Wave equation with $ p(x,t)$-Laplacian and damping term: Blow-up of solutions

Stanislav Antontsev

doi:10.1016/j.crme.2011.09.001

Wave equation with

p (x, t)

-Laplacian and damping term: Blow-up of solutions
[Equation des ondes avec

p (x, t)

-Laplacian et un tèrme dissipatif : Blow-up des solutions]

Stanislav Antontsev ¹

¹ CMAF, University of Lisbon, Av. Prof. Gama Pinto, 2, 1649-003 Lisbon, Portugal

Comptes Rendus. Mécanique, Volume 339 (2011) no. 12, pp. 751-755.

Résumé
Abstract

On considère le problème de Dirichlet pour lʼéquation

u_{t t} = div (a (x, t) {| \nabla u |}^{p (x, t) - 2} \nabla u) + α △ u_{t} + b (x, t) {| u |}^{σ (x, t) - 2} u

où

α ⩾ 0

est une constante,

a (x, t)

b (x, t)

sont des coefficients variables et

p (x, t), σ (x, t)

sont des exposants nonlinéaires. Sous conditions appropriées on étudie le temps fini de blow-up des solutions.

We study the Dirichlet problem for equation

u_{t t} = div (a (x, t) {| \nabla u |}^{p (x, t) - 2} \nabla u) + α △ u_{t} + b (x, t) {| u |}^{σ (x, t) - 2} u

in which α is a nonnegative constant, the coefficients

a (x, t)

b (x, t)

and the exponents of nonlinearity

p (x, t)

σ (x, t)

are given functions. Under suitable conditions on the data, we study the finite time blow-up of the solutions.

Reçu le : 2011-07-17
Accepté le : 2011-09-09
Publié le : 2011-09-29

DOI : 10.1016/j.crme.2011.09.001

Keywords: Waves, Nonlinear wave equation, Energy estimates, Variable nonlinearity, Nonstandard growth conditions, Blow-up
Mot clés : Ondes, Équation des ondes nonlinéaire, Estimations énergétiques, Nonlinéarités variables, Croissance non standard, Blow-up

Affiliations des auteurs :

Stanislav Antontsev ¹

¹ CMAF, University of Lisbon, Av. Prof. Gama Pinto, 2, 1649-003 Lisbon, Portugal

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     author = {Stanislav Antontsev},
     title = {Wave equation with $ p(x,t)${-Laplacian} and damping term: {Blow-up} of solutions},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {751--755},
     publisher = {Elsevier},
     volume = {339},
     number = {12},
     year = {2011},
     doi = {10.1016/j.crme.2011.09.001},
     language = {en},
}

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Stanislav Antontsev. Wave equation with $ p(x,t)$-Laplacian and damping term: Blow-up of solutions. Comptes Rendus. Mécanique, Volume 339 (2011) no. 12, pp. 751-755. doi : 10.1016/j.crme.2011.09.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.09.001/

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