Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains

Laister, Robert; Robinson, James C.; Sierżęga, Mikolaj

doi:10.1016/j.crma.2014.05.010

Partial differential equations

Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains
[Non-existence de solutions locales pour les équations de la chaleur semi-linéaires de type Osgood dans des domaines bornés]

Présenté par : the Editorial Board

Laister, Robert ¹ ; Robinson, James C.² ; Sierżęga, Mikolaj ²

¹ Department of Engineering Design and Mathematics, University of the West of England, Bristol BS16 1QY, UK
² Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK

Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 621-626.

Résumé
Abstract

Nous établissons un résultat de non-existence locale pour l'équation $u_{t} - Δ u = f (u)$ avec des conditions aux limites de Dirichlet sur un domaine borné lisse $Ω \subset R^{n}$ et des données initiales dans $L^{q} (Ω)$ lorsque le terme de source f est non décroissant et ${\lim \sup}_{s \to \infty} s^{- γ} f (s) = \infty$ pour un exposant $γ > q (1 + 2 / n)$ . Ceci nous permet de construire un f localement Lipschitz qui satisfait la condition de Osgood $\int_{1}^{\infty} 1 / f (s) d s = \infty$ , ce qui garantit l'existence globale pour des données initiales dans $L^{\infty} (Ω)$ , de telle sorte que pour chaque q tel que $1 \leq q < \infty$ il existe une condition initiale non négative $u_{0} \in L^{q} (Ω)$ pour laquelle le problème semi-linéaire correspondant n'admet pas de solution locale en temps ( « blow-up immédiat »).

We establish a local non-existence result for the equation $u_{t} - Δ u = f (u)$ with Dirichlet boundary conditions on a smooth bounded domain $Ω \subset R^{n}$ and initial data in $L^{q} (Ω)$ when the source term f is non-decreasing and ${\lim \sup}_{s \to \infty} s^{- γ} f (s) = \infty$ for some exponent $γ > q (1 + 2 / n)$ . This allows us to construct a locally Lipschitz f satisfying the Osgood condition $\int_{1}^{\infty} 1 / f (s) d s = \infty$ , which ensures global existence for initial data in $L^{\infty} (Ω)$ , such that for every q with $1 \leq q < \infty$ there is a non-negative initial condition $u_{0} \in L^{q} (Ω)$ for which the corresponding semilinear problem has no local-in-time solution (‘immediate blow-up’).

Reçu le : 2014-04-28
Accepté le : 2014-05-29
Publié le : 2014-06-21

DOI : 10.1016/j.crma.2014.05.010

Affiliations des auteurs :

Laister, Robert ¹ ; Robinson, James C. ² ; Sierżęga, Mikolaj ²

¹ Department of Engineering Design and Mathematics, University of the West of England, Bristol BS16 1QY, UK
² Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK

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     author = {Laister, Robert and Robinson, James C. and Sier\.z\k{e}ga, Mikolaj},
     title = {Non-existence of local solutions of semilinear heat equations of {Osgood} type in bounded domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {621--626},
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Laister, Robert; Robinson, James C.; Sierżęga, Mikolaj. Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains. Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 621-626. doi : 10.1016/j.crma.2014.05.010. http://www.numdam.org/articles/10.1016/j.crma.2014.05.010/

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