logo CRAS
Comptes Rendus. Mathématique
Partial Differential Equations
Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source
Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 23-32.

Consider a class of integrodifferential of parabolic equations involving variable source with Dirichlet boundary condition

ut=Δu-0tgt-sΔux,sds+|u|p(x)-2u.

By means energy methods, we obtain a lower bound for blow-up time of the solution if blow-up occurs. Furthermore, assuming the initial energy is negative we establish a new blow-up criterion and give an upper bound for blow-up time of the solution.

Considérons une classe d’équations intégro-différentielles paraboliques comprenant une source variable et avec condition de Dirichlet au bord

ut=Δu-0tgt-sΔux,sds+|u|p(x)-2u.

À l’aide des méthodes d’énergie nous obtenons une borne inférieure pour le temps où intervient une éventuelle explosion de la solution. De plus, en supposant que l’énergie initiale est négative nous établissons un nouveau critère pour l’explosion et nous donnons une borne supérieure pour le temps d’explosion de la solution.

Received : 2019-04-06
Revised : 2019-09-25
Accepted : 2019-12-16
Published online : 2020-03-19
DOI : https://doi.org/10.5802/crmath.8
@article{CRMATH_2020__358_1_23_0,
     author = {Abita Rahmoune},
     title = {Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source},
     journal = {Comptes Rendus. Math\'ematique},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     pages = {23-32},
     doi = {10.5802/crmath.8},
     language = {en},
     url={comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_1_23_0/}
}
Abita Rahmoune. Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source. Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 23-32. doi : 10.5802/crmath.8. https://comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_1_23_0/

[1] E. Acerbi; G. Mingione Regularity results for stationary eletrorheological fluids, Arch. Ration. Mech. Anal., Tome 164 (2002), 065008, pp. 213-259 | Article

[2] G. Akagi; M. Ôtani Evolutions inclusions governed by subdifferentials in reflexive Banach spaces, J. Evol. Equ., Tome 4 (2004), pp. 519-541 | Article | MR 2105275 | Zbl 1081.34059

[3] S. N. Antonsev Blow up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., Tome 234 (2010), 1320004, pp. 2633-2645 | Article | MR 2652114 | Zbl 1266.53082

[4] L. Diening; P. Hästo; P. Harjulehto; M. Růžička Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Tome 2017, Springer, 2011 | MR 2790542 | Zbl 1222.46002

[5] L. Diening; M. Růžička Calderon Zygmund operators on generalized Lebesgue spaces L p(x) (Ω) and problems related to fluid dynamics, J. Reine Angew. Math., Tome 563 (2003), pp. 197-220 | Zbl 1072.76071

[6] X. Fan; J. Shen; D. Zhao Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal. Appl., Tome 262 (2001), pp. 749-760

[7] R. Ferreira; A. de Pablo; M. Pérez-LLanos Critical exponents for a semilinear parabolic equation with variable reaction, Proc. R. Soc. Edinb., Sect. A, Math., Tome 142 (2012) no. 5, pp. 1027-1042 | Article | MR 2981022 | Zbl 1270.35070

[8] Y. Fu The existence of solutions for elliptic systems with nonuniform growth, Stud. Math., Tome 151 (2002), pp. 227-246 | Article | MR 1917835 | Zbl 1007.35023

[9] O. Kovàcik; J. Rákosnik On spaces L p(x) (Ω) and W 1,p(x) (Ω), Czech. Math. J., Tome 41 (1991), pp. 592-618

[10] J. L. Lions Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1966 | Zbl 0189.40603

[11] Shuying Tian Bounds for blow-up time in a semilinear parabolic problem with viscoelastic term., Comput. Math. Appl., Tome 2 (2017), pp. 87-94 | MR 3679861 | Zbl 1390.35176