2011/05/09 by Vasilii V. Kurta, Kurta, Vasilii V. · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1105.1810
arxiv created 2011/05/09 · arxiv updated 2011/05/11
We establish a Liouville comparison principle for entire sub- and super-solutions of the equation (∗) wt-Δp (w) = |w|q-1w in the half-space \mathbb S= \mathbb R1+× \mathbb Rn, where n≥ 1, q>0 and Δp (w):=divx(|∇x w|p-2∇x w), 1<p≤ 2. In our study we impose neither restrictions on the behaviour of entire sub- and super-solutions on the hyper-plane t=0, nor any growth conditions on the behavior of them or any of their partial derivatives at infinity. We prove that if 1<q≤ p-1+\frac pn, and u and v are, respectively, an entire weak super- and an entire weak sub-solution of (∗) in \Bbb S which belong, only locally in \Bbb S, to the corresponding Sobolev space and are such that u≤ v, then u≡ v. The result is sharp. As direct corollaries we obtain both new and known Fujita-type and Liouville-type results.