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

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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,

1.	Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan
2.	Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560, Japan
3.	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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889

References:

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[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.

show all references

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.

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