# American Institute of Mathematical Sciences

September  2014, 34(9): 3831-3846. doi: 10.3934/dcds.2014.34.3831

## Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

 1 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

Received  July 2013 Revised  December 2013 Published  March 2014

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.
Citation: Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
##### References:
 [1] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, Math. Methods Appl. Sci. (to appear)., (). Google Scholar [2] M. D'Abbicco and S. Lucente, A modified test function method for damped wave equations,, Adv. Nonlinear Stud. 13 (2013), 13 (2013), 867. Google Scholar [3] M. D'Abbicco, S. Lucente and M. Reissig, Semi-Linear wave equations with effective damping,, Chin. Ann. Math., 34 (2013), 345. doi: 10.1007/s11401-013-0773-0. Google Scholar [4] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109. Google Scholar [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, Differential Integral Equations, 17 (2004), 637. Google Scholar [6] T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations,, J. Differential Equations, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034. Google Scholar [7] M. Ikeda and Y. Wakasugi, A note on the lifespan of solutions to the semilinear damped wave equation,, Proc. Amer. Math. Soc. (to appear)., (). Google Scholar [8] R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbfR^N$ with noncompactly supported initial data,, Nonliear Anal., 61 (2005), 1189. doi: 10.1016/j.na.2005.01.097. Google Scholar [9] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential,, Funkcialaj Ekvacioj, 52 (2009), 411. doi: 10.1619/fesi.52.411. Google Scholar [10] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential,, J. Math. Soc. Japan, 65 (2013), 183. doi: 10.2969/jmsj/06510183. Google Scholar [11] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations,, Comm. Pure Appl. Math., 33 (1980), 501. doi: 10.1002/cpa.3160330403. Google Scholar [12] J. S. Kenigson and J. J. Kenigson, Energy decay estimates for the dissipative wave equation with space-time dependent potential,, Math. Meth. Appl. Sci., 34 (2011), 48. doi: 10.1002/mma.1330. Google Scholar [13] M. Khader, Global existence for the dissipative wave equations with space-time dependent potential,, Nonlinear Anal., 81 (2013), 87. doi: 10.1016/j.na.2012.10.015. Google Scholar [14] H. Kuiper, Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems,, Electron. J. Differential Equations, 2003 (2003), 1. Google Scholar [15] J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping,, Discrete Contin. Dyn. Syst., 32 (2012), 4307. doi: 10.3934/dcds.2012.32.4307. Google Scholar [16] P. Marcati and K. Nishihara, The $L^p-L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media,, J. Differential Equations, 191 (2003), 445. doi: 10.1016/S0022-0396(03)00026-3. Google Scholar [17] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (): 169. doi: 10.2977/prims/1195190962. Google Scholar [18] K. Mochizuki, Scattering theory for wave equations with dissipative terms,, Publ. Res. Inst. Math. Sci., 12 (1976), 383. doi: 10.2977/prims/1195190721. Google Scholar [19] T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem,, J. Math. Soc. Japan, 56 (2004), 585. doi: 10.2969/jmsj/1191418647. Google Scholar [20] K. Nishihara, $L^p-L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application,, Math. Z., 244 (2003), 631. Google Scholar [21] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464. doi: 10.1006/jdeq.2000.3933. Google Scholar [22] G. Todorova and B. Yordanov, Weighted $L^2$-estimates for dissipative wave equations with variable coefficients,, J. Differential Equations, 246 (2009), 4497. doi: 10.1016/j.jde.2009.03.020. Google Scholar [23] Y. Wakasugi, Small data global existence for the semilinear wave equation with space-time dependent damping,, J. Math. Anal. Appl., 393 (2012), 66. doi: 10.1016/j.jmaa.2012.03.035. Google Scholar [24] Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping,, Trends in Mathematics, (2014), 375. doi: 10.1007/978-3-319-02550-6_19. Google Scholar [25] J. Wirth, Solution representations for a wave equation with weak dissipation,, Math. Meth. Appl. Sci., 27 (2004), 101. doi: 10.1002/mma.446. Google Scholar [26] J. Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation,, J. Differential Equations, 222 (2006), 487. doi: 10.1016/j.jde.2005.07.019. Google Scholar [27] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation,, J. Differential Equations, 232 (2007), 74. doi: 10.1016/j.jde.2006.06.004. Google Scholar [28] H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves,, Bull. Sci. Math., 124 (2000), 415. doi: 10.1016/S0007-4497(00)00141-X. Google Scholar [29] Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case,, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109. doi: 10.1016/S0764-4442(01)01999-1. Google Scholar [30] Y. Zhou, Life span of classical solutions to $u_{t t} - u_{x x} = |u|^{1+\alpha}$,, Chinese Ann. Math. Ser. B, 13 (1992), 230. Google Scholar

