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Global bifurcation for the Hénon problem
On the strauss index of semilinear tricomi equation
1. | School of Mathematics, Southeast University, Nanjing 211189, China |
2. | Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany |
3. | School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing 210023, China |
$ u $ |
$ \begin{equation*} \partial_t^2 u-t\, \Delta u = |u|^p, \quad \big(u(0, \cdot), \partial_t u(0, \cdot)\big) = (u_0, u_1), \end{equation*} $ |
$ t>0 $ |
$ x\in \mathbb R^n $ |
$n\geq2$ |
$ p>1 $ |
$ u_i\in C_0^{\infty}( \mathbb R^n) $ |
$ i = 0, 1 $ |
$\|u(t, \cdot)\|_{L^\infty_x(\mathbb R)}$ |
$u$ |
$t$ |
$ u $ |
$ p>5 $ |
References:
[1] |
J. Barros Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator I, II, III, Duke Math. J., 98 (1999), 465–483; 111 (2002), 561–584; 117 (2003), 385–387.
doi: 10.1215/S0012-7094-99-09814-9. |
[2] |
M. Beals, Singularities due to cusp interactions in nonlinear waves in "Nonlinear Hyperbolic Equations and Field Theory", Pitman Res. Notes Math. Ser., Longman Sci. Tech. Harlow, 253 (1992), 36–51. |
[3] |
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surv. Appl. Math., Vol.3, Chapman and Hall, London, 1958. |
[4] |
M. D'Abbicco,
The threshold of effective damping for semilinear wave equations, Math. Meth. Appl. Sci., 6 (2015), 1032-1045.
doi: 10.1002/mma.3126. |
[5] |
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953. |
[6] |
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953. |
[7] |
A. Galstian,
Global existence for the one-dimensional second order semilinear hyperbolic equations, J. Math. Anal. Appl., 344 (2008), 76-98.
doi: 10.1016/j.jmaa.2008.02.022. |
[8] |
R. T. Glassey,
Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.
doi: 10.1007/BF01162066. |
[9] |
R. T. Glassey,
Existence in the large for $\Box$u = F(u) in two space dimensions, Math. Z., 178 (1981), 233-261.
doi: 10.1007/BF01262042. |
[10] |
V. Georgiev, H. Lindblad and C. D. Sogge,
Weighted Strichartz estimates and global existence for semi-linear wave equations, Amer. J. Math., 119 (1997), 1291-1319.
doi: 10.1353/ajm.1997.0038. |
[11] |
D. K. Gvazava,
The global solution of the Tricomi problem for a class of nonlinear mixed differential equations, Differ. Equ., 3 (1967), 1-4.
|
[12] |
D. Y. He, I. Witt and H. C. Yin,
On the global solution problem of semilinear generalized Tricomi equations, I, Calc. Var. Partial Differ. Equ., 56 (2017), 1-24.
doi: 10.1007/s00526-017-1125-9. |
[13] |
D. Y. He, I. Witt and H. C. Yin, On the global solution problem of semilinear generalized Tricomi equations, II, preprint, arXiv: 1611.07606, to appear in Pac. J. Math., 2020.
doi: 10.1007/s00526-017-1125-9. |
[14] |
D. Y. He, I. Witt and H. C. Yin,
On semilinear Tricomi equations with critical exponents or in two space dimensions, J. Differ. Equ., 263 (2017), 8102-8137.
doi: 10.1016/j.jde.2017.08.033. |
[15] |
D. Y. He, I. Witt and H. C. Yin, Finite time blowup for the 1-D semilinear Tricomi equation with critical exponent, preprint, 2019. |
[16] |
F. John,
Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscr. Math., 28 (1979), 235-265.
doi: 10.1007/BF01647974. |
[17] |
J. Lin and Z. Tu, Lifespan of semilinear generalized Tricomi equation with Strauss type exponent, preprint, arXiv: 1903.11351. |
[18] |
H. Lindblad and C. D. Sogge,
On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.
doi: 10.1006/jfan.1995.1075. |
[19] |
D. Lupo, C. S. Morawetz and K. R. Payne,
On closed boundary value problems for equations of mixed elliptic-hyperbolic type, Commun. Pure Appl. Math., 60 (2007), 1319-1348.
doi: 10.1002/cpa.20169. |
[20] |
D. Lupo and K. R. Payne,
Spectral bounds for Tricomi problems and application to semilinear existence and existence with uniqueness results, J. Differ. Equ., 184 (2002), 139-162.
