Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source

Abita Rahmoune

doi:10.5802/crmath.8

Équations aux dérivées partielles

Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source
[Borne supérieure pour le temps d’explosion d’une classe d’équations intégro-différentielles de type parabolique avec une source variable]

Abita Rahmoune ¹

¹ Department of technical sciences, Laghouat University, Algeria

Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 23-32.

Résumé
Abstract

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

u_{t} = Δ u - \int_{0}^{t} g (t - s) Δ u (x, s) d s + {| u |}^{p (x) - 2} u .

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.

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

u_{t} = Δ u - \int_{0}^{t} g (t - s) Δ u (x, s) d s + {| u |}^{p (x) - 2} u .

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.

Reçu le : 2019-04-06
Révisé le : 2019-09-25
Accepté le : 2019-12-16
Publié le : 2020-03-18

DOI : 10.5802/crmath.8

Affiliations des auteurs :

Abita Rahmoune ¹

¹ Department of technical sciences, Laghouat University, Algeria

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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},
     pages = {23--32},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     doi = {10.5802/crmath.8},
     language = {en},
}

TY  - JOUR
AU  - Abita Rahmoune
TI  - Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 23
EP  - 32
VL  - 358
IS  - 1
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.8
LA  - en
ID  - CRMATH_2020__358_1_23_0
ER  -

%0 Journal Article
%A Abita Rahmoune
%T Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source
%J Comptes Rendus. Mathématique
%D 2020
%P 23-32
%V 358
%N 1
%I Académie des sciences, Paris
%R 10.5802/crmath.8
%G en
%F 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/articles/10.5802/crmath.8/

Bibliographie
Cité par

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

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

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

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

[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., Volume 563 (2003), pp. 197-220 | Zbl

[6] X. Fan; J. Shen; D. Zhao Sobolev embedding theorems for spaces $W^{k, p (x)} (Ω)$ , J. Math. Anal. Appl., Volume 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., Volume 142 (2012) no. 5, pp. 1027-1042 | DOI | MR | Zbl

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

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

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

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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Blow-up, non-extinction and exponential growth of solutions to a fourth-order parabolic equation

Khadijeh Baghaei

C. R. Méca (2022)