Revista de la
Unión Matemática Argentina
Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations
Mohamed Amine Kerker
Volume 63, no. 2 (2022), pp. 293–303    

https://doi.org/10.33044/revuma.2099

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Abstract

We consider a class of semilinear wave equations with both strongly and nonlinear weakly damped terms, \[ u_{tt}-\Delta u-\omega\Delta u_t+\mu\vert u_t\vert^{m-2}u_t=\vert u\vert^{p-2}u, \] associated with initial and Dirichlet boundary conditions. Under certain conditions, we show that any solution with arbitrarily high positive initial energy blows up in finite time if $m < p$. Furthermore, we obtain a lower bound for the blow-up time.

References

  1. K. Baghaei, Lower bounds for the blow-up time in a superlinear hyperbolic equation with linear damping term, Comput. Math. Appl. 73 (2017), no. 4, 560–564. MR 3606353.
  2. Y. Boukhatem and B. Benabderrahmane, Blow up of solutions for a semilinear hyperbolic equation, Electron. J. Qual. Theory Differ. Equ. 2012, no. 40, 12 pp. MR 2920963.
  3. F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 23 (2006), no. 2, 185–207. MR 2201151.
  4. V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994), no. 2, 295–308. MR 1273304.
  5. S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions, Adv. Differential Equations 13 (2008), no. 11-12, 1051–1074. MR 2483130.
  6. B. Guo and F. Liu, A lower bound for the blow-up time to a viscoelastic hyperbolic equation with nonlinear sources, Appl. Math. Lett. 60 (2016), 115–119. MR 3505862.
  7. M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear viscoelastic problem with arbitrary positive initial energy, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 (2013), no. 6, 657–665. MR 3134526.
  8. H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138–146. MR 0399682.
  9. H. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc. 129 (2001), no. 3, 793–805. MR 1792187.
  10. S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr. 231 (2001), no. 1, 105–111. MR 1866197.
  11. S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl. 320 (2006), no. 2, 902–915. MR 2226003.
  12. G. A. Philippin, Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys. 66 (2015), no. 1, 129–134. MR 3304710.
  13. H. Song, Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation, Nonlinear Anal. Real World Appl. 26 (2015), 306–314. MR 3384338.
  14. G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $R^n$, J. Math. Anal. Appl. 303 (2005), no. 1, 242–257. MR 2113879.
  15. E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), no. 2, 155–182. MR 1719145.
  16. Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite time blow up, Commun. Pure Appl. Anal. 18 (2019), no. 3, 1351–1358. MR 3917710.
  17. S. Yu, On the strongly damped wave equation with nonlinear damping and source terms, Electron. J. Qual. Theory Differ. Equ. 2009, no. 39, 18 pp. MR 2511292.
  18. J. Zhou, Lower bounds for blow-up time of two nonlinear wave equations, Appl. Math. Lett. 45 (2015), 64–68. MR 3316963.