This paper deals with the lower bound for blow-up solutions to a nonlinear viscoelastic hyperbolic equation. An inverse Hölder inequality with the correction constant is employed to overcome the difficulty caused by the failure of the embedding inequality ${W}_{0}^{1,r}(\mathrm{\Omega})\hookrightarrow {L}^{2\alpha -2}(\mathrm{\Omega})(\frac{{r}^{\u204e}+2}{2}<\alpha <{r}^{\u204e}=\frac{Nr}{N-r})$ and the lack of a version of the Gagliardo–Nirenberg inequality. Moreover, a lower bound for blow-up time is obtained by establishing a first-order differential inequality. This result is a continuation of an earlier work [1].

Accepted:

Published online:

Bin Guo ^{1}

@article{CRMECA_2017__345_6_370_0, author = {Bin Guo}, title = {An inverse {H\"older} inequality and its application in lower bound estimates for blow-up time}, journal = {Comptes Rendus. M\'ecanique}, pages = {370--377}, publisher = {Elsevier}, volume = {345}, number = {6}, year = {2017}, doi = {10.1016/j.crme.2017.04.002}, language = {en}, }

Bin Guo. An inverse Hölder inequality and its application in lower bound estimates for blow-up time. Comptes Rendus. Mécanique, Volume 345 (2017) no. 6, pp. 370-377. doi : 10.1016/j.crme.2017.04.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.04.002/

[1] Blow-up of solutions to quasilinear hyperbolic equations with $p(x,t)$-Laplacian operator and positive initial energy, C. R. Mecanique, Volume 342 (2014), pp. 513-519

[2] Regularity results for stationary eletrorheological fluids, Arch. Ration. Mech. Anal., Volume 164 (2002), pp. 213-259

[3] Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal., Volume 93 (2013), pp. 62-77

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

[5] Wave equation with $p(x,t)$-Laplacian and damping: blow-up of solutions, C. R. Mecanique, Volume 339 (2011), pp. 751-755

[6] 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

[7] Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., Volume 66 (2015), pp. 129-134

[8] A lower bound for the blow-up time to a damped semilinear wave equation, Appl. Math. Lett., Volume 37 (2014), pp. 22-25

[9] Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett., Volume 45 (2015), pp. 64-68

[10] Lower bounds for blow-up time in a nonlinear parabolic problems, J. Math. Anal. Appl., Volume 354 (2009), pp. 394-396

[11] Lower bound for blow-up time in parabolic problems under Neumann conditions, Appl. Anal., Volume 85 (2006), pp. 1301-1311

[12] Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity, Ann. Mat. Pura Appl. (4), Volume 191 (2012), pp. 551-562

[13] Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, Germany, 2011

[14] Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)$-Laplace operator and a non-local term, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 2, pp. 715-730

*Cited by Sources: *

Comments - Policy