Global existence for semilinear damped wave equations in relation with the Strauss conjecture

Mengyun Liu; Chengbo Wang

doi:10.3934/dcds.2020058

Previous Article
Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system
DCDS Home
This Issue
Next Article
On mean field systems with multi-classes

February 2020, 40(2): 709-724. doi: 10.3934/dcds.2020058

Global existence for semilinear damped wave equations in relation with the Strauss conjecture

Mengyun Liu and Chengbo Wang ^,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

^* Corresponding author: Chengbo Wang

Received July 2018 Revised March 2019 Published November 2019

Fund Project: The authors were supported by NSFC 11971428.

We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $ n $ dimensional nontrapping asymptotically Euclidean manifolds, when $ n = 3, 4 $ as well as two dimensional Euclidean space. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.

Keywords: Global existence, weighted strichartz estimates, damped wave equations, local energy estimates, Strauss conjecture.

Mathematics Subject Classification: Primary: 35L05, 35L15, 35L71; Secondary: 35B33, 58J45.

Citation: Mengyun Liu, Chengbo Wang. Global existence for semilinear damped wave equations in relation with the Strauss conjecture. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 709-724. doi: 10.3934/dcds.2020058

References:

[1]	J. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations, 35 (2010), 23-67. doi: 10.1080/03605300903396601. Google Scholar
[2]	M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 (2015), 5040-5073. doi: 10.1016/j.jde.2015.06.018. Google Scholar
[3]	D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. doi: 10.1515/FORM.2011.009. Google Scholar
[4]	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., 62 (2019), 157-189. Google Scholar
[5]	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. Google Scholar
[6]	R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340. doi: 10.1007/BF01162066. Google Scholar
[7]	K. Hidano, J. Metcalfe, H. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809. doi: 10.1090/S0002-9947-09-05053-3. Google Scholar
[8]	M. Ikeda and M. Sobajima, Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data, Math. Ann., 372 (2018), 1017-1040. doi: 10.1007/s00208-018-1664-1. Google Scholar
[9]	J. Jiang, C. Wang and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal., 11 (2012), 1723-1752. doi: 10.3934/cpaa.2012.11.1723. Google Scholar
[10]	F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403. Google Scholar
[11]	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
[12]	N. Lai and H. Takamura, Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case, Nonlinear Anal., 168 (2018), 222-237. doi: 10.1016/j.na.2017.12.008. Google Scholar
[13]	Ni. Lai, H. Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377-5394. doi: 10.1016/j.jde.2017.06.017. Google Scholar
[14]	N. Lai and Y. Zhou, The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions, J. Math. Pures Appl., 123 (2019), 229-243. doi: 10.1016/j.matpur.2018.04.009. Google Scholar
[15]	T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t = \|u\|^{1+\alpha}$, Discrete Contin. Dynam. Systems, 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. Google Scholar
[16]	J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst., 32 (2012), 4307-4320. doi: 10.3934/dcds.2012.32.4307. Google Scholar
[17]	H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and C. Wang, The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661. doi: 10.1007/s00208-014-1006-x. Google Scholar
[18]	J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math., preprint, arXiv: 1703.08064, (2017) Google Scholar
[19]	J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353 (2012), 1183-1237. doi: 10.1007/s00208-011-0714-8. Google Scholar
[20]	J. Metcalfe and C. Wang, The Strauss conjecture on asymptotically flat space-times, SIAM J. Math. Anal., 49 (2017), 4579-4594. doi: 10.1137/16M1074886. Google Scholar
[21]	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
[22]	C. D. Sogge, Lectures on Non-linear Wave Equations, International Press, Boston, MA, second edition, 2008. Google Scholar
[23]	C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32. doi: 10.1007/s11854-010-0023-2. Google Scholar
[24]	D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807. doi: 10.1090/S0002-9947-00-02750-1. Google Scholar
[25]	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
[26]	Z. Tu and J. Lin, Life-span of semilinear wave equations with scale-invariant damping: Critical strauss exponent case, Differential Integral Equations, 32 (2019), 249-264. Google Scholar
[27]	K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differential Equations, 266 (2019), 5360-5376. doi: 10.1016/j.jde.2018.10.028. Google Scholar
[28]	K. Wakasa and B. Yordanov, On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case, Nonlinear Anal., 180 (2019), 67-74. doi: 10.1016/j.na.2018.09.012. Google Scholar
[29]	C. Wang, The Glassey conjecture on asymptotically flat manifolds, Trans. Amer. Math. Soc., 367 (2015), 7429-7451. doi: 10.1090/S0002-9947-2014-06423-4. Google Scholar
[30]	C. Wang, Long-time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174. doi: 10.1080/03605302.2017.1345939. Google Scholar
[31]	C. Wang, Recent progress on the strauss conjecture and related problems., SCIENTIA SINICA Mathematica, 48 (2018), 111-130. Google Scholar
[32]	C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566. doi: 10.1016/j.jmaa.2011.01.053. Google Scholar
[33]	C. Wang and X. Yu, Recent works on the Strauss conjecture, Contemp. Math., 581 (2012), 235-256. doi: 10.1090/conm/581/11497. Google Scholar
[34]	C. Wang and X. Yu, Global existence of null-form wave equations on small asymptotically Euclidean manifolds, J. Funct. Anal., 266 (2014), 5676-5708. doi: 10.1016/j.jfa.2014.02.028. Google Scholar
[35]	S. Yang, Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal., 209 (2013), 683-728. doi: 10.1007/s00205-013-0631-y. Google Scholar
[36]	S. Yang, On the quasilinear wave equations in time dependent inhomogeneous media, J. Hyperbolic Differ. Equ., 13 (2016), 273-330. doi: 10.1142/S0219891616500090. Google Scholar
[37]	B. Yordanov and Q. 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
[38]	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