doi: 10.1006/jdeq.2001.4139. |
[21] |
D. Lupo and K. R. Payne,
Critical exponents for semilinear equations of mixed elliptic-hyperbolic and degenerate types, Commun. Pure Appl. Math., 56 (2003), 403-424.
doi: 10.1002/cpa.3031. |
[22] |
D. Lupo and K. R. Payne,
Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types, Duke Math. J., 127 (2005), 251-290.
doi: 10.1215/S0012-7094-04-12722-8. |
[23] |
C. S. Morawetz,
Mixed equations and transonic flow, J. Hyperbolic Differ. Equ., 1 (2004), 1-26.
doi: 10.1142/S0219891604000081. |
[24] |
Z. P. Ruan, I. Witt and H. C. Yin, On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data, Commun. Contemp. Math., 17 (2015), 49 pp.
doi: 10.1142/S021919971450028X. |
[25] |
Z. P. Ruan, I. Witt and H. C. Yin,
On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains, J. Differ. Equ., 259 (2015), 7406-7462.
doi: 10.1016/j.jde.2015.08.025. |
[26] |
Z. P. Ruan, I. Witt and H. C. Yin,
Minimal regularity solutions of semilinear generalized Tricomi equations, Pac. J. Math., 296 (2018), 181-226.
doi: 10.2140/pjm.2018.296.181. |
[27] |
J. Schaeffer,
The equation $u_tt-\Delta u = |u|^p$ for the critical value of p, Proc. R. Soc. Edinb., 101 (1985), 31-44.
doi: 10.1017/S0308210500026135. |
[28] |
T. Sideris,
Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differ. Equ., 52 (1984), 378-406.
doi: 10.1016/0022-0396(84)90169-4. |
[29] |
C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Math., Vol. 105, 1993.
doi: 10.1017/CBO9780511530029. |
[30] |
W. Strauss,
Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.
doi: 10.1016/0022-1236(81)90063-X. |
[31] |
K. Taniguchi and Y. Tozaki,
A hyperbolic equation with double characteristics which has a solution with branching singularities, Math. Jpn., 25 (1980), 279-300.
|
[32] |
K. Yagdjian,
A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differ. Equ., 206 (2004), 227-252.
doi: 10.1016/j.jde.2004.07.028. |
[33] |
K. Yagdjian,
Global existence in the Cauchy problem for nonlinear wave equations with variable speed of propagation, Oper. Theory Adv. Appl., 159 (2005), 301-385.
doi: 10.1007/3-7643-7386-5_4. |
[34] |
K. Yagdjian,
Global existence for the $n$-dimensional semilinear Tricomi-type equations, Commun. Partial Differ. Equ., 31 (2006), 907-944.
doi: 10.1080/03605300500361511. |
[35] |
K. Yagdjian,
The self-similar solutions of the Tricomi-type equations, Z. angew. Math. Phys., 58 (2007), 612-645.
doi: 10.1007/s00033-006-5099-2. |
[36] |
B. Yordanov and Q. S. Zhang,
Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.
doi: 10.1016/j.jfa.2005.03.012. |
[37] |
Y. Zhou,
Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Differ. Equ., 8 (1995), 135-144.
|
show all references
References:
[1] |
J. Barros Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator I, II, III, Duke Math. J., 98 (1999), 465–483; 111 (2002), 561–584; 117 (2003), 385–387.
doi: 10.1215/S0012-7094-99-09814-9. |
[2] |
M. Beals, Singularities due to cusp interactions in nonlinear waves in "Nonlinear Hyperbolic Equations and Field Theory", Pitman Res. Notes Math. Ser., Longman Sci. Tech. Harlow, 253 (1992), 36–51. |
[3] |
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surv. Appl. Math., Vol.3, Chapman and Hall, London, 1958. |
[4] |
M. D'Abbicco,
The threshold of effective damping for semilinear wave equations, Math. Meth. Appl. Sci., 6 (2015), 1032-1045.
doi: 10.1002/mma.3126. |
[5] |
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953. |
[6] |
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953. |
[7] |
A. Galstian,
Global existence for the one-dimensional second order semilinear hyperbolic equations, J. Math. Anal. Appl., 344 (2008), 76-98.
doi: 10.1016/j.jmaa.2008.02.022. |
[8] |
R. T. Glassey,
Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.
doi: 10.1007/BF01162066. |
[9] |
R. T. Glassey,
Existence in the large for $\Box$u = F(u) in two space dimensions, Math. Z., 178 (1981), 233-261.
