July  2015, 14(4): 1533-1545. doi: 10.3934/cpaa.2015.14.1533

Remarks on global solutions of dissipative wave equations with exponential nonlinear terms

1. 

Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560

Received  July 2013 Revised  March 2014 Published  April 2015

The Cauchy problem for dissipative wave equations with exponential type nonlinear terms is considered in the energy space in two spatial dimensions. The nonlinear terms have a singularity at the origin, and global solutions are shown based on the Gagliardo-Nirenberg type inequality.
Citation: Makoto Nakamura. Remarks on global solutions of dissipative wave equations with exponential nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1533-1545. doi: 10.3934/cpaa.2015.14.1533
References:
[1]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compositio Math.}, 53 (1984), 259.

[2]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[3]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 330 (2000), 437. doi: 10.1016/S0764-4442(00)00201-9.

[4]

J. L. Chern and C. S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities on domains with the singularity on the boundary,, \emph{Arch. Rational Mech. Anal.}, 197 (2010), 401. doi: 10.1007/s00205-009-0269-y.

[5]

H. Egnell, Positive solutions of semilinear equations in cones,, \emph{Trans. Amer. Math. Soc.}, 330 (1992), 191. doi: 10.2307/2154160.

[6]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 21 (2004), 767. doi: 10.1016/j.anihpc.2003.07.002.

[7]

N. Ghoussoub and F. Robert, Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth,, \emph{IMRP Int. Math. Res. Pap.}, 21867 (2006), 1.

[8]

N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities,, \emph{Geom. Funct. Anal.}, 16 (2006), 1201. doi: 10.1007/s00039-006-0579-2.

[9]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations,, \emph{Comm. Math. Phys.}, 123 (1989), 535.

[10]

N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, \emph{Differential Integral Equations}, 17 (2004), 637.

[11]

N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation in the subcritical case,, \emph{J. Differential Equations}, 207 (2004), 161. doi: 10.1016/j.jde.2004.06.018.

[12]

J. Hernández, F. J. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 137 (2007), 41. doi: 10.1017/S030821050500065X.

[13]

T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations,, \emph{J. Differential Equations}, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034.

[14]

C. H. Hsia, C. S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg,, \emph{J. Funct. Anal.}, 259 (2010), 1816. doi: 10.1016/j.jfa.2010.05.004.

[15]

R. Ikehata, Y. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^N$ with lower power nonlinearities,, \emph{J. Math. Soc. Japan}, 56 (2004), 365. doi: 10.2969/jmsj/1191418635.

[16]

M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form,, preprint., ().

[17]

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,, \emph{J. Math. Soc. Japan}, 47 (1995), 617. doi: 10.2969/jmsj/04740617.

[18]

T. T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t=| u| ^{1+\alpha}$,, \emph{Discrete Contin. Dynam. Systems}, 1 (1995), 503. doi: 10.3934/dcds.1995.1.503.

[19]

C. S. Lin, Interpolation inequalities with weights,, \emph{Comm. Partial Differential Equations}, 11 (1986), 1515. doi: 10.1080/03605308608820473.

[20]

C. S. Lin and H. Wadade, Minimizing problems for the Hardy-Sobolev type inequality with the singularity on the boundary,, \emph{Tohoku Mathematical Journal}, (). doi: 10.2748/tmj/1332767341.

[21]

C. S. Lin and Z. Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities,, \emph{Proc. Amer. Math. Soc.}, 132 (2004), 1685. doi: 10.1090/S0002-9939-04-07245-4.

[22]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, \emph{Publ. Res. Inst. Math. Sci.}, 12 (): 169.

[23]

S. Nagayasu and H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin,, \emph{J. Funct. Anal.}, 258 (2010), 3725. doi: 10.1016/j.jfa.2010.02.015.

[24]

M. Nakamura, Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces,, \emph{Reviews in Mathematical Physics}, 23 (2011), 903. doi: 10.1142/S0129055X11004473.

[25]

M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, \emph{J. Funct. Anal.}, 150 (1998), 364. doi: 10.1006/jfan.1997.3236.

