric assumptions, highlighting the roles of $p$, $q$, and curvature through intrinsic geometric inequalities.

Abstract

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 $p>1, q>0$ and $pq\geq 1$ 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 weak solution for any $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$.

Figures (3)

• Figure 1: Cylinders $Q$ and $Q^{\prime }$
• Figure 2: Cylinders $Q_{k}$
• Figure 3: Range of $p, q$

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Definition 1
• Definition 2
• Lemma 2.1
• Lemma 2.2
• Definition 3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3