show all references

References:

[1]	J. Bony and D. Häfner, The semilinear wave equation on asymptotically Euclidean manifolds, Comm. Partial Differential Equations, 35 (2010), 23-67. doi: 10.1080/03605300903396601. Google Scholar
[2]	M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 (2015), 5040-5073. doi: 10.1016/j.jde.2015.06.018. Google Scholar
[3]	D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. doi: 10.1515/FORM.2011.009. Google Scholar
[4]	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., 62 (2019), 157-189. Google Scholar
[5]	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. Google Scholar
[6]	R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340. doi: 10.1007/BF01162066. Google Scholar
[7]	K. Hidano, J. Metcalfe, H. Smith, C. D. Sogge and Y. Zhou, On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles, Trans. Amer. Math. Soc., 362 (2010), 2789-2809. doi: 10.1090/S0002-9947-09-05053-3. Google Scholar
[8]	M. Ikeda and M. Sobajima, Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data, Math. Ann., 372 (2018), 1017-1040. doi: 10.1007/s00208-018-1664-1. Google Scholar
[9]	J. Jiang, C. Wang and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal., 11 (2012), 1723-1752. doi: 10.3934/cpaa.2012.11.1723. Google Scholar
[10]	F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403. Google Scholar
[11]	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
[12]	N. Lai and H. Takamura, Blow-up for semilinear damped wave equations with subcritical exponent in the scattering case, Nonlinear Anal., 168 (2018), 222-237. doi: 10.1016/j.na.2017.12.008. Google Scholar
[13]	Ni. Lai, H. Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent, J. Differential Equations, 263 (2017), 5377-5394. doi: 10.1016/j.jde.2017.06.017. Google Scholar
[14]	N. Lai and Y. Zhou, The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions, J. Math. Pures Appl., 123 (2019), 229-243. doi: 10.1016/j.matpur.2018.04.009. Google Scholar
[15]	T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t = \|u\|^{1+\alpha}$, Discrete Contin. Dynam. Systems, 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. Google Scholar
[16]	J. Lin, K. Nishihara and J. Zhai, Critical exponent for the semilinear wave equation with time-dependent damping, Discrete Contin. Dyn. Syst., 32 (2012), 4307-4320. doi: 10.3934/dcds.2012.32.4307. Google Scholar
[17]	H. Lindblad, J. Metcalfe, C. D. Sogge, M. Tohaneanu and C. Wang, The Strauss conjecture on Kerr black hole backgrounds, Math. Ann., 359 (2014), 637-661. doi: 10.1007/s00208-014-1006-x. Google Scholar
[18]	J. Metcalfe, J. Sterbenz and D. Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math., preprint, arXiv: 1703.08064, (2017) Google Scholar
[19]	J. Metcalfe and D. Tataru, Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353 (2012), 1183-1237. doi: 10.1007/s00208-011-0714-8. Google Scholar
[20]	J. Metcalfe and C. Wang, The Strauss conjecture on asymptotically flat space-times, SIAM J. Math. Anal., 49 (2017), 4579-4594. doi: 10.1137/16M1074886. Google Scholar
[21]	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
[22]	C. D. Sogge, Lectures on Non-linear Wave Equations, International Press, Boston, MA, second edition, 2008. Google Scholar
[23]	C. D. Sogge and C. Wang, Concerning the wave equation on asymptotically Euclidean manifolds, J. Anal. Math., 112 (2010), 1-32. doi: 10.1007/s11854-010-0023-2. Google Scholar
[24]	D. Tataru, Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc., 353 (2001), 795-807. doi: 10.1090/S0002-9947-00-02750-1. Google Scholar
[25]	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
