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October 2015, 35(10): 4889-4903. doi: 10.3934/dcds.2015.35.4889
Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms
Michinori Ishiwata 1, , Makoto Nakamura 2, and Hidemitsu Wadade 3,
Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan
Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan
Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192, Japan
Received October 2014 Revised January 2015 Published April 2015
The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights. Time local and global solutions are shown to exist in the energy class. The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.
Keywords: Brézis-Gallouët-Wainger inequality., Caffarelli-Kohn-Nirenberg inequality, weighted Trudinger-Moser inequality, Klein-Gordon equation.
Mathematics Subject Classification: Primary: 35L70; Secondary: 35L1.
Citation: 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
S. Adachi and K. Tanaka, A scale-invariant form of Trudinger-Moser inequality and its best exponent,, Proc. Amer. Math. Soc., 1102 (1999), 148. Google Scholar
F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.1090/S0894-0347-1989-1002633-4. Google Scholar
L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth,, Comm. Partial Differential Equations, 36 (2011), 797. doi: 10.1080/03605302.2010.534684. Google Scholar
A. A. Baraket, Local existence and estimations for a semilinear wave equation in two dimension space,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7 (2004), 1. Google Scholar
P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. doi: 10.1090/S0002-9947-1984-0742415-3. Google Scholar
C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, (1976). Google Scholar
M. Bouchekif and A. Matallah, Multiple positive solutions for elliptic equations involving a concave term and critical Sobolev-Hardy exponent,, Appl. Math. Lett., 22 (2009), 268. doi: 10.1016/j.aml.2008.03.024. Google Scholar
P. Brenner, $L_p$-estimates of difference schemes for strictly hyperbolic systems with nonsmooth data,, SIAM J. Numer. Anal., 14 (1977), 1126. doi: 10.1137/0714078. Google Scholar
H. Brézis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential,, Selecta Math. (N.S.), 11 (2005), 1. doi: 10.1007/s00029-005-0003-z. Google Scholar
H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: 10.1016/0362-546X(80)90068-1. Google Scholar
H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Partial Differential Equations, 5 (1980), 773. doi: 10.1080/03605308008820154. Google Scholar
N. Burq, F. Planchon, J. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential,, J. Funct. Anal., 203 (2003), 519. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 1267. doi: 10.3934/cpaa.2014.13.1267. Google Scholar
J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions,, J. Hyperbolic Differ. Equ., 6 (2009), 549. doi: 10.1142/S0219891609001927. Google Scholar
Z. Gan, Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential,, Commun. Pure Appl. Anal., 8 (2009), 1541. doi: 10.3934/cpaa.2009.8.1541. Google Scholar
J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Differential Equations, 144 (1998), 441. doi: 10.1006/jdeq.1997.3375. Google Scholar
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation. II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 15. Google Scholar
J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50. doi: 10.1006/jfan.1995.1119. Google Scholar
T-S. Hsu and H-L. Lin, Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev-Hardy exponents,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 617. doi: 10.1017/S0308210509000729. Google Scholar
S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity,, Comm. Pure Appl. Math., 59 (2006), 1639. doi: 10.1002/cpa.20127. Google Scholar
S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant,, Proc. Amer. Math. Soc., 135 (2007), 87. doi: 10.1090/S0002-9939-06-08240-2. Google Scholar
S. Ibrahim and R. Jrad, Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain,, J. Differential Equations, 250 (2011), 3740. doi: 10.1016/j.jde.2011.01.008. Google Scholar
S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Well posedness and unconditional non uniqueness for a 2D semilinear heat equation,, preprint., (). Google Scholar
M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 297. doi: 10.1016/j.anihpc.2013.03.004. Google Scholar
T. Kato, Schrödinger operators with singular potentials,, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 13 (1972), 135. doi: 10.1007/BF02760233. Google Scholar
J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma,, Phys. Fluids, 20 (1977), 1176. doi: 10.1063/1.861679. Google Scholar
K. Morii, T. Sato and H. Wadade, Brézis-Gallouët-Wainger inequality with a double logarithmic term on a bounded domain and its sharp constants,, Math. Inequal. Appl., 14 (2011), 295. doi: 10.7153/mia-14-24. Google Scholar
K. Morii, T. Sato and H. Wadade, Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: its sharp constants and extremal functions,, Nonlinear Anal., 73 (2010), 1747. doi: 10.1016/j.na.2010.05.012. Google Scholar
K. Morii, T. Sato, Y. Sawano and H. Wadade, Sharp constants of Brézis-Gallouët-Wainger type inequalities with a double logarithmic term on bounded domains in Besov and Triebel-Lizorkin spaces,, Boundary Value Problems, (2010). Google Scholar
J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077. Google Scholar
S. Nagayasu and H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin,, J. Funct. Anal., 258 (2010), 3725. doi: 10.1016/j.jfa.2010.02.015. Google Scholar
M. Nakamura, Small global solutions for nonlinear complex Ginzburg-Landau equations and nonlinear dissipative wave equations in Sobolev spaces,, Reviews in Mathematical Physics, 23 (2011), 903. doi: 10.1142/S0129055X11004473. Google Scholar
M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order,, J. Funct. Anal., 155 (1998), 364. doi: 10.1006/jfan.1997.3236. Google Scholar
M. Nakamura and T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth,, Math. Z., 231 (1999), 479. doi: 10.1007/PL00004737. Google Scholar
M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces,, Publ. Res. Inst. Math. Sci., 37 (2001), 255. doi: 10.2977/prims/1145477225. Google Scholar
T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equation,, Nonlinear Anal., 14 (1990), 765. doi: 10.1016/0362-546X(90)90104-O. Google Scholar
T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem,, J. Math. Anal. Appl., 155 (1991), 531. doi: 10.1016/0022-247X(91)90017-T. Google Scholar
T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012. Google Scholar
T. Ozawa, Characterization of Trudinger's inequality,, J. Inequal. Appl., 1 (1997), 369. doi: 10.1155/S102558349700026X. Google Scholar
I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term,, Arch. Rational Mech. Anal., 129 (1995), 201. doi: 10.1007/BF00383673. Google Scholar
J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth,, Internat. Math. Res. Notices, (1994). doi: 10.1155/S1073792894000346. Google Scholar
R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, Indiana Univ. Math. J., 21 (1972), 841. Google Scholar
M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. Henri Poincaré, 5 (1988), 425. Google Scholar
M. Struwe, The critical nonlinear wave equation in two space dimensions,, J. Eur. Math. Soc. (JEMS), 15 (2013), 1805. doi: 10.4171/JEMS/404. Google Scholar
N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar
B. Wang, Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$,, Discrete Contin. Dynam. Systems, 5 (1999), 753. doi: 10.3934/dcds.1999.5.753. Google Scholar
Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy,, Proc. Amer. Math. Soc., 136 (2008), 3477. doi: 10.1090/S0002-9939-08-09514-2. Google Scholar
Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011
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
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
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
Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212
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
Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064
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
Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378
Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741
Jijiang Sun, Shiwang Ma. Infinitely many sign-changing solutions for the Brézis-Nirenberg problem. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2317-2330. doi: 10.3934/cpaa.2014.13.2317
Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455
Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031
Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
Michinori Ishiwata Makoto Nakamura Hidemitsu Wadade