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June  2022, 11(3): 781-792. doi: 10.3934/eect.2021025

The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China

2. 

School of Mathematics, Jilin University, Changchun, Jilin, 130012, China

* Corresponding author: Menglan Liao

Received  August 2020 Revised  April 2021 Published  June 2022 Early access  May 2021

This paper deals with the following viscoelastic wave equation with a strong damping and logarithmic nonlinearity:
$ u_{tt}-\Delta u+\int_0^tg(t-s)\Delta u(s)ds-\Delta u_t = |u|^{p-2}u\ln|u|. $
A finite time blow-up result is proved for high initial energy. Meanwhile, the lifespan of the weak solution is discussed. The present results in this paper complement and improve the previous work that is obtained by Ha and Park [Adv. Differ. Equ., (2020) 2020: 235].
Citation: Menglan Liao. The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 781-792. doi: 10.3934/eect.2021025
References:
[1]

H. ChenP. Luo and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[2]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[3]

M. Del Pino and J. Dolbeault, Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $p$-Laplacian, C. R. Acad. Sci. Paris Ser. I Math., 334 (2002), 365-370.  doi: 10.1016/S1631-073X(02)02225-2.

[4]

H. Di, Y. Shang and Z. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl., 51 (2020), 102968. doi: 10.1016/j.nonrwa.2019.102968.

[5]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré–AN, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.

[6]

Y. GuoM. A. RammahaS. SakuntasathienE. S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.  doi: 10.1016/j.jde.2014.07.009.

[7]

Y. GuoM. A. Rammaha and S. Sakuntasathien, Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities, J. Differential Equations, 262 (2017), 1956-1979.  doi: 10.1016/j.jde.2016.10.037.

[8]

T. G. Ha and S.-H. Park, Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity, Adv. Differ. Equ., (2020), Paper No. 235, 17 pp. doi: 10.1186/s13662-020-02694-x.

[9]

Y. Han and Q. Li, Lifespan of solutions to a damped plate equation with logarithmic nonlinearity, Evol. Equ. Control Theory. doi: 10.3934/eect.2020101.

[10]

C. N. Le and X. T. Le, Global solution and blow-up for a class of $p$–Laplacian evolution equations with logarithmic nonlinearity, Acta. Appl. Math., 151 (2017), 149-169.  doi: 10.1007/s10440-017-0106-5.

[11]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[12]

M. Liao and W. Gao, Blow-up phenomena for a nonlocal $p$–Laplace equation with Neumann boundary conditions, Arch. Math., 108 (2017), 313-324.  doi: 10.1007/s00013-016-0986-z.

[13]

L. Ma and Z. B. Zhong, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Meth. Appl. Sci., 41 (2018), 2639-2653.  doi: 10.1002/mma.4766.

[14]

S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.  doi: 10.1016/j.jmaa.2005.07.022.

[15]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.

[16]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[17]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015.

[18]

F. SunL. Liu and Y. Wu, Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level, Appl. Anal., 98 (2019), 2308-2327.  doi: 10.1080/00036811.2018.1460812.

[19]

G. Zu and B. Guo, Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy, Evol. Equ. Control Theory, 10 (2021), 259-270.  doi: 10.3934/eect.2020065.

show all references

References:
[1]

H. ChenP. Luo and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84-98.  doi: 10.1016/j.jmaa.2014.08.030.

[2]

H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.  doi: 10.1016/j.jde.2015.01.038.

[3]

M. Del Pino and J. Dolbeault, Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $p$-Laplacian, C. R. Acad. Sci. Paris Ser. I Math., 334 (2002), 365-370.  doi: 10.1016/S1631-073X(02)02225-2.

[4]

H. Di, Y. Shang and Z. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl., 51 (2020), 102968. doi: 10.1016/j.nonrwa.2019.102968.

[5]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré–AN, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.

[6]

Y. GuoM. A. RammahaS. SakuntasathienE. S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.  doi: 10.1016/j.jde.2014.07.009.

[7]

Y. GuoM. A. Rammaha and S. Sakuntasathien, Blow-up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities, J. Differential Equations, 262 (2017), 1956-1979.  doi: 10.1016/j.jde.2016.10.037.

[8]

T. G. Ha and S.-H. Park, Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity, Adv. Differ. Equ., (2020), Paper No. 235, 17 pp. doi: 10.1186/s13662-020-02694-x.

[9]

Y. Han and Q. Li, Lifespan of solutions to a damped plate equation with logarithmic nonlinearity, Evol. Equ. Control Theory. doi: 10.3934/eect.2020101.

[10]

C. N. Le and X. T. Le, Global solution and blow-up for a class of $p$–Laplacian evolution equations with logarithmic nonlinearity, Acta. Appl. Math., 151 (2017), 149-169.  doi: 10.1007/s10440-017-0106-5.

[11]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[12]

M. Liao and W. Gao, Blow-up phenomena for a nonlocal $p$–Laplace equation with Neumann boundary conditions, Arch. Math., 108 (2017), 313-324.  doi: 10.1007/s00013-016-0986-z.

[13]

L. Ma and Z. B. Zhong, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Meth. Appl. Sci., 41 (2018), 2639-2653.  doi: 10.1002/mma.4766.

[14]

S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.  doi: 10.1016/j.jmaa.2005.07.022.

[15]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.  doi: 10.1088/0951-7715/19/7/001.

[16]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.

[17]

H. Song and C. Zhong, Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 3877-3883.  doi: 10.1016/j.nonrwa.2010.02.015.

[18]

F. SunL. Liu and Y. Wu, Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level, Appl. Anal., 98 (2019), 2308-2327.  doi: 10.1080/00036811.2018.1460812.

[19]

G. Zu and B. Guo, Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy, Evol. Equ. Control Theory, 10 (2021), 259-270.  doi: 10.3934/eect.2020065.

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