[26]

M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth,, \emph{Math. Z.}, 231 (1999), 479. doi: 10.1007/PL00004737.

[27]

M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces,, \emph{Publications of R.I.M.S., 37 (2001), 255.

[28]

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,, \emph{Math. Z.}, 214 (1993), 325. doi: 10.1007/BF02572407.

[29]

T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem,, \emph{J. Math. Soc. Japan}, 56 (2004), 585. doi: 10.2969/jmsj/1191418647.

[30]

K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application,, \emph{Math. Z.}, 244 (2003), 631.

[31]

K. Nishihara, $\ $, Sugaku, 62 (2010), 20.

[32]

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth,, \emph{Internat. Math. Res. Notices}, (1994). doi: 10.1155/S1073792894000346.

[33]

R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, \emph{Indiana Univ. Math. J.}, 21 (): 841.

[34]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, \emph{J. Differential Equations}, 174 (2001), 464. doi: 10.1006/jdeq.2000.3933.

[35]

H. Yang and J. Chen, A result on Hardy-Sobolev critical elliptic equations with boundary singularities,, \emph{Commun. Pure Appl. Anal.}, 6 (2007), 191. doi: 10.3934/cpaa.2007.6.191.

[36]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case,, \emph{C. R. Acad. Sci. Paris Ser. I Math.}, 333 (2001), 109. doi: 10.1016/S0764-4442(01)01999-1.

show all references

References:
[1]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compositio Math.}, 53 (1984), 259.

[2]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, \emph{Comm. Pure Appl. Math.}, 54 (2001), 229. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[3]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities,, \emph{C. R. Acad. Sci. Paris S\'er. I Math.}, 330 (2000), 437. doi: 10.1016/S0764-4442(00)00201-9.

[4]

J. L. Chern and C. S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities on domains with the singularity on the boundary,, \emph{Arch. Rational Mech. Anal.}, 197 (2010), 401. doi: 10.1007/s00205-009-0269-y.

[5]

H. Egnell, Positive solutions of semilinear equations in cones,, \emph{Trans. Amer. Math. Soc.}, 330 (1992), 191. doi: 10.2307/2154160.

[6]

N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 21 (2004), 767. doi: 10.1016/j.anihpc.2003.07.002.

[7]

N. Ghoussoub and F. Robert, Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth,, \emph{IMRP Int. Math. Res. Pap.}, 21867 (2006), 1.

[8]

N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities,, \emph{Geom. Funct. Anal.}, 16 (2006), 1201. doi: 10.1007/s00039-006-0579-2.

[9]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations,, \emph{Comm. Math. Phys.}, 123 (1989), 535.

[10]

N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, \emph{Differential Integral Equations}, 17 (2004), 637.

[11]

N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation in the subcritical case,, \emph{J. Differential Equations}, 207 (2004), 161. doi: 10.1016/j.jde.2004.06.018.

[12]

J. Hernández, F. J. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 137 (2007), 41. doi: 10.1017/S030821050500065X.

[13]

T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations,, \emph{J. Differential Equations}, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034.

[14]

C. H. Hsia, C. S. Lin and H. Wadade, Revisiting an idea of Brézis and Nirenberg,, \emph{J. Funct. Anal.}, 259 (2010), 1816. doi: 10.1016/j.jfa.2010.05.004.

[15]

R. Ikehata, Y. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbbR^N$ with lower power nonlinearities,, \emph{J. Math. Soc. Japan}, 56 (2004), 365. doi: 10.2969/jmsj/1191418635.

[16]

M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form,, preprint., ().

[17]

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,, \emph{J. Math. Soc. Japan}, 47 (1995), 617. doi: 10.2969/jmsj/04740617.

[18]

T. T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t=| u| ^{1+\alpha}$,, \emph{Discrete Contin. Dynam. Systems}, 1 (1995), 503. doi: 10.3934/dcds.1995.1.503.

[19]

C. S. Lin, Interpolation inequalities with weights,, \emph{Comm. Partial Differential Equations}, 11 (1986), 1515. doi: 10.1080/03605308608820473.

