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October  2021, 41(10): 4545-4566. doi: 10.3934/dcds.2021048

## On the critical decay for the wave equation with a cubic convolution in 3D

 1 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan 2 Department of Creative Engineering, National Institute of Technology, Kushiro College, 2-32-1 Otanoshike-Nishi, Kushiro-Shi, Hokkaido 084-0916, Japan

* Corresponding author: Tomoyuki Tanaka

Received  September 2020 Revised  January 2021 Published  October 2021 Early access  March 2021

Fund Project: The first author is supported by JSPS KAKENHI Grant Number JP20J12750. The second author is supported by the Grant-in-Aid for Scientific Research (B) (No.18H01132), the Grant-in-Aid for Scientific Research (B) (No.19H01795) and Young Scientists Research (No. 20K14351), Japan Society for the Promotion of Science

We consider the wave equation with a cubic convolution
 $\partial_{t}^2 u-\Delta u = (|x|^{- \gamma}*u^2)u$
in three space dimensions. Here,
 $0< \gamma<3$
and
 $*$
stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then
 $\gamma\ge2$
assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when
 $2\le \gamma<3$
. In this paper, we consider the Cauchy problem for
 $2\le \gamma<3$
in the space-time weighted
 $L^ \infty$
space in which functions have critical decay rate. When
 $\gamma = 2$
, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in [13]). When
 $2< \gamma<3$
, we also prove unique global existence of solutions for small data.
Citation: Tomoyuki Tanaka, Kyouhei Wakasa. On the critical decay for the wave equation with a cubic convolution in 3D. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4545-4566. doi: 10.3934/dcds.2021048
##### References:
 [1] R. Agemi and H. Takamura, The lifespan of classical solutions to nonlinear wave equations in two space dimensions, Hokkaido Math. J., 21 (1992), 517-542.  doi: 10.14492/hokmj/1381413726. [2] R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for $p$-$q$ systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766. [3] F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decaying initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487.  doi: 10.1080/03605308608820470. [4] V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.  doi: 10.1353/ajm.1997.0038. [5] R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066. [6] R. Glassey, Existence in the large for $\Box u = f(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042. [7] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcial. Ekvac., 43 (2000), 559-588. [8] F. John, Plane Waves and Spherical Means, Applied to Partial Differential Equations, Interscience Publishers, Inc., New York, 1955. [9] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [10] M. Kato and M. Sakuraba, Global existence and blow-up for semilinear damped wave equations in three space dimensions, Nonlinear Anal., 182 (2019), 209-225.  doi: 10.1016/j.na.2018.12.013. [11] J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math., 10 (1957), 523-530.  doi: 10.1002/cpa.3160100404. [12] H. Kubo, On the critical decay and power for semilinear wave equations in odd space dimensions, Discrete Contin. Dyn. Syst., 2 (1996), 173-190.  doi: 10.3934/dcds.1996.2.173. [13] H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications, Dispersive nonlinear problems in mathematical physics, 123–148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004. [14] H. Kubo and K. Kubota, Asymptotic behavior of radially symmetric solutions of $\Box u = |u|^p$ for super critical values $p$ in even space dimensions, Jpn. J. Math., 24 (1998), 191-256.  