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Archiv der Mathematik

, Volume 100, Issue 2, pp 167–178 | Cite as

A Liouville comparison principle for sub- and super-solutions of the equation w t −Δ p (w) =  |w| q-1 w

  • Vasilii V. Kurta
Article
  • 147 Downloads

Abstract

We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation (*) w t −Δ p (w) =  |w| q-1 w in the half-space \({\mathbb {S}= \mathbb {R}^1_+\times \mathbb {R}^n}\) , where n ≥ 1, q > 0, and \({ \Delta_p(w) := {\rm div}_x \left(|\nabla_x w|^{p-2}\nabla_x w \right)}\) , 1 < p ≤ 2. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions of (*) on the hyper-plane t = 0, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if \({1< q \leq p-1+\frac {p}{n}}\) and u and v are, respectively, an entire weak super-solution and an entire weak sub-solution of (*) in \({\mathbb {S}}\) which belong, only locally in \({\mathbb {S}}\) , to the corresponding Sobolev space and are such that u ≥  v, then uv. The result is sharp. As direct corollaries we obtain known Fujita-type and Liouville-type theorems.

Mathematics Subject Classification

Primary 35K92 Secondary 35B44 35B51 35B53 35K15 

Keywords

Blow-up Comparison principle Entire weak sub-solution Entire weak super-solution Fujita theorem Liouville theorem p-Laplacian operator Quasilinear parabolic equation 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.American Mathematical Society, Mathematical ReviewsAnn ArborUSA

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