Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux

Han, Yuzhu; Gao, Wenjie; Li, Haixia

doi:10.1016/j.crma.2015.07.003

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

Han, Yuzhu ¹ ; Gao, Wenjie ¹ ; Li, Haixia ²

¹ School of Mathematics, Jilin University, Changchun 130012, PR China
² School of Mathematics, Changchun Normal University, Changchun 130032, PR China

Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 825-830.

Résumé
Abstract

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

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

u_{t} - Δ u + \int_{0}^{t} g (t - s) Δ u (x, s) d s = 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 : 2014-06-19
Accepté le : 2015-07-09
Publié le : 2015-08-04

DOI : 10.1016/j.crma.2015.07.003

Affiliations des auteurs :

Han, Yuzhu ¹ ; Gao, Wenjie ¹ ; Li, Haixia ²

¹ School of Mathematics, Jilin University, Changchun 130012, PR China
² School of Mathematics, Changchun Normal University, Changchun 130032, PR China

@article{CRMATH_2015__353_9_825_0,
     author = {Han, Yuzhu and Gao, Wenjie and Li, Haixia},
     title = {Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {825--830},
     publisher = {Elsevier},
     volume = {353},
     number = {9},
     year = {2015},
     doi = {10.1016/j.crma.2015.07.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.07.003/}
}

TY  - JOUR
AU  - Han, Yuzhu
AU  - Gao, Wenjie
AU  - Li, Haixia
TI  - Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 825
EP  - 830
VL  - 353
IS  - 9
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.07.003/
DO  - 10.1016/j.crma.2015.07.003
LA  - en
ID  - CRMATH_2015__353_9_825_0
ER  -

%0 Journal Article
%A Han, Yuzhu
%A Gao, Wenjie
%A Li, Haixia
%T Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux
%J Comptes Rendus. Mathématique
%D 2015
%P 825-830
%V 353
%N 9
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.07.003/
%R 10.1016/j.crma.2015.07.003
%G en
%F CRMATH_2015__353_9_825_0

Han, Yuzhu; Gao, Wenjie; Li, Haixia. Blow-up of solutions to a semilinear heat equation with a viscoelastic term and a nonlinear boundary flux. Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 825-830. doi : 10.1016/j.crma.2015.07.003. http://www.numdam.org/articles/10.1016/j.crma.2015.07.003/

Bibliographie
Cité par

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

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

[3] Fang, Z.B.; Liu, L.R. 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] Fang, Z.B.; Sun, L. 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] Fila, M. Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carol., Volume 30 (1989), pp. 479-484

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

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

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

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

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

[11] Hu, B.; Yin, H.M. 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] Messaoudi, S.A. 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] Messaoudi, S.A. Blow up of solutions of a semilinear heat equation with a memory term, Abstr. Appl. Anal., Volume 2 (2005), pp. 87-94

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

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

[16] Walter, W. 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] Weissler, F.B. An $L^{\infty}$ blow-up estimate for a nonlinear heat equation, Commun. Pure Appl. Math., Volume 38 (1985), pp. 291-295

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

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

Cité par Sources :

^☆ 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).