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July  2019, 18(4): 1967-2008. doi: 10.3934/cpaa.2019090

## $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

 1 Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan 2 Center for Advanced Intelligence Project, RIKEN, Japan 3 Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan 4 Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan 5 Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan

Received  August 2018 Revised  October 2018 Published  January 2019

We study the Cauchy problem of the damped wave equation
 \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*}
and give sharp
 $L^p$
-
 $L^q$
estimates of the solution for
 $1\le q \le p < \infty\ (p\neq 1)$
with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in
 $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$
with
 $r \in (1,2]$
,
 $s\ge 0$
, and
 $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$
, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power
 $1+\frac{2r}{n}$
, while it is known that the critical power
 $1+\frac{2}{n}$
belongs to the blow-up region when
 $r = 1$
. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.
Citation: Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar [2] J. Chen, D. Fan and C. Zhang, Space-time estimates on damped fractional wave equation, Abstr. Appl. Anal., (2014). doi: 10.1155/2014/428909.  Google Scholar [3] J. Chen, D. Fan and C. Zhang, Estimates for damped fractional wave equations and applications, Electronic Journal of Differential Equations, 2015 (2015), 1-14.   Google Scholar [4] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar [5] K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, Funkcial. Ekvac., to appear, arXiv: 1609.01035v2. Google Scholar [6] K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys., 57 (2016), 1-8.  doi: 10.1063/1.4960725.  Google Scholar [7] M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser, Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.  Google Scholar [8] L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. ⅹⅷ+638 pp. doi: 10.1007/978-1-4939-1194-3.  Google Scholar [9] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652.   Google Scholar [10] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 1165-1185.  doi: 10.1090/S0002-9947-05-03818-3.  Google Scholar [11] 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.  Google Scholar [12] N. Hayashi and P. I. Naumkin, Damped wave equation with a critical nonlinearity in higher space dimensions, J. Math. Appl. Anal., 446 (2017), 801-822.  doi: 10.1016/j.jmaa.2016.09.005.  Google Scholar [13] 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.  Google Scholar [14] M. Ikeda, T. Inui and Y. Wakasugi, The Cauchy problem for the nonlinear damped wave equation with slowly decaying data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), no. 2, Art. 10, 53 pp. doi: 10.1007/s00030-017-0434-1.  Google Scholar [15] R. Ikehata, Y. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbf R^N$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373.   Google Scholar [16] 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.   Google Scholar [17] R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbf R^N$, J. Math. Anal. Appl., 269 (2002), 87-97.  doi: 10.1016/S0022-247X(02)00021-5.  Google Scholar [18] R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbf R^N$ with noncompactly supported initial data, Nonlinear Anal., 61 (2005), 1189-1208.  doi: 10.1016/j.na.2005.01.097.  Google Scholar [19] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.  doi: 10.4064/sm-143-2-175-197.  Google Scholar [20] 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.  Google Scholar [21] M. Kirane and M. Qafsaoui, Fujita's exponent for a semilinear wave equation with linear damping, Adv. Nonlinear Stud., 2 (2002), 41-49.  doi: 10.1515/ans-2002-0103.  Google Scholar [22] T.-T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha}$, Discrete Contin. Dynam. Syst., 1 (1995), 503-520.  doi: 10.3934/dcds.1995.1.503.  Google Scholar [23] 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.  Google Scholar [24] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.  doi: 10.2977/prims/1195190962.  Google Scholar [25] A. Miyachi, On some estimates for the wave equation in $L^p$ and $H^p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 331-354.   Google Scholar [26] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342.  doi: 10.1007/BF02572407.  Google Scholar [27] 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.  Google Scholar [28] T. Narazaki, Global solutions to the Cauchy problem for a system of damped wave equations, Differential and Integral Equations, 24 (2011), 569-600.   Google Scholar [29] T. Narazaki and K. Nishihara, Asymptotic behavior of solutions for the damped wave equation with slowly decaying data, J. Math. Anal. Appl., 338 (2008), 803-819.  doi: 10.1016/j.jmaa.2007.05.068.  Google Scholar [30] 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.  doi: 10.1007/s00209-003-0516-0.  Google Scholar [31] J. C. Peral, $L^p$ estimates for the wave equation, J. Funct. Anal., 36 (1980), 114-145.  doi: 10.1016/0022-1236(80)90110-X.  Google Scholar [32] S. Sakata and Y. Wakasugi, Movement of time-delayed hot spots in Euclidean space, Math. Z., 285 (2017), 1007-1040.  doi: 10.1007/s00209-016-1735-5.  Google Scholar [33] S. Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa, 24 (1970), 331-348.   Google Scholar [34] 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.  Google Scholar [35] F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.  Google Scholar [36] 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.  Google Scholar [37] 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.  Google Scholar

