Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms

Michinori Ishiwata; Makoto Nakamura; Hidemitsu Wadade

doi:10.3934/dcds.2015.35.4889

Article Contents

2015, Volume 35, Issue 10: 4889-4903. Doi: 10.3934/dcds.2015.35.4889

This issue Previous Article Existence of Neumann and singular solutions of the fast diffusion equation Next Article Wave extension problem for the fractional Laplacian

Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms

1.
Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Machikaneyama-cho, Toyonaka, Osaka 560-8531
2.
Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560
3.
Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa 920-1192

Received: September 30, 2014

Revised: December 31, 2014

Published: September 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L70; Secondary: 35L15.

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References

[1]	S. Adachi and K. Tanaka, A scale-invariant form of Trudinger-Moser inequality and its best exponent, Proc. Amer. Math. Soc., 1102 (1999), 148-153.
[2]	F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773.doi: 10.1090/S0894-0347-1989-1002633-4.
[3]	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), 797-818.doi: 10.1080/03605302.2010.534684.
[4]	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), 1-21.
[5]	P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.doi: 10.1090/S0002-9947-1984-0742415-3.
[6]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.
[7]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
[8]	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), 268-275.doi: 10.1016/j.aml.2008.03.024.
[9]	P. Brenner, $L_p$-estimates of difference schemes for strictly hyperbolic systems with nonsmooth data, SIAM J. Numer. Anal., 14 (1977), 1126-1144.doi: 10.1137/0714078.
[10]	H. Brézis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential, Selecta Math. (N.S.), 11 (2005), 1-7.doi: 10.1007/s00029-005-0003-z.
[11]	H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.doi: 10.1016/0362-546X(80)90068-1.
[12]	H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.doi: 10.1080/03605308008820154.
[13]	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), 519-549.doi: 10.1016/S0022-1236(03)00238-6.
[14]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
[15]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.doi: 10.3934/cpaa.2014.13.1267.
[16]	J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions, J. Hyperbolic Differ. Equ., 6 (2009), 549-575.doi: 10.1142/S0219891609001927.
[17]	Z. Gan, Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential, Commun. Pure Appl. Anal., 8 (2009), 1541-1554.doi: 10.3934/cpaa.2009.8.1541.
[18]	J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.doi: 10.1006/jdeq.1997.3375.
[19]	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), 15-35.
[20]	J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.doi: 10.1006/jfan.1995.1119.
[21]	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), 617-633.doi: 10.1017/S0308210509000729.
[22]	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), 1639-1658.doi: 10.1002/cpa.20127.
[23]	S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc., 135 (2007), 87-97.doi: 10.1090/S0002-9939-06-08240-2.
[24]	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), 3740-3771.doi: 10.1016/j.jde.2011.01.008.
[25]	S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Well posedness and unconditional non uniqueness for a 2D semilinear heat equation, preprint.
[26]	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), 297-314.doi: 10.1016/j.anihpc.2013.03.004.
[27]	T. Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math., 13 (1972), 135-148 (1973).doi: 10.1007/BF02760233.
[28]	J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma, Phys. Fluids, 20 (1977), 1176-1179.doi: 10.1063/1.861679.
[29]	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), 295-312.doi: 10.7153/mia-14-24.
[30]	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), 1747-1766.doi: 10.1016/j.na.2010.05.012.
[31]	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), Art. ID 584521, 38 pp.
[32]	J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092.
[33]	S. Nagayasu and H. Wadade, Characterization of the critical Sobolev space on the optimal singularity at the origin, J. Funct. Anal., 258 (2010), 3725-3757.doi: 10.1016/j.jfa.2010.02.015.
[34]	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), 903-931.doi: 10.1142/S0129055X11004473.
[35]	M. Nakamura and T. Ozawa, Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal., 155 (1998), 364-380.doi: 10.1006/jfan.1997.3236.
[36]	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), 479-487.doi: 10.1007/PL00004737.
[37]	M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces, Publ. Res. Inst. Math. Sci., 37 (2001), 255-293.doi: 10.2977/prims/1145477225.
[38]	T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equation, Nonlinear Anal., 14 (1990), 765-769.doi: 10.1016/0362-546X(90)90104-O.
[39]	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), 531-540.doi: 10.1016/0022-247X(91)90017-T.
[40]	T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.doi: 10.1006/jfan.1995.1012.
[41]	T. Ozawa, Characterization of Trudinger's inequality, J. Inequal. Appl., 1 (1997), 369-374.doi: 10.1155/S102558349700026X.
[42]	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), 201-224.doi: 10.1007/BF00383673.
[43]	J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, (1994), 303ff., approx. 7 pp.doi: 10.1155/S1073792894000346.
[44]	R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J., 21 (1972), 841-842.
[45]	M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. Henri Poincaré, Analyse nonlinéaire, 5 (1988), 425-464.
[46]	M. Struwe, The critical nonlinear wave equation in two space dimensions, J. Eur. Math. Soc. (JEMS), 15 (2013), 1805-1823.doi: 10.4171/JEMS/404.
[47]	N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.
[48]	B. Wang, Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$, Discrete Contin. Dynam. Systems, 5 (1999), 753-763.doi: 10.3934/dcds.1999.5.753.
[49]	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), 3477-3482.doi: 10.1090/S0002-9939-08-09514-2.