# American Institute of Mathematical Sciences

December  2012, 32(12): 4307-4320. doi: 10.3934/dcds.2012.32.4307

## Critical exponent for the semilinear wave equation with time-dependent damping

 1 Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R., China 2 Faculty of Political Science and Economics, Waseda University, Tokyo 169-8050, Japan 3 Department of Mathematics, Zhejiang University, Hangzhou 310027

Received  May 2011 Revised  May 2012 Published  August 2012

We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$\left\{ \begin{array}{ll} u_{tt} - \Delta u + b(t)u_t=|u|^{\rho}, & (t,x) \in \mathbb{R}^+ \times \mathbb{R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x), & x \in \mathbb{R}^N. \end{array}\right. (*)$$ When $b(t)=b_0(t+1)^{-\beta}$ with $b_0>0$ and $-1 < \beta <1$ and $\int_{{\bf R}^N} u_i(x)\,dx >0\,(i=0,1)$, we show that the time-global solution of ($*$) does not exist provided that $1<\rho \leq \rho_F(N):= 1+2/N$ (Fujita exponent). On the other hand, when $\rho_F(N)<\rho<\frac{N+2}{[N-2]_+}:= \left\{ \begin{array}{ll} \infty & (N=1,2), \\ (N+2)/(N-2) & (N \ge 3), \end{array} \right.$ the small data global existence of solution has been recently proved in [K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math. 34 (2011), 327-343] provided that $0 \le \beta<1$. We can prove the small data global existence even if $-1<\beta<0$. Thus, we conclude that the Fujita exponent $\rho_F(N)$ is still critical even in the time-dependent damping case. For the proofs we apply the weighted energy method and the method of test functions by [Qi S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001), 109--114].
Citation: Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
##### References:
 [1] 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, Sec. 1., 13 (1966), 109-124. [2] Y. Han 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. [3] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254. [4] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652. [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation in the subcritical case, J. Differential Equations, 207 (2004), 161-194. doi: 10.1016/j.jde.2004.06.018. [6] N. Hayashi, E. I. Kaikina and P. I. Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl., 334 (2007), 1400-1425. doi: 10.1016/j.jmaa.2007.01.021. [7] 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. [8] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651. doi: 10.1016/S0022-0396(02)00008-6. [9] R. Ikehata and K. Nishihara, Diffusion phenomenon for second order linear evolution equations, Studia Math., 158 (2003), 153-161. doi: 10.4064/sm158-2-4. [10] R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations, 226 (2006), 1-29. doi: 10.1016/j.jde.2006.01.002. [11] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential, Funk. Ekvac., 52 (2009), 411-435. doi: 10.1619/fesi.52.411. [12] R. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197. [13] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617. [14] T.-T. Li and Yi. Zhou, Breakdown of solutions to $\square u +u_t = u^{1+\alpha}$, Discrete Cont. Dynam. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. [15] J. Lin, K. Nishihara and J. Zhai, $L^2$ estimates of solutions for the damped wave equations with space-time dependent damping term, J. Differential Equations, 248 (2010), 403-422. doi: 10.1016/j.jde.2009.09.022. [16] J. Lin, K. Nishihara and J. Zhai, Decay property of solutions for damped wave equations with space-time dependent damping term, J. Math. Anal. Appl., 374 (2011), 602-614. doi: 10.1016/j.jmaa.2010.09.032. [17] J. Lin and J. Zhai, Blow-up of the solution for semilinear damped wave equation with time-dependent damping, to appear in Commun. Contemp. Math.. [18] 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. [19] 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. [20] K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan 58 (2006), 805-836. doi: 10.2969/jmsj/1156342039. [21] K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Commun. Partial Differential Equations, 35 (2010), 1402-1418. doi: 10.1080/03605302.2010.490285. [22] K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math., 34 (2011), 327-343. doi: 10.3836/tjm/1327931389. [23] K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421. doi: 10.1016/j.jmaa.2009.06.065. [24] K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl., 313 (2006), 598-610. doi: 10.1016/j.jmaa.2005.08.059. [25] P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Disc. Cont. Dyn. Sys. S, 2 (2009), 609-629. doi: 10.3934/dcdss.2009.2.609. [26] M. Reissig, $L_p$-$L_q$ decay estimates for wave equations with time-dependent coefficients, J. Nonlinear Math. Phys., 11 (2004), 534-548. doi: 10.2991/jnmp.2004.11.4.9. [27] 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. [28] G. Todorova and B. Yordanov, Nonlinear dissipative wave equations with potential, AMS Contemporary Mathematics, 426 (2007), 317-337. doi: 10.1090/conm/426/08196. [29] 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. [30] 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. [31] T. Yamazaki, Asymptotic behavior for abstract wave equations with decaying dissipation, Adv. Differential Equations, 11 (2006), 419-456. [32] T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation, Adv. Stud. Pure Math., 47 (2007), 363-381. [33] Qi. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris, 333 (2001), 109-114. doi: 10.1016/S0764-4442(01)01999-1. [34] Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbb{R}^N2$, Appl. Math. Lett., 18 (2005), 281-286. doi: 10.1016/j.aml.2003.07.018.

