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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 |
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 (1976/77), 169-189.
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 (1976/77), 169-189.
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. |
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