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.

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.

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.

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
Mots-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 
@article{CRMECA_2014__342_9_513_0,
     author = {Bin Guo and Wenjie Gao},
     title = {Blow-up of solutions to quasilinear hyperbolic equations with $ p(x,t)${-Laplacian} and positive initial energy},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {513--519},
     publisher = {Elsevier},
     volume = {342},
     number = {9},
     year = {2014},
     doi = {10.1016/j.crme.2014.06.001},
     language = {en},
}
TY  - JOUR
AU  - Bin Guo
AU  - Wenjie Gao
TI  - Blow-up of solutions to quasilinear hyperbolic equations with $ p(x,t)$-Laplacian and positive initial energy
JO  - Comptes Rendus. Mécanique
PY  - 2014
SP  - 513
EP  - 519
VL  - 342
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crme.2014.06.001
LA  - en
ID  - CRMECA_2014__342_9_513_0
ER  - 
%0 Journal Article
%A Bin Guo
%A Wenjie Gao
%T Blow-up of solutions to quasilinear hyperbolic equations with $ p(x,t)$-Laplacian and positive initial energy
%J Comptes Rendus. Mécanique
%D 2014
%P 513-519
%V 342
%N 9
%I Elsevier
%R 10.1016/j.crme.2014.06.001
%G en
%F CRMECA_2014__342_9_513_0
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/

[1] E. Acerbi; G. Mingione Regularity results for stationary electrorheological fluids, Arch. Ration. Mech. Anal., Volume 164 (2002), pp. 213-259

[2] L. Diening; P. Harjulehto; P. Hästö; M. Rûžička Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, Germany, 2011

[3] M. Ruzicka Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000

[4] V. Georgiev; G. Todorova Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equ., Volume 109 (1994), pp. 295-308

[5] S.N. Antonsev; J. Ferreira Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal., Volume 93 (2013), pp. 62-77

[6] S.N. Antontsev Wave equation with p(x,t)-Laplacian and damping: blow-up of solutions, C. R. Mecanique, Volume 339 (2011), pp. 751-755

[7] B. Guo; W.J. Gao Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplacian operator | arXiv

[8] M. Mihǎilescu; P. Pucci; V. Rǎdulescu Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., Volume 340 (2008), pp. 687-698

[9] H.A. Levine Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt=−Au+F(u), Trans. Amer. Math. Soc., Volume 192 (1974), pp. 1-21

Cité par Sources :

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

Commentaires - Politique