Characterizes metric measure spaces satisfying parabolic Harnack inequalities for doubly nonlinear equations via volume doubling and Poincaré inequalities using estimates on a related Cauchy problem.
Existence results for Leibenson's equation on Riemannian manifolds
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abstract
We consider on an arbitrary Riemannian manifold $M$ the \textit{Leibenson equation} $\partial _{t}u=\Delta _{p}u^{q}$, 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=\Delta _{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 $u_{0}\in L^{1}(M)\cap L^{\infty}(M)$.
fields
math.AP 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
A certain upper bound for weak solutions of the Leibenson equation on Riemannian manifolds is equivalent to a Euclidean-type Sobolev inequality.
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Doubling measures, Poincar\'e inequalities and parabolic Harnack inequalities for a doubly nonlinear equation
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