doi: 10.4099/math1924.24.191. [15] H. Kubo and M. Ohta, On the global behavior of classical solutions to coupled systems of semilinear wave equations, New trends in the theory of hyperbolic equations, Oper. Theory Adv. Appl., 159, Adv. Partial Differ. Equ. (Basel), Birkhäuser, Basel, 2005. doi: 10.1007/3-7643-7386-5_2. [16] K. Kubota, Existence of a global solution to a semi-linear wave equation with initial data of non-compact support in low space dimensions, Hokkaido Math. J., 22 (1993), 123-180.  doi: 10.14492/hokmj/1381413170. [17] N.-A. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381.  doi: 10.1016/j.jfa.2014.05.020. [18] H. Lindblad, Blow-up for solutions of $\Box u = |u|^p$ with small initial data, Comm. Partial Differential Equations, 15 (1990), 757-821.  doi: 10.1080/03605309908820708. [19] H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.  doi: 10.1353/ajm.1996.0042. [20] G. P. Menzala and W. A. Strauss, On a wave equation with a cubic convolution,, J. Differential Equations, 43 (1982), 93-105.  doi: 10.1016/0022-0396(82)90076-6. [21] M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. in Partial Differential Equations, 12 (1987), 677-700.  doi: 10.1080/03605308708820506. [22] J. Schaeffer, The equation $u_tt-\Delta u = |u|^{p}$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31-44.  doi: 10.1017/S0308210500026135. [23] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4. [24] H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations, 8 (1995), 647-661. [25] H. Takamura, Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations, Nonlinear Anal., 125 (2015), 227-240.  doi: 10.1016/j.na.2015.05.024. [26] H. Takamura, H. Uesaka and K. Wakasa, Blow-up theorem for semilinear wave equations with non-zero initial position, J. Differential Equations, 249 (2010), 914-930.  doi: 10.1016/j.jde.2010.01.010. [27] H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions, J. Differential Equations, 251 (2011), 1157-1171.  doi: 10.1016/j.jde.2011.03.024. [28] K. Tsutaya, A global existence theorem for semilinear wave equations with data of non compact support in two space dimensions, Comm. Partial Differential Equations, 17 (1992), 1925-1954.  doi: 10.1080/03605309208820909. [29] K. Tsutaya, Global existence theorem for semilinear wave equations with non-compact data in two space dimensions, J. Differential Equations, 104 (1993), 332-360.  doi: 10.1006/jdeq.1993.1076. [30] K. Tsutaya, Global existence and the lifespan of solutions of semilinear wave equations with data of non compact support in three space dimensions, Funkcial. Ekvac., 37 (1994), 1-18. [31] K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. Vol. 1, 2, Kluwer Acad. Publ., Dordrecht, 2003. [32] K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type, J. Math. Anal. Appl., 411 (2014), 719-731.  doi: 10.1016/j.jmaa.2013.10.021. [33] K. Wakasa, The lifespan of solutions to wave equations with weighted nonlinear terms in one space dimension, Hokkaido Math. J., 46 (2017), 257-276.  doi: 10.14492/hokmj/1498788020. [34] B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012. [35] Y. Zhou, Blow up of classical solutions to $\Box u=|u|^{1+\alpha}$ in three space dimensions, J. Partial Differential Equations, 5 (1992), 21-32. [36] Y. Zhou, Life span of classical solutions to $\Box u=|u|^{p}$ in two space dimensions, Chin. Ann. Math. Ser. B, 14 (1993), 225-236. [37] Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x. [38] Y. Zhou and W. Han, Life-span of solutions to critical semilinear wave equations, Comm. Partial Differential Equations, 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

