# 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 & Continuous Dynamical Systems - A, 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, 13 (1966), 109.   Google Scholar [2] Y. Han 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 [3] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503.  doi: 10.3792/pja/1195519254.  Google Scholar [4] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, Differential Integral Equations, 17 (2004), 637.   Google Scholar [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation in the subcritical case,, J. Differential Equations, 207 (2004), 161.  doi: 10.1016/j.jde.2004.06.018.  Google Scholar [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.  doi: 10.1016/j.jmaa.2007.01.021.  Google Scholar [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.  doi: 10.1016/j.jde.2004.03.034.  Google Scholar [8] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain,, J. Differential Equations, 186 (2002), 633.  doi: 10.1016/S0022-0396(02)00008-6.  Google Scholar [9] R. Ikehata and K. Nishihara, Diffusion phenomenon for second order linear evolution equations,, Studia Math., 158 (2003), 153.  doi: 10.4064/sm158-2-4.  Google Scholar [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.  doi: 10.1016/j.jde.2006.01.002.  Google Scholar [11] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential,, Funk. Ekvac., 52 (2009), 411.  doi: 10.1619/fesi.52.411.  Google Scholar [12] R. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations,, Studia Math., 143 (2000), 175.   Google Scholar [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.  doi: 10.2969/jmsj/04740617.  Google Scholar [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.  doi: 10.3934/dcds.1995.1.503.  Google Scholar [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.  doi: 10.1016/j.jde.2009.09.022.  Google Scholar [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.  doi: 10.1016/j.jmaa.2010.09.032.  Google Scholar [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.., ().   Google Scholar [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.  doi: 10.2969/jmsj/1191418647.  Google Scholar [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.   Google Scholar [20] K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space,, J. Math. Soc. Japan 58 (2006), 58 (2006), 805.  doi: 10.2969/jmsj/1156342039.  Google Scholar [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.  doi: 10.1080/03605302.2010.490285.  Google Scholar [22] K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping,, Tokyo J. Math., 34 (2011), 327.  doi: 10.3836/tjm/1327931389.  Google Scholar [23] K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations,, J. Math. Anal. Appl., 360 (2009), 412.  doi: 10.1016/j.jmaa.2009.06.065.  Google Scholar [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.  doi: 10.1016/j.jmaa.2005.08.059.  Google Scholar [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.  doi: 10.3934/dcdss.2009.2.609.  Google Scholar [26] M. Reissig, $L_p$-$L_q$ decay estimates for wave equations with time-dependent coefficients,, J. Nonlinear Math. Phys., 11 (2004), 534.  doi: 10.2991/jnmp.2004.11.4.9.  Google Scholar [27] 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 [28] G. Todorova and B. Yordanov, Nonlinear dissipative wave equations with potential,, AMS Contemporary Mathematics, 426 (2007), 317.  doi: 10.1090/conm/426/08196.  Google Scholar [29] 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 [30] 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 [31] T. Yamazaki, Asymptotic behavior for abstract wave equations with decaying dissipation,, Adv. Differential Equations, 11 (2006), 419.   Google Scholar [32] T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation,, Adv. Stud. Pure Math., 47 (2007), 363.   Google Scholar [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.  doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar [34] Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbbR^N$,, Appl. Math. Lett., 18 (2005), 281.  doi: 10.1016/j.aml.2003.07.018.  Google Scholar

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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, 13 (1966), 109.   Google Scholar [2] Y. Han 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 [3] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equations,, Proc. Japan Acad., 49 (1973), 503.  doi: 10.3792/pja/1195519254.  Google Scholar [4] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, Differential Integral Equations, 17 (2004), 637.   Google Scholar [5] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation in the subcritical case,, J. Differential Equations, 207 (2004), 161.  doi: 10.1016/j.jde.2004.06.018.  Google Scholar [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.  doi: 10.1016/j.jmaa.2007.01.021.  Google Scholar [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.  doi: 10.1016/j.jde.2004.03.034.  Google Scholar [8] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain,, J. Differential Equations, 186 (2002), 633.  doi: 10.1016/S0022-0396(02)00008-6.  Google Scholar [9] R. Ikehata and K. Nishihara, Diffusion phenomenon for second order linear evolution equations,, Studia Math., 158 (2003), 153.  doi: 10.4064/sm158-2-4.  Google Scholar [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.  doi: 10.1016/j.jde.2006.01.002.  Google Scholar [11] R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential,, Funk. Ekvac., 52 (2009), 411.  doi: 10.1619/fesi.52.411.  Google Scholar [12] R. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations,, Studia Math., 143 (2000), 175.   Google Scholar [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.  doi: 10.2969/jmsj/04740617.  Google Scholar [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.  doi: 10.3934/dcds.1995.1.503.  Google Scholar [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.  doi: 10.1016/j.jde.2009.09.022.  Google Scholar [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.  doi: 10.1016/j.jmaa.2010.09.032.  Google Scholar [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.., ().   Google Scholar [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.  doi: 10.2969/jmsj/1191418647.  Google Scholar [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.   Google Scholar [20] K. Nishihara, Global asymptotics for the damped wave equation with absorption in higher dimensional space,, J. Math. Soc. Japan 58 (2006), 58 (2006), 805.  doi: 10.2969/jmsj/1156342039.  Google Scholar [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.  doi: 10.1080/03605302.2010.490285.  Google Scholar [22] K. Nishihara, Asymptotic behavior of solutions to the semilinear wave equation with time-dependent damping,, Tokyo J. Math., 34 (2011), 327.  doi: 10.3836/tjm/1327931389.  Google Scholar [23] K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations,, J. Math. Anal. Appl., 360 (2009), 412.  doi: 10.1016/j.jmaa.2009.06.065.  Google Scholar [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.  doi: 10.1016/j.jmaa.2005.08.059.  Google Scholar [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.  doi: 10.3934/dcdss.2009.2.609.  Google Scholar [26] M. Reissig, $L_p$-$L_q$ decay estimates for wave equations with time-dependent coefficients,, J. Nonlinear Math. Phys., 11 (2004), 534.  doi: 10.2991/jnmp.2004.11.4.9.  Google Scholar [27] 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 [28] G. Todorova and B. Yordanov, Nonlinear dissipative wave equations with potential,, AMS Contemporary Mathematics, 426 (2007), 317.  doi: 10.1090/conm/426/08196.  Google Scholar [29] 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 [30] 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 [31] T. Yamazaki, Asymptotic behavior for abstract wave equations with decaying dissipation,, Adv. Differential Equations, 11 (2006), 419.   Google Scholar [32] T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation,, Adv. Stud. Pure Math., 47 (2007), 363.   Google Scholar [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.  doi: 10.1016/S0764-4442(01)01999-1.  Google Scholar [34] Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbbR^N$,, Appl. Math. Lett., 18 (2005), 281.  doi: 10.1016/j.aml.2003.07.018.  Google Scholar
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