Comptes Rendus
no. 4
Lower bound estimates of blow-up time for a quasilinear hyperbolic equation with superlinear sources
Ge Zu1 ; Fang Li1  
1 School of Mathematics, Jilin University, Changchun 130012, PR China
Comptes Rendus. Mécanique, Volume 348 (2020) no. 4, pp. 307-313.

This paper deals with the lower bound for blow-up solutions to a quasilinear hyperbolic equation with strong damping. An inverse Hölder inequality with a correction constant is employed to overcome the difficulty caused by the failure of the embedding inequality. Moreover, a lower bound for blow-up time is obtained by constructing a new control functional with a small dissipative term and by applying an inverse Hölder inequality as well as energy inequalities. This result gives a positive answer to the open problem presented in [1].

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Révisé le :
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Publié le :
DOI : 10.5802/crmeca.9
Mots clés : Inverse Hölder inequality, Energy estimate method, Energy inequality, Lower bound estimate, Quasilinear hyperbolic equation
Ge Zu 1 ; Fang Li 1

1 School of Mathematics, Jilin University, Changchun 130012, PR China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Ge Zu; Fang Li. Lower bound estimates of blow-up time for a quasilinear hyperbolic equation with superlinear sources. Comptes Rendus. Mécanique, Volume 348 (2020) no. 4, pp. 307-313. doi : 10.5802/crmeca.9. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.9/

[1] B. Guo An inverse Hölder inequality and its application in lower bound estimates for blow-up time, C. R. Mec., Volume 345 (2017), pp. 370-377 | DOI

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

[3] L. Diening; P. Harjulehto; P. Hästö; M. Rûžička Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer-Verlag, Heidelberg, 2011 | DOI | MR | Zbl

[4] M. Ruzicka Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin Heidelberg, 2000 | DOI | MR | Zbl

[5] D. H. Sattinger On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., Volume 30 (1968), pp. 148-172 | DOI | MR | Zbl

[6] H. A. Levine Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = -Au + F(u), Trans. Am. Math. Soc., Volume 192 (1974), pp. 1-21 | DOI | MR | Zbl

[7] V. Georgiev; G. Todorova Existence of a solution of the wave equation with nonlinear damping and source terms, J. Diff. Equ., Volume 109 (1994), pp. 295-308 | DOI | MR | Zbl

[8] H. T. Song; D. S. Xue Blow-up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., Volume 109 (2014), pp. 245-251 | DOI | MR | Zbl

[9] B. Guo; F. Liu A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett., Volume 60 (2016), pp. 115-119 | DOI | MR | Zbl

[10] L. E. Payne; J. C. Song; P. W. Schaefer Lower bounds for blow-up time in a nonlinear parabolic problems, J. Math. Anal. Appl., Volume 354 (2009), pp. 394-396 | DOI | MR

[11] J. Zhou Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett., Volume 45 (2015), pp. 64-68 | DOI | MR | Zbl

[12] L. L. Sun; B. Guo; W. J. Gao A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., Volume 37 (2014), pp. 22-25 | MR | Zbl

[13] S. N. Antontsev Wave equation with p(x,t)-Laplacian and damping: Existence and blow-up, Diff. Equ. Appl., Volume 3 (2011), pp. 503-525 | DOI | MR | Zbl

[14] S. N. Antontsev Wave equation with p(x,t)-Laplacian and damping: Blow-up of solutions, C. R. Mec., Volume 339 (2011), pp. 751-755 | DOI

[15] B. Guo; W. J. Gao Blow-up of solutions to quasilinear hyperbolic equations with p(x,t)-Laplacian operator and positive initial energy, C. R. Mec., Volume 342 (2014), pp. 513-519 | MR | Zbl

[16] S. A. Messaoudi; A. A. Talahmeh A blow-up result for a nonlinear wave equation with variable exponent nonlinearities, Appl. Anal., Volume 969 (2017), pp. 1509-1515 | DOI | MR | Zbl

[17] S. A. Messaoudi; A. A. Talahmeh Blow up solutions of a quasilinear wave equation with variable exponent nonlinearities, Math. Meth. Appl. Sci., Volume 40 (2017), pp. 6976-6986 | DOI | Zbl

[18] H. A. Levine Remarks on the growth and nonexistence of solutions to nonlinear wave equations, A Seminar on PDEs-1973, Rutgers Univ., New Brunswick, N.J., 1973, pp. 59-70

[19] J. M. Ball Remarks on blow-up and nonexistence theorems for nonlinear evolution equations., Quart. J. Math. Oxford Ser., Volume 28 (1977) no. 112, pp. 473-486 | DOI | MR | Zbl

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