[26]	Z. Tu and J. Lin, Life-span of semilinear wave equations with scale-invariant damping: Critical strauss exponent case, Differential Integral Equations, 32 (2019), 249-264. Google Scholar
[27]	K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Differential Equations, 266 (2019), 5360-5376. doi: 10.1016/j.jde.2018.10.028. Google Scholar
[28]	K. Wakasa and B. Yordanov, On the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case, Nonlinear Anal., 180 (2019), 67-74. doi: 10.1016/j.na.2018.09.012. Google Scholar
[29]	C. Wang, The Glassey conjecture on asymptotically flat manifolds, Trans. Amer. Math. Soc., 367 (2015), 7429-7451. doi: 10.1090/S0002-9947-2014-06423-4. Google Scholar
[30]	C. Wang, Long-time existence for semilinear wave equations on asymptotically flat space-times, Comm. Partial Differential Equations, 42 (2017), 1150-1174. doi: 10.1080/03605302.2017.1345939. Google Scholar
[31]	C. Wang, Recent progress on the strauss conjecture and related problems., SCIENTIA SINICA Mathematica, 48 (2018), 111-130. Google Scholar
[32]	C. Wang and X. Yu, Concerning the Strauss conjecture on asymptotically Euclidean manifolds, J. Math. Anal. Appl., 379 (2011), 549-566. doi: 10.1016/j.jmaa.2011.01.053. Google Scholar
[33]	C. Wang and X. Yu, Recent works on the Strauss conjecture, Contemp. Math., 581 (2012), 235-256. doi: 10.1090/conm/581/11497. Google Scholar
[34]	C. Wang and X. Yu, Global existence of null-form wave equations on small asymptotically Euclidean manifolds, J. Funct. Anal., 266 (2014), 5676-5708. doi: 10.1016/j.jfa.2014.02.028. Google Scholar
[35]	S. Yang, Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal., 209 (2013), 683-728. doi: 10.1007/s00205-013-0631-y. Google Scholar
[36]	S. Yang, On the quasilinear wave equations in time dependent inhomogeneous media, J. Hyperbolic Differ. Equ., 13 (2016), 273-330. doi: 10.1142/S0219891616500090. Google Scholar
[37]	B. Yordanov and Q. 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
[38]	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

[1]	Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723
[2]	Jason Metcalfe, David Spencer. Global existence for a coupled wave system related to the Strauss conjecture. Communications on Pure & Applied Analysis, 2018, 17 (2) : 593-604. doi: 10.3934/cpaa.2018032
[3]	Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210
[4]	Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134
[5]	Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143
[6]	Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905
[7]	Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050
[8]	Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100
[9]	Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358
[10]	Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[11]	Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070
[12]	Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024
[13]	Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020364
[14]	Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[15]	Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177
[16]	Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023
[17]	Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
[18]	Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041
[19]	John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[20]	Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729

American Institute of Mathematical Sciences

Global existence for semilinear damped wave equations in relation with the Strauss conjecture

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Global existence for semilinear damped wave equations in relation with the Strauss conjecture

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors