Comptes Rendus
no. 9
Blow-up of solutions to quasilinear hyperbolic equations with p(x,t)-Laplacian and positive initial energy
[Explosion de solutions d'équations hyperboliques quasi linéaires avec p(x,t)-laplacien et énergie initiale positive]
Bin Guo1, 2  ; Wenjie Gao1
1 School of Mathematics, Jilin University, Changchun 130012, China
2 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Comptes Rendus. Mécanique, Volume 342 (2014) no. 9, pp. 513-519.

Le but de cet article est d'étudier un problème aux limites initial et homogène défini par une équation hyperbolique quasi linéaire avec un p(x,t)-Laplacien et une énergie initiale positive. Les auteurs montrent que la solution explose dans un temps fini sous certaines conditions sur la valeur initiale, les exposants et les coefficients de l'équation. Les résultats généralisent et améliorent celui de S.N. Antonsev (2011) [6]. En outre, les conditions de positivité de l'intégrale pour les données initiales et le caractère borné de pt(x,t) sont supprimées.

The aim of this paper is to study an initial and homogeneous boundary value problem to a quasilinear hyperbolic equation with a p(x,t)-Laplacian and a positive initial energy. The authors prove that the solution blows up in a finite time under some conditions on the initial value, the exponents and the coefficients in the equation. The results generalize and improve that of S.N. Antonsev (2011) [6]. Besides, the conditions of positivity of the integral to the initial data and the boundedness of pt(x,t) are removed.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2014.06.001
Keywords: Quasilinear hyperbolic, Blow-up in finite time, Positive initial energy
Mot clés : Quasilinéaire hyperbolique, Explosion en temps fini, Énergie initiale positive
Bin Guo 1, 2 ; Wenjie Gao 1

1 School of Mathematics, Jilin University, Changchun 130012, China
2 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 
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Bin Guo; Wenjie Gao. Blow-up of solutions to quasilinear hyperbolic equations with $ p(x,t)$-Laplacian and positive initial energy. Comptes Rendus. Mécanique, Volume 342 (2014) no. 9, pp. 513-519. doi : 10.1016/j.crme.2014.06.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2014.06.001/

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☆ The project is supported by NSFC (11271154, 11301211), by Fundamental Research Funds of Jilin University (450060501317) and by the 985 program of Jilin University.

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