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

Partial Differential Equations

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

Abita Rahmoune ¹

¹ Department of technical sciences, Laghouat University, Algeria

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

Abstract
Résumé

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.

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.

Received: 2019-04-06
Revised: 2019-09-25
Accepted: 2019-12-16
Published online: 2020-03-18

DOI: 10.5802/crmath.8

Author's affiliations:

Abita Rahmoune ¹

¹ Department of technical sciences, Laghouat University, Algeria

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     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},
}

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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/

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