Existence results for Leibenson's equation on Riemannian manifolds

Published 28 Jan 2026 in math.AP and math.DG | (2601.20640v1)

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 $u_{0}\in L^{1}(M)\cap L^{{\infty}(M)$.}