A Liouville comparison principle for entire sub- and super-solutions of the equation $ {u}_{t}-{\mathrm{\Delta }}_{p}(u)={|u|}^{q-1}u$

Kurta, Vasilii V.

doi:10.1016/j.crma.2011.06.006

Partial Differential Equations

A Liouville comparison principle for entire sub- and super-solutions of the equation

u_{t} - Δ_{p} (u) = {| u |}^{q - 1} u

[Sur un critère de comparaison de type de Liouville pour des sous- et super-solutions entières de lʼéquation

u_{t} - Δ_{p} (u) = {| u |}^{q - 1} u

]

Présenté par : Haïm Brezis

Kurta, Vasilii V.¹

¹ Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, MI 48107-8604, USA

Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 773-776.

Résumé
Abstract

Nous établissons un critère de comparaison de type de Liouville pour des sous- et super-solutions entières de lʼéquation $(⁎)$ $w_{t} - Δ_{p} (w) = {| w |}^{q - 1} w$ dans le demi-espace $S = R_{+}^{1} \times R^{n}$ , où $n ⩾ 1$ , $q > 0$ et $Δ_{p} (w) : = {div}_{x} ({| \nabla_{x} w |}^{p - 2} \nabla_{x} w)$ , $1 < p ⩽ 2$ . Dans notre étude, nous nʼimposons ni des restrictions sur le comportement des sous- ou super-solutions entières sur le hyper-plan $t = 0$ , ni des conditions de croissance sur le comportement à lʼinfini de ces solutions ou de leurs dérivées partielles. Nous démontrons que si $1 < q ⩽ p - 1 + \frac{p}{n}$ , et u et v constituent, respectivement, une super-solution faible entière et une sous-solution faible entière de (⁎) dans $S$ qui appartiennent, localement en $S$ , à lʼespace de Sobolev approprié, et qui sont telles que $u ⩽ v$ , alors $u \equiv v$ . Ce résultat est précis. Comme corollaires immédiats, nous obtenons des nouveaux résultats, ainsi que des résultats connus de type Fujita et Liouville.

We establish a Liouville comparison principle for entire sub- and super-solutions of the equation $(⁎)$ $w_{t} - Δ_{p} (w) = {| w |}^{q - 1} w$ in the half-space $S = R_{+}^{1} \times R^{n}$ , where $n ⩾ 1$ , $q > 0$ and $Δ_{p} (w) : = {div}_{x} ({| \nabla_{x} w |}^{p - 2} \nabla_{x} w)$ , $1 < p ⩽ 2$ . 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 their behaviour or on that of any of their partial derivatives at infinity. We prove that if $1 < q ⩽ p - 1 + \frac{p}{n}$ , and u and v are, respectively, an entire weak super-solution and an entire weak sub-solution of (⁎) in $S$ which belong, only locally in $S$ , to the corresponding Sobolev space and are such that $u ⩽ v$ , then $u \equiv v$ . The result is sharp. As direct corollaries we obtain both new and known Fujita-type and Liouville-type results.

Reçu le : 2011-02-28
Accepté le : 2011-06-07
Publié le : 2011-06-30

DOI : 10.1016/j.crma.2011.06.006

Affiliations des auteurs :

Kurta, Vasilii V. ¹

¹ Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, MI 48107-8604, USA

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     title = {A {Liouville} comparison principle for entire sub- and super-solutions of the equation $ {u}_{t}-{\mathrm{\Delta }}_{p}(u)={|u|}^{q-1}u$},
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Kurta, Vasilii V. A Liouville comparison principle for entire sub- and super-solutions of the equation $ {u}_{t}-{\mathrm{\Delta }}_{p}(u)={|u|}^{q-1}u$. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 773-776. doi : 10.1016/j.crma.2011.06.006. http://www.numdam.org/articles/10.1016/j.crma.2011.06.006/

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