# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021025
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## 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 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 & Control Theory, doi: 10.3934/eect.2021025
##### References:
 [1] H. Chen, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. 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.  Google Scholar [7] Y. Guo, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] F. Sun, L. 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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] H. Chen, P. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. 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.  Google Scholar [7] Y. Guo, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] F. Sun, L. 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.  Google Scholar [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.  Google Scholar
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