# 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 and Continuous Dynamical Systems, 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)., (). [2] M. D'Abbicco and S. Lucente, A modified test function method for damped wave equations, Adv. Nonlinear Stud. 13 (2013), 867-892. [3] M. D'Abbicco, S. Lucente and M. Reissig, Semi-Linear wave equations with effective damping, Chin. Ann. Math., Ser. B, 34 (2013), 345-380. doi: 10.1007/s11401-013-0773-0. [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), 109-124. [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652. [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), 82-118. doi: 10.1016/j.jde.2004.03.034. [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)., (). [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), 1189-1208. doi: 10.1016/j.na.2005.01.097. [9] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential, Funkcialaj Ekvacioj, 52 (2009), 411-435. doi: 10.1619/fesi.52.411. [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), 183-236. doi: 10.2969/jmsj/06510183. [11] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505. doi: 10.1002/cpa.3160330403. [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), 48-62. doi: 10.1002/mma.1330. [13] M. Khader, Global existence for the dissipative wave equations with space-time dependent potential, Nonlinear Anal., 81 (2013), 87-100. doi: 10.1016/j.na.2012.10.015. [14] H. Kuiper, Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems, Electron. J. Differential Equations, 2003 (2003), 1-11. [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), 4307-4320. doi: 10.3934/dcds.2012.32.4307. [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), 445-469. doi: 10.1016/S0022-0396(03)00026-3. [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. [18] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci., 12 (1976), 383-390. doi: 10.2977/prims/1195190721. [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), 585-626. doi: 10.2969/jmsj/1191418647. [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), 631-649. [21] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. [22] G. Todorova and B. Yordanov, Weighted $L^2$-estimates for dissipative wave equations with variable coefficients, J. Differential Equations, 246 (2009), 4497-4518. doi: 10.1016/j.jde.2009.03.020. [23] Y. Wakasugi, Small data global existence for the semilinear wave equation with space-time dependent damping, J. Math. Anal. Appl., 393 (2012), 66-79. doi: 10.1016/j.jmaa.2012.03.035. [24] Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, Trends in Mathematics, (2014), 375-390. doi: 10.1007/978-3-319-02550-6_19. [25] J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Meth. Appl. Sci., 27 (2004), 101-124. doi: 10.1002/mma.446. [26] J. Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations, 222 (2006), 487-514. doi: 10.1016/j.jde.2005.07.019. [27] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74-103. doi: 10.1016/j.jde.2006.06.004. [28] H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433. doi: 10.1016/S0007-4497(00)00141-X. [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), 109-114. doi: 10.1016/S0764-4442(01)01999-1. [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), 230-243.

show all references

##### References:
 [1] M. D'Abbicco, The threshold of effective damping for semilinear wave equations,, Math. Methods Appl. Sci. (to appear)., (). [2] M. D'Abbicco and S. Lucente, A modified test function method for damped wave equations, Adv. Nonlinear Stud. 13 (2013), 867-892. [3] M. D'Abbicco, S. Lucente and M. Reissig, Semi-Linear wave equations with effective damping, Chin. Ann. Math., Ser. B, 34 (2013), 345-380. doi: 10.1007/s11401-013-0773-0. [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), 109-124. [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652. [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), 82-118. doi: 10.1016/j.jde.2004.03.034. [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)., (). [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), 1189-1208. doi: 10.1016/j.na.2005.01.097. [9] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential, Funkcialaj Ekvacioj, 52 (2009), 411-435. doi: 10.1619/fesi.52.411. [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), 183-236. doi: 10.2969/jmsj/06510183. [11] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505. doi: 10.1002/cpa.3160330403. [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), 48-62. doi: 10.1002/mma.1330. [13] M. Khader, Global existence for the dissipative wave equations with space-time dependent potential, Nonlinear Anal., 81 (2013), 87-100. doi: 10.1016/j.na.2012.10.015. [14] H. Kuiper, Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems, Electron. J. Differential Equations, 2003 (2003), 1-11. [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), 4307-4320. doi: 10.3934/dcds.2012.32.4307. [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), 445-469. doi: 10.1016/S0022-0396(03)00026-3. [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. [18] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci., 12 (1976), 383-390. doi: 10.2977/prims/1195190721. [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), 585-626. doi: 10.2969/jmsj/1191418647. [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), 631-649. [21] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. [22] G. Todorova and B. Yordanov, Weighted $L^2$-estimates for dissipative wave equations with variable coefficients, J. Differential Equations, 246 (2009), 4497-4518. doi: 10.1016/j.jde.2009.03.020. [23] Y. Wakasugi, Small data global existence for the semilinear wave equation with space-time dependent damping, J. Math. Anal. Appl., 393 (2012), 66-79. doi: 10.1016/j.jmaa.2012.03.035. [24] Y. Wakasugi, Critical exponent for the semilinear wave equation with scale invariant damping, Trends in Mathematics, (2014), 375-390. doi: 10.1007/978-3-319-02550-6_19. [25] J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Meth. Appl. Sci., 27 (2004), 101-124. doi: 10.1002/mma.446. [26] J. Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations, 222 (2006), 487-514. doi: 10.1016/j.jde.2005.07.019. [27] J. Wirth, Wave equations with time-dependent dissipation II. Effective dissipation, J. Differential Equations, 232 (2007), 74-103. doi: 10.1016/j.jde.2006.06.004. [28] H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433. doi: 10.1016/S0007-4497(00)00141-X. [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), 109-114. doi: 10.1016/S0764-4442(01)01999-1. [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), 230-243.
 [1] Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 [2] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [3] Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351 [4] Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105 [5] Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143 [6] Ahmad Z. Fino, Mohamed Ali Hamza. Blow-up of solutions to semilinear wave equations with a time-dependent strong damping. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022006 [7] Júlia Matos. Unfocused blow up solutions of semilinear parabolic equations. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 905-928. doi: 10.3934/dcds.1999.5.905 [8] Tayeb Hadj Kaddour, Michael Reissig. Blow-up results for effectively damped wave models with nonlinear memory. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2687-2707. doi: 10.3934/cpaa.2020239 [9] John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31 [10] Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 [11] Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082 [12] Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 [13] Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585 [14] Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733 [15] Mengyun Liu, Chengbo Wang. Global existence for semilinear damped wave equations in relation with the Strauss conjecture. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 709-724. doi: 10.3934/dcds.2020058 [16] Kyouhei Wakasa. The lifespan of solutions to semilinear damped wave equations in one space dimension. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1265-1283. doi: 10.3934/cpaa.2016.15.1265 [17] Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719 [18] Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065 [19] Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations and Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 [20] Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

2020 Impact Factor: 1.392