[20]

C. S. Lin and H. Wadade, Minimizing problems for the Hardy-Sobolev type inequality with the singularity on the boundary,, \emph{Tohoku Mathematical Journal}, (). doi: 10.2748/tmj/1332767341.

[21]

C. S. Lin and Z. Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities,, \emph{Proc. Amer. Math. Soc.}, 132 (2004), 1685. doi: 10.1090/S0002-9939-04-07245-4.

[22]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, \emph{Publ. Res. Inst. Math. Sci.}, 12 (): 169.

[23]

S. Nagayasu and H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin,, \emph{J. Funct. Anal.}, 258 (2010), 3725. doi: 10.1016/j.jfa.2010.02.015.

[24]

M. Nakamura, Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces,, \emph{Reviews in Mathematical Physics}, 23 (2011), 903. doi: 10.1142/S0129055X11004473.

[25]

M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, \emph{J. Funct. Anal.}, 150 (1998), 364. doi: 10.1006/jfan.1997.3236.

[26]

M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth,, \emph{Math. Z.}, 231 (1999), 479. doi: 10.1007/PL00004737.

[27]

M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces,, \emph{Publications of R.I.M.S., 37 (2001), 255.

[28]

M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,, \emph{Math. Z.}, 214 (1993), 325. doi: 10.1007/BF02572407.

[29]

T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem,, \emph{J. Math. Soc. Japan}, 56 (2004), 585. doi: 10.2969/jmsj/1191418647.

[30]

K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application,, \emph{Math. Z.}, 244 (2003), 631.

[31]

K. Nishihara, $\ $, Sugaku, 62 (2010), 20.

[32]

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth,, \emph{Internat. Math. Res. Notices}, (1994). doi: 10.1155/S1073792894000346.

[33]

R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, \emph{Indiana Univ. Math. J.}, 21 (): 841.

[34]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping,, \emph{J. Differential Equations}, 174 (2001), 464. doi: 10.1006/jdeq.2000.3933.

[35]

H. Yang and J. Chen, A result on Hardy-Sobolev critical elliptic equations with boundary singularities,, \emph{Commun. Pure Appl. Anal.}, 6 (2007), 191. doi: 10.3934/cpaa.2007.6.191.

[36]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case,, \emph{C. R. Acad. Sci. Paris Ser. I Math.}, 333 (2001), 109. doi: 10.1016/S0764-4442(01)01999-1.

[1]

B. Abdellaoui, I. Peral. On quasilinear elliptic equations related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2003, 2 (4) : 539-566. doi: 10.3934/cpaa.2003.2.539

[2]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697

[3]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. Kinetic & Related Models, 2017, 10 (1) : 33-59. doi: 10.3934/krm.2017002

[4]

Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629

[5]

Mayte Pérez-Llanos. Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 1975-2005. doi: 10.3934/cpaa.2016024

[6]

Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889

[7]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

[8]

Yacheng Liu, Runzhang Xu. Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 171-189. doi: 10.3934/dcdsb.2007.7.171

[9]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

[10]

Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975

[11]

Alexandre Nolasco de Carvalho, Jan W. Cholewa, Tomasz Dlotko. Damped wave equations with fast growing dissipative nonlinearities. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1147-1165. doi: 10.3934/dcds.2009.24.1147

[12]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293

[13]

Hideo Kubo. Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Conference Publications, 2007, 2007 (Special) : 602-613. doi: 10.3934/proc.2007.2007.602

[14]

Nakao Hayashi, Pavel I. Naumkin. Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities. Inverse Problems & Imaging, 2007, 1 (2) : 391-398. doi: 10.3934/ipi.2007.1.391

[15]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

[16]

Olivier Guibé, Anna Mercaldo. Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Communications on Pure & Applied Analysis, 2008, 7 (1) : 163-192. doi: 10.3934/cpaa.2008.7.163

[17]

José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-30. doi: 10.3934/dcdsb.2019079

[18]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11

[19]

E. B. Dynkin. A new inequality for superdiffusions and its applications to nonlinear differential equations. Electronic Research Announcements, 2004, 10: 68-77.

[20]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]