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##### References:
 [1] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, Math. Methods Appl. Sci. (to appear)., (). Google Scholar [2] M. D'Abbicco and S. Lucente, A modified test function method for damped wave equations,, Adv. Nonlinear Stud. 13 (2013), 13 (2013), 867. Google Scholar [3] M. D'Abbicco, S. Lucente and M. Reissig, Semi-Linear wave equations with effective damping,, Chin. Ann. Math., 34 (2013), 345. doi: 10.1007/s11401-013-0773-0. Google Scholar [4] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109. Google Scholar [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, Differential Integral Equations, 17 (2004), 637. Google Scholar [6] T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations,, J. Differential Equations, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034. Google Scholar [7] M. Ikeda and Y. Wakasugi, A note on the lifespan of solutions to the semilinear damped wave equation,, Proc. Amer. Math. Soc. (to appear)., (). Google Scholar [8] R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbfR^N$ with noncompactly supported initial data,, Nonliear Anal., 61 (2005), 1189. doi: 10.1016/j.na.2005.01.097. Google Scholar [9] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential,, Funkcialaj Ekvacioj, 52 (2009), 411. doi: 10.1619/fesi.52.411. Google Scholar [10] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay rate of the energy for wave equations with critical potential,, J. Math. Soc. Japan, 65 (2013), 183. doi: 10.2969/jmsj/06510183. Google Scholar [11] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations,, Comm. Pure Appl. Math., 33 (1980), 501. doi: 10.1002/cpa.3160330403. Google Scholar [12] J. S. Kenigson and J. J. Kenigson, Energy decay estimates for the dissipative wave equation with space-time dependent potential,, Math. Meth. Appl. Sci., 34 (2011), 48. doi: 10.1002/mma.1330. Google Scholar [13] M. Khader, Global existence for the dissipative wave equations with space-time dependent potential,, Nonlinear Anal., 81 (2013), 87. doi: 10.1016/j.na.2012.10.015. Google Scholar [14] H. Kuiper, Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems,, Electron. J. Differential Equations, 2003 (2003), 1. Google Scholar [15] J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping,, Discrete Contin. Dyn. Syst., 32 (2012), 4307. doi: 10.3934/dcds.2012.32.4307. Google Scholar [16] P. Marcati and K. Nishihara, The $L^p-L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media,, J. Differential Equations, 191 (2003), 445. doi: 10.1016/S0022-0396(03)00026-3. Google Scholar [17] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. Res. Inst. Math. Sci., 12 (): 169. doi: 10.2977/prims/1195190962. Google Scholar [18] K. Mochizuki, Scattering theory for wave equations with dissipative terms,, Publ. Res. Inst. Math. Sci., 12 (1976), 383. doi: 10.2977/prims/1195190721. Google Scholar [19] T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem,, J. Math. Soc. Japan, 56 (2004), 585. doi: 10.2969/jmsj/1191418647. Google Scholar [20] K. Nishihara, $L^p-L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application,, Math. Z., 244 (2003), 631. Google Scholar [21] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, J. Differential Equations, 174 (2001), 464. doi: 10.1006/jdeq.2000.3933. Google Scholar [22] G. Todorova and B. Yordanov, Weighted $L^2$-estimates for dissipative wave equations with variable coefficients,, J. Differential Equations, 246 (2009), 4497. doi: 10.1016/j.jde.2009.03.020. Google Scholar [23] Y. Wakasugi, Small data global existence for the semilinear wave equation with space-time dependent damping,, J. Math. Anal. Appl., 393 (2012), 66. doi: 10.1016/j.jmaa.2012.03.035. Google Scholar [24] Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping,, Trends in Mathematics, (2014), 375. doi: 10.1007/978-3-319-02550-6_19. Google Scholar [25] J. Wirth, Solution representations for a wave equation with weak dissipation,, Math. Meth. Appl. Sci., 27 (2004), 101. doi: 10.1002/mma.446. Google Scholar [26] J. Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation,, J. Differential Equations, 222 (2006), 487. doi: 10.1016/j.jde.2005.07.019. Google Scholar [27] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation,, J. Differential Equations, 232 (2007), 74. doi: 10.1016/j.jde.2006.06.004. Google Scholar [28] H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves,, Bull. Sci. Math., 124 (2000), 415. doi: 10.1016/S0007-4497(00)00141-X. Google Scholar [29] Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case,, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109. doi: 10.1016/S0764-4442(01)01999-1. Google Scholar [30] Y. Zhou, Life span of classical solutions to $u_{t t} - u_{x x} = |u|^{1+\alpha}$,, Chinese Ann. Math. Ser. B, 13 (1992), 230. Google Scholar
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