# American Institute of Mathematical Sciences

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:
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Soc.}, 330 (1992), 191. doi: 10.2307/2154160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, \emph{Differential Integral Equations}, 17 (2004), 637. Google Scholar [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. Google Scholar [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. 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,, \emph{J. Differential Equations}, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034. Google Scholar [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. Google Scholar [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. Google Scholar [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., (). Google Scholar [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. Google Scholar [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. Google Scholar [19] C. S. Lin, Interpolation inequalities with weights,, \emph{Comm. Partial Differential Equations}, 11 (1986), 1515. doi: 10.1080/03605308608820473. Google Scholar [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. Google Scholar [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. Google Scholar [22] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, \emph{Publ. Res. Inst. Math. Sci.}, 12 (): 169. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] K. Nishihara, $\$, Sugaku, 62 (2010), 20. Google Scholar [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. Google Scholar [33] R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, \emph{Indiana Univ. Math. J.}, 21 (): 841. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, \emph{Compositio Math.}, 53 (1984), 259. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] H. Egnell, Positive solutions of semilinear equations in cones,, \emph{Trans. Amer. Math. Soc.}, 330 (1992), 191. doi: 10.2307/2154160. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities,, \emph{Differential Integral Equations}, 17 (2004), 637. Google Scholar [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. Google Scholar [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. 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,, \emph{J. Differential Equations}, 203 (2004), 82. doi: 10.1016/j.jde.2004.03.034. Google Scholar [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. Google Scholar [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. Google Scholar [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., (). Google Scholar [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. Google Scholar [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. Google Scholar [19] C. S. Lin, Interpolation inequalities with weights,, \emph{Comm. Partial Differential Equations}, 11 (1986), 1515. doi: 10.1080/03605308608820473. Google Scholar [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. Google Scholar [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. Google Scholar [22] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, \emph{Publ. Res. Inst. Math. Sci.}, 12 (): 169. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] K. Nishihara, $\$, Sugaku, 62 (2010), 20. Google Scholar [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. Google Scholar [33] R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, \emph{Indiana Univ. Math. J.}, 21 (): 841. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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