show all references

##### References:
 [1] R. Agemi and H. Takamura, The lifespan of classical solutions to nonlinear wave equations in two space dimensions, Hokkaido Math. J., 21 (1992), 517-542.  doi: 10.14492/hokmj/1381413726. [2] R. Agemi, Y. Kurokawa and H. Takamura, Critical curve for $p$-$q$ systems of nonlinear wave equations in three space dimensions, J. Differential Equations, 167 (2000), 87-133.  doi: 10.1006/jdeq.2000.3766. [3] F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decaying initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487.  doi: 10.1080/03605308608820470. [4] V. Georgiev, H. Lindblad and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math., 119 (1997), 1291-1319.  doi: 10.1353/ajm.1997.0038. [5] R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066. [6] R. Glassey, Existence in the large for $\Box u = f(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042. [7] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcial. Ekvac., 43 (2000), 559-588. [8] F. John, Plane Waves and Spherical Means, Applied to Partial Differential Equations, Interscience Publishers, Inc., New York, 1955. [9] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [10] M. Kato and M. Sakuraba, Global existence and blow-up for semilinear damped wave equations in three space dimensions, Nonlinear Anal., 182 (2019), 209-225.  doi: 10.1016/j.na.2018.12.013. [11] J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math., 10 (1957), 523-530.  doi: 10.1002/cpa.3160100404. [12] H. Kubo, On the critical decay and power for semilinear wave equations in odd space dimensions, Discrete Contin. Dyn. Syst., 2 (1996), 173-190.  doi: 10.3934/dcds.1996.2.173. [13] H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications, Dispersive nonlinear problems in mathematical physics, 123–148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004. [14] H. Kubo and K. Kubota, Asymptotic behavior of radially symmetric solutions of $\Box u = |u|^p$ for super critical values $p$ in even space dimensions, Jpn. J. Math., 24 (1998), 191-256.  doi: 10.4099/math1924.24.191. [15] H. Kubo and M. Ohta, On the global behavior of classical solutions to coupled systems of semilinear wave equations, New trends in the theory of hyperbolic equations, Oper. Theory Adv. Appl., 159, Adv. Partial Differ. Equ. (Basel), Birkhäuser, Basel, 2005. doi: 10.1007/3-7643-7386-5_2. [16] K. Kubota, Existence of a global solution to a semi-linear wave equation with initial data of non-compact support in low space dimensions, Hokkaido Math. J., 22 (1993), 123-180.  doi: 10.14492/hokmj/1381413170. [17] N.-A. Lai and Y. Zhou, An elementary proof of Strauss conjecture, J. Funct. Anal., 267 (2014), 1364-1381.  doi: 10.1016/j.jfa.2014.05.020. [18] H. Lindblad, Blow-up for solutions of $\Box u = |u|^p$ with small initial data, Comm. Partial Differential Equations, 15 (1990), 757-821.  doi: 10.1080/03605309908820708. [19] H. Lindblad and C. D. Sogge, Long-time existence for small amplitude semilinear wave equations, Amer. J. Math., 118 (1996), 1047-1135.  doi: 10.1353/ajm.1996.0042. [20] G. P. Menzala and W. A. Strauss, On a wave equation with a cubic convolution,, J. Differential Equations, 43 (1982), 93-105.  doi: 10.1016/0022-0396(82)90076-6. [21] M. A. Rammaha, Finite-time blow-up for nonlinear wave equations in high dimensions, Comm. in Partial Differential Equations, 12 (1987), 677-700.  doi: 10.1080/03605308708820506. [22] J. Schaeffer, The equation $u_tt-\Delta u = |u|^{p}$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh Sect. A, 101 (1985), 31-44.  doi: 10.1017/S0308210500026135. [23] T. C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations, 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4. [24] H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations, 8 (1995), 647-661. [25] H. Takamura, Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations, Nonlinear Anal., 125 (2015), 227-240.  doi: 10.1016/j.na.2015.05.024. [26] H. Takamura, H. Uesaka and K. Wakasa, Blow-up theorem for semilinear wave equations with non-zero initial position, J. Differential Equations, 249 (2010), 914-930.  doi: 10.1016/j.jde.2010.01.010. [27] H. Takamura and K. Wakasa, The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions, J. Differential Equations, 251 (2011), 1157-1171.  doi: 10.1016/j.jde.2011.03.024. [28] K. Tsutaya, A global existence theorem for semilinear wave equations with data of non compact support in two space dimensions, Comm. Partial Differential Equations, 17 (1992), 1925-1954.  doi: 10.1080/03605309208820909. [29] K. Tsutaya, Global existence theorem for semilinear wave equations with non-compact data in two space dimensions, J. Differential Equations, 104 (1993), 332-360.  doi: 10.1006/jdeq.1993.1076. [30] K. Tsutaya, Global existence and the lifespan of solutions of semilinear wave equations with data of non compact support in three space dimensions, Funkcial. Ekvac., 37 (1994), 1-18. [31] K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. Vol. 1, 2, Kluwer Acad. Publ., Dordrecht, 2003. [32] K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type, J. Math. Anal. Appl., 411 (2014), 719-731.  doi: 10.1016/j.jmaa.2013.10.021. [33] K. Wakasa, The lifespan of solutions to wave equations with weighted nonlinear terms in one space dimension, Hokkaido Math. J., 46 (2017), 257-276.  doi: 10.14492/hokmj/1498788020. [34] B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012. [35] Y. Zhou, Blow up of classical solutions to $\Box u=|u|^{1+\alpha}$ in three space dimensions, J. Partial Differential Equations, 5 (1992), 21-32. [36] Y. Zhou, Life span of classical solutions to $\Box u=|u|^{p}$ in two space dimensions, Chin. Ann. Math. Ser. B, 14 (1993), 225-236. [37] Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x. [38] Y. Zhou and W. Han, Life-span of solutions to critical semilinear wave equations, Comm. Partial Differential Equations, 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.
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