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  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 & Continuous Dynamical Systems, 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.  Google Scholar

[2]

R. AgemiY. 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.  Google Scholar

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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.  Google Scholar

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V. GeorgievH. 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.  Google Scholar

[5]

R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.  Google Scholar

[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.  Google Scholar

[7]

K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcial. Ekvac., 43 (2000), 559-588.   Google Scholar

[8]

F. John, Plane Waves and Spherical Means, Applied to Partial Differential Equations, Interscience Publishers, Inc., New York, 1955. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math., 10 (1957), 523-530.  doi: 10.1002/cpa.3160100404.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations, 8 (1995), 647-661.   Google Scholar

[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.  Google Scholar

[26]

H. TakamuraH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. AgemiY. 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.  Google Scholar

[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.  Google Scholar

[4]

V. GeorgievH. 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.  Google Scholar

[5]

R. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.  Google Scholar

[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.  Google Scholar

[7]

K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcial. Ekvac., 43 (2000), 559-588.   Google Scholar

[8]

F. John, Plane Waves and Spherical Means, Applied to Partial Differential Equations, Interscience Publishers, Inc., New York, 1955. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. B. Keller, On solutions of nonlinear wave equations, Comm. Pure Appl. Math., 10 (1957), 523-530.  doi: 10.1002/cpa.3160100404.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

H. Takamura, Blow-up for semilinear wave equations with slowly decaying data in high dimensions, Differential Integral Equations, 8 (1995), 647-661.   Google Scholar

[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.  Google Scholar

[26]

H. TakamuraH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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