Comptes Rendus
no. 9
Partial differential equations
Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux
[Explosion de solutions de l'équation de la chaleur semi-linéaire avec un terme viscoélastique et un flux de limite non linéaire]

Présenté par : the Editorial Board

Yuzhu Han1 ; Wenjie Gao1 ; Haixia Li2
1 School of Mathematics, Jilin University, Changchun 130012, PR China
2 School of Mathematics, Changchun Normal University, Changchun 130032, PR China
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 825-830.

Dans cet article, on étudie une équation de la chaleur semi-linéaire

ut−Δu+∫0tg(t−s)Δu(x,s)ds=0
avec un terme viscoélastique et un flux non linéaire sur la limite. En définissant une modifiée fonctionnelle d'énergie et en utilisant un argument de concavité, un résultat d'explosion des solutions avec énergie initiale négative est prouvé.

In this article, we study a semilinear heat equation

ut−Δu+∫0tg(t−s)Δu(x,s)ds=0
with a viscoelastic term and a nonlinear flux on the boundary. By defining a modified energy functional and using a concavity argument, a blow-up result for solutions with negative initial energy is proved.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.07.003
Yuzhu Han 1 ; Wenjie Gao 1 ; Haixia Li 2

1 School of Mathematics, Jilin University, Changchun 130012, PR China
2 School of Mathematics, Changchun Normal University, Changchun 130032, PR China 
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Yuzhu Han; Wenjie Gao; Haixia Li. Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux. Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 825-830. doi : 10.1016/j.crma.2015.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.07.003/

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

[2] R.A. Admas; J.J.F. Fournier Sobolev Spaces, Elsevier, 2003 (access online via)

[3] Z.B. Fang; L.R. Liu Global existence and uniform energy decay rates for the semilinear parabolic equation with a memory term and mixed boundary condition, Abstr. Appl. Anal., Volume 2013 (2013) (Article ID 532935)

[4] Z.B. Fang; L. Sun Blow up of solutions with positive initial energy for the nonlocal semilinear heat equation, J. Korean Soc. Ind. Appl. Math., Volume 16 (2012), pp. 235-242

[5] M. Fila Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carol., Volume 30 (1989), pp. 479-484

[6] M. Fila; P. Quittner The blowup rate for the heat equation with a nonlinear boundary condition, Math. Methods Appl. Sci., Volume 14 (1991), pp. 197-205

[7] A. Friedman; B. McLeod Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J., Volume 34 (1985), pp. 425-477

[8] Y. Giga; R.V. Kohn Asymptotic self-similar blowup of semilinear heat equations, Commun. Pure Appl. Math., Volume 38 (1985), pp. 297-319

[9] Y. Giga; R.V. Kohn Characterizing blowup using similarity variables, Indiana Univ. Math. J., Volume 36 (1987), pp. 425-447

[10] Y. Giga; R.V. Kohn Nondegeneracy of blowup for semilinear heat equations, Commun. Pure Appl. Math., Volume 42 (1989), pp. 845-884

[11] B. Hu; H.M. Yin The profile near blowup time for solutions of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc., Volume 346 (1994), pp. 117-135

[12] S.A. Messaoudi Blow-up of solutions of a semilinear heat equation with a visco-elastic term, Prog. Nonlinear Differ. Equ. Appl., Volume 64 (2005), pp. 351-356

[13] S.A. Messaoudi Blow up of solutions of a semilinear heat equation with a memory term, Abstr. Appl. Anal., Volume 2 (2005), pp. 87-94

[14] P. Pucci; J. Serrin Asymptotic stability for nonlinear parabolic systems, Energy Methods in Continuum Mechanics, Springer, The Netherlands, 1996, pp. 66-74

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

[16] W. Walter On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition, SIAM J. Math. Anal., Volume 6 (1975), pp. 85-90

[17] F.B. Weissler An L∞ blow-up estimate for a nonlinear heat equation, Commun. Pure Appl. Math., Volume 38 (1985), pp. 291-295

[18] H.M. Yin On parabolic Volterra equations in several space dimensions, SIAM J. Math. Anal., Volume 22 (1991), pp. 1723-1737

[19] H.M. Yin Weak and classical solutions of some Volterra integro-differential equations, Commun. Partial Differ. Equ., Volume 17 (1992), pp. 1369-1385

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☆ The project is supported by NSFC (11271154, 11401252), by Science and Technology Development Project of Jilin Province (20150201058NY) and by the 985 program of Jilin University. The first author is also supported by Fundamental Research Funds of Jilin University (450060501179).

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