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- Publisher
- Springer Journals
- Copyright
- Copyright © 2017 by Sociedade Brasileira de Matemática
- Subject
- Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
- ISSN
- 1678-7544
- eISSN
- 1678-7714
- DOI
- 10.1007/s00574-017-0057-5
- Publisher site
- See Article on Publisher Site
Abstract
Let (M, g) be a n-dimensional smooth compact Riemannian manifold without boundary with $$n\ge 2$$ n ≥ 2 . We prove that the optimal Riemannian p-entropy inequality $$\begin{aligned} \int _M |u|^p\log (|u|^p) \; dv_g\le \dfrac{n}{\tau }\log \left[ {\mathcal {A}}(p,\tau )\left( \int _M |\nabla _g u|^p\; dv_g\right) ^{\frac{\tau }{p}}+{\mathcal {B}}(p)\right] \end{aligned}$$ ∫ M | u | p log ( | u | p ) d v g ≤ n τ log A ( p , τ ) ∫ M | ∇ g u | p d v g τ p + B ( p ) is valid for every $$u\in H^{1,p}(M)$$ u ∈ H 1 , p ( M ) with $$\Vert u\Vert _p=1$$ ‖ u ‖ p = 1 where $$p>1$$ p > 1 and $$1\le \tau < \min \{2,p\}$$ 1 ≤ τ < min { 2 , p } or $$\tau = p \le 2$$ τ = p ≤ 2 . Also, we investigated the relationship between this optimal inequality and the hypercontractivity property for the non-linear evolution equation $$\begin{aligned} u_t=\Delta _p\left( u^{\frac{1}{p-1}}\right) ,\quad x\in M,\ t>0. \end{aligned}$$ u t = Δ p u 1 p - 1 , x ∈ M , t > 0 . When $$\tau =p\le 2$$ τ = p ≤ 2 we find explicit estimates of the time asymptotic behavior of their solutions with initial data in the spaces $$L^q(M)$$ L q ( M ) , $$1\le q<\infty $$ 1 ≤ q < ∞ .
Journal
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Nov 4, 2017