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

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

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.
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
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. Google Scholar

[2]

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

[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. doi: 10.1080/03605302.2010.534684. Google Scholar

[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. Google Scholar

[5]

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

[6]

C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar

[7]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, (1976). Google Scholar

[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. doi: 10.1016/j.aml.2008.03.024. Google Scholar

[9]

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

[10]

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

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H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: 10.1016/0362-546X(80)90068-1. Google Scholar

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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

[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. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar

[14]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar

[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. doi: 10.3934/cpaa.2014.13.1267. Google Scholar

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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

[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. doi: 10.3934/cpaa.2009.8.1541. Google Scholar

[18]

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

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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

[20]

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

[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. doi: 10.1017/S0308210509000729. Google Scholar

[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. doi: 10.1002/cpa.20127. Google Scholar

[23]

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

[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. doi: 10.1016/j.jde.2011.01.008. Google Scholar

[25]

S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Well posedness and unconditional non uniqueness for a 2D semilinear heat equation,, preprint., (). Google Scholar

[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. doi: 10.1016/j.anihpc.2013.03.004. Google Scholar

[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, 13 (1972), 135. doi: 10.1007/BF02760233. Google Scholar

[28]

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

[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. doi: 10.7153/mia-14-24. Google Scholar

[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. doi: 10.1016/j.na.2010.05.012. Google Scholar

[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). Google Scholar

[32]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077. Google Scholar

[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. doi: 10.1016/j.jfa.2010.02.015. Google Scholar

[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. doi: 10.1142/S0129055X11004473. Google Scholar

[35]

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

[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. doi: 10.1007/PL00004737. Google Scholar

[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. doi: 10.2977/prims/1145477225. Google Scholar

[38]

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

[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. doi: 10.1016/0022-247X(91)90017-T. Google Scholar

[40]

T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012. Google Scholar

[41]

T. Ozawa, Characterization of Trudinger's inequality,, J. Inequal. Appl., 1 (1997), 369. doi: 10.1155/S102558349700026X. Google Scholar

[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. doi: 10.1007/BF00383673. Google Scholar

[43]

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

[44]

R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, Indiana Univ. Math. J., 21 (1972), 841. Google Scholar

[45]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. Henri Poincaré, 5 (1988), 425. Google Scholar

[46]

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

[47]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar

[48]

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

[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. doi: 10.1090/S0002-9939-08-09514-2. Google Scholar

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. Google Scholar

[2]

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

[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. doi: 10.1080/03605302.2010.534684. Google Scholar

[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. Google Scholar

[5]

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

[6]

C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar

[7]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Grundlehren der Mathematischen Wissenschaften, (1976). Google Scholar

[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. doi: 10.1016/j.aml.2008.03.024. Google Scholar

[9]

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

[10]

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

[11]

H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: 10.1016/0362-546X(80)90068-1. Google Scholar

[12]

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

[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. doi: 10.1016/S0022-1236(03)00238-6. Google Scholar

[14]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights,, Compositio Math., 53 (1984), 259. Google Scholar

[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. doi: 10.3934/cpaa.2014.13.1267. Google Scholar

[16]

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

[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. doi: 10.3934/cpaa.2009.8.1541. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[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. doi: 10.1017/S0308210509000729. Google Scholar

[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. doi: 10.1002/cpa.20127. Google Scholar

[23]

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

[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. doi: 10.1016/j.jde.2011.01.008. Google Scholar

[25]

S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Well posedness and unconditional non uniqueness for a 2D semilinear heat equation,, preprint., (). Google Scholar

[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. doi: 10.1016/j.anihpc.2013.03.004. Google Scholar

[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, 13 (1972), 135. doi: 10.1007/BF02760233. Google Scholar

[28]

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

[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. doi: 10.7153/mia-14-24. Google Scholar

[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. doi: 10.1016/j.na.2010.05.012. Google Scholar

[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). Google Scholar

[32]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077. Google Scholar

[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. doi: 10.1016/j.jfa.2010.02.015. Google Scholar

[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. doi: 10.1142/S0129055X11004473. Google Scholar

[35]

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

[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. doi: 10.1007/PL00004737. Google Scholar

[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. doi: 10.2977/prims/1145477225. Google Scholar

[38]

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

[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. doi: 10.1016/0022-247X(91)90017-T. Google Scholar

[40]

T. Ozawa, On critical cases of Sobolev's inequalities,, J. Funct. Anal., 127 (1995), 259. doi: 10.1006/jfan.1995.1012. Google Scholar

[41]

T. Ozawa, Characterization of Trudinger's inequality,, J. Inequal. Appl., 1 (1997), 369. doi: 10.1155/S102558349700026X. Google Scholar

[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. doi: 10.1007/BF00383673. Google Scholar

[43]

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

[44]

R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities,, Indiana Univ. Math. J., 21 (1972), 841. Google Scholar

[45]

M. Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces,, Ann. Inst. Henri Poincaré, 5 (1988), 425. Google Scholar

[46]

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

[47]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473. Google Scholar

[48]

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

[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. doi: 10.1090/S0002-9939-08-09514-2. Google Scholar

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