show all references

##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar [2] J. Chen, D. Fan and C. Zhang, Space-time estimates on damped fractional wave equation, Abstr. Appl. Anal., (2014). doi: 10.1155/2014/428909.  Google Scholar [3] J. Chen, D. Fan and C. Zhang, Estimates for damped fractional wave equations and applications, Electronic Journal of Differential Equations, 2015 (2015), 1-14.   Google Scholar [4] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar [5] K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, Funkcial. Ekvac., to appear, arXiv: 1609.01035v2. Google Scholar [6] K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys., 57 (2016), 1-8.  doi: 10.1063/1.4960725.  Google Scholar [7] M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser, Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.  Google Scholar [8] L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. ⅹⅷ+638 pp. doi: 10.1007/978-1-4939-1194-3.  Google Scholar [9] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652.   Google Scholar [10] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 1165-1185.  doi: 10.1090/S0002-9947-05-03818-3.  Google Scholar [11] 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.  Google Scholar [12] N. Hayashi and P. I. Naumkin, Damped wave equation with a critical nonlinearity in higher space dimensions, J. Math. Appl. Anal., 446 (2017), 801-822.  doi: 10.1016/j.jmaa.2016.09.005.  Google Scholar [13] 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.  Google Scholar [14] M. Ikeda, T. Inui and Y. Wakasugi, The Cauchy problem for the nonlinear damped wave equation with slowly decaying data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), no. 2, Art. 10, 53 pp. doi: 10.1007/s00030-017-0434-1.  Google Scholar [15] R. Ikehata, Y. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbf R^N$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373.   Google Scholar [16] 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.   Google Scholar [17] R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbf R^N$, J. Math. Anal. Appl., 269 (2002), 87-97.  doi: 10.1016/S0022-247X(02)00021-5.  Google Scholar [18] R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbf R^N$ with noncompactly supported initial data, Nonlinear Anal., 61 (2005), 1189-1208.  doi: 10.1016/j.na.2005.01.097.  Google Scholar [19] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.  doi: 10.4064/sm-143-2-175-197.  Google Scholar [20] 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.  Google Scholar [21] M. Kirane and M. Qafsaoui, Fujita's exponent for a semilinear wave equation with linear damping, Adv. Nonlinear Stud., 2 (2002), 41-49.  doi: 10.1515/ans-2002-0103.  Google Scholar [22] T.-T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha}$, Discrete Contin. Dynam. Syst., 1 (1995), 503-520.  doi: 10.3934/dcds.1995.1.503.  Google Scholar [23] 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.  Google Scholar [24] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189.  doi: 10.2977/prims/1195190962.  Google Scholar [25] A. Miyachi, On some estimates for the wave equation in $L^p$ and $H^p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 331-354.   Google Scholar [26] M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342.  doi: 10.1007/BF02572407.  Google Scholar [27] 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.  Google Scholar [28] T. Narazaki, Global solutions to the Cauchy problem for a system of damped wave equations, Differential and Integral Equations, 24 (2011), 569-600.   Google Scholar [29] T. Narazaki and K. Nishihara, Asymptotic behavior of solutions for the damped wave equation with slowly decaying data, J. Math. Anal. Appl., 338 (2008), 803-819.  doi: 10.1016/j.jmaa.2007.05.068.  Google Scholar [30] 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.  doi: 10.1007/s00209-003-0516-0.  Google Scholar [31] J. C. Peral, $L^p$ estimates for the wave equation, J. Funct. Anal., 36 (1980), 114-145.  doi: 10.1016/0022-1236(80)90110-X.  Google Scholar [32] S. Sakata and Y. Wakasugi, Movement of time-delayed hot spots in Euclidean space, Math. Z., 285 (2017), 1007-1040.  doi: 10.1007/s00209-016-1735-5.  Google Scholar [33] S. Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa, 24 (1970), 331-348.   Google Scholar [34] 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.  Google Scholar [35] F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.  Google Scholar [36] 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.  Google Scholar [37] 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.  Google Scholar
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