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##### References:
 [1] 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, Sec. 1., 13 (1966), 109-124. [2] Y. Han 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. [3] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-505. doi: 10.3792/pja/1195519254. [4] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652. [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation in the subcritical case, J. Differential Equations, 207 (2004), 161-194. doi: 10.1016/j.jde.2004.06.018. [6] N. Hayashi, E. I. Kaikina and P. I. Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl., 334 (2007), 1400-1425. doi: 10.1016/j.jmaa.2007.01.021. [7] 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. [8] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651. doi: 10.1016/S0022-0396(02)00008-6. [9] R. Ikehata and K. Nishihara, Diffusion phenomenon for second order linear evolution equations, Studia Math., 158 (2003), 153-161. doi: 10.4064/sm158-2-4. [10] R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations, 226 (2006), 1-29. doi: 10.1016/j.jde.2006.01.002. [11] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential, Funk. Ekvac., 52 (2009), 411-435. doi: 10.1619/fesi.52.411. [12] R. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197. [13] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617. [14] T.-T. Li and Yi. Zhou, Breakdown of solutions to $\square u +u_t = u^{1+\alpha}$, Discrete Cont. Dynam. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. [15] J. Lin, K. Nishihara and J. Zhai, $L^2$ estimates of solutions for the damped wave equations with space-time dependent damping term, J. Differential Equations, 248 (2010), 403-422. doi: 10.1016/j.jde.2009.09.022. [16] J. Lin, K. Nishihara and J. Zhai, Decay property of solutions for damped wave equations with space-time dependent damping term, J. Math. Anal. Appl., 374 (2011), 602-614. doi: 10.1016/j.jmaa.2010.09.032. [17] J. Lin and J. Zhai, Blow-up of the solution for semilinear damped wave equation with time-dependent damping, to appear in Commun. Contemp. Math.. [18] 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. [19] 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. [20] K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space, J. Math. Soc. Japan 58 (2006), 805-836. doi: 10.2969/jmsj/1156342039. [21] K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Commun. Partial Differential Equations, 35 (2010), 1402-1418. doi: 10.1080/03605302.2010.490285. [22] K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping, Tokyo J. Math., 34 (2011), 327-343. doi: 10.3836/tjm/1327931389. [23] K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421. doi: 10.1016/j.jmaa.2009.06.065. [24] K. Nishihara and H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl., 313 (2006), 598-610. doi: 10.1016/j.jmaa.2005.08.059. [25] P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Disc. Cont. Dyn. Sys. S, 2 (2009), 609-629. doi: 10.3934/dcdss.2009.2.609. [26] M. Reissig, $L_p$-$L_q$ decay estimates for wave equations with time-dependent coefficients, J. Nonlinear Math. Phys., 11 (2004), 534-548. doi: 10.2991/jnmp.2004.11.4.9. [27] 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. [28] G. Todorova and B. Yordanov, Nonlinear dissipative wave equations with potential, AMS Contemporary Mathematics, 426 (2007), 317-337. doi: 10.1090/conm/426/08196. [29] 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. [30] 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. [31] T. Yamazaki, Asymptotic behavior for abstract wave equations with decaying dissipation, Adv. Differential Equations, 11 (2006), 419-456. [32] T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation, Adv. Stud. Pure Math., 47 (2007), 363-381. [33] Qi. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris, 333 (2001), 109-114. doi: 10.1016/S0764-4442(01)01999-1. [34] Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbb{R}^N2$, Appl. Math. Lett., 18 (2005), 281-286. doi: 10.1016/j.aml.2003.07.018.
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