doi: 10.1007/BF01262042. |
[10] |
V. Georgiev, H. Lindblad and C. D. Sogge,
Weighted Strichartz estimates and global existence for semi-linear wave equations, Amer. J. Math., 119 (1997), 1291-1319.
doi: 10.1353/ajm.1997.0038. |
[11] |
D. K. Gvazava,
The global solution of the Tricomi problem for a class of nonlinear mixed differential equations, Differ. Equ., 3 (1967), 1-4.
|
[12] |
D. Y. He, I. Witt and H. C. Yin,
On the global solution problem of semilinear generalized Tricomi equations, I, Calc. Var. Partial Differ. Equ., 56 (2017), 1-24.
doi: 10.1007/s00526-017-1125-9. |
[13] |
D. Y. He, I. Witt and H. C. Yin, On the global solution problem of semilinear generalized Tricomi equations, II, preprint, arXiv: 1611.07606, to appear in Pac. J. Math., 2020.
doi: 10.1007/s00526-017-1125-9. |
[14] |
D. Y. He, I. Witt and H. C. Yin,
On semilinear Tricomi equations with critical exponents or in two space dimensions, J. Differ. Equ., 263 (2017), 8102-8137.
doi: 10.1016/j.jde.2017.08.033. |
[15] |
D. Y. He, I. Witt and H. C. Yin, Finite time blowup for the 1-D semilinear Tricomi equation with critical exponent, preprint, 2019. |
[16] |
F. John,
Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscr. Math., 28 (1979), 235-265.
doi: 10.1007/BF01647974. |
[17] |
J. Lin and Z. Tu, Lifespan of semilinear generalized Tricomi equation with Strauss type exponent, preprint, arXiv: 1903.11351. |
[18] |
H. Lindblad and C. D. Sogge,
On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130 (1995), 357-426.
doi: 10.1006/jfan.1995.1075. |
[19] |
D. Lupo, C. S. Morawetz and K. R. Payne,
On closed boundary value problems for equations of mixed elliptic-hyperbolic type, Commun. Pure Appl. Math., 60 (2007), 1319-1348.
doi: 10.1002/cpa.20169. |
[20] |
D. Lupo and K. R. Payne,
Spectral bounds for Tricomi problems and application to semilinear existence and existence with uniqueness results, J. Differ. Equ., 184 (2002), 139-162.
doi: 10.1006/jdeq.2001.4139. |
[21] |
D. Lupo and K. R. Payne,
Critical exponents for semilinear equations of mixed elliptic-hyperbolic and degenerate types, Commun. Pure Appl. Math., 56 (2003), 403-424.
doi: 10.1002/cpa.3031. |
[22] |
D. Lupo and K. R. Payne,
Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types, Duke Math. J., 127 (2005), 251-290.
doi: 10.1215/S0012-7094-04-12722-8. |
[23] |
C. S. Morawetz,
Mixed equations and transonic flow, J. Hyperbolic Differ. Equ., 1 (2004), 1-26.
doi: 10.1142/S0219891604000081. |
[24] |
Z. P. Ruan, I. Witt and H. C. Yin, On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data, Commun. Contemp. Math., 17 (2015), 49 pp.
doi: 10.1142/S021919971450028X. |
[25] |
Z. P. Ruan, I. Witt and H. C. Yin,
On the existence of low regularity solutions to semilinear generalized Tricomi equations in mixed type domains, J. Differ. Equ., 259 (2015), 7406-7462.
doi: 10.1016/j.jde.2015.08.025. |
[26] |
Z. P. Ruan, I. Witt and H. C. Yin,
Minimal regularity solutions of semilinear generalized Tricomi equations, Pac. J. Math., 296 (2018), 181-226.
doi: 10.2140/pjm.2018.296.181. |
[27] |
J. Schaeffer,
The equation $u_tt-\Delta u = |u|^p$ for the critical value of p, Proc. R. Soc. Edinb., 101 (1985), 31-44.
doi: 10.1017/S0308210500026135. |
[28] |
T. Sideris,
Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differ. Equ., 52 (1984), 378-406.
doi: 10.1016/0022-0396(84)90169-4. |
[29] |
C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Math., Vol. 105, 1993.
doi: 10.1017/CBO9780511530029. |
[30] |
W. Strauss,
Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.
doi: 10.1016/0022-1236(81)90063-X. |
[31] |
K. Taniguchi and Y. Tozaki,
A hyperbolic equation with double characteristics which has a solution with branching singularities, Math. Jpn., 25 (1980), 279-300.
|
[32] |
K. Yagdjian,
A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differ. Equ., 206 (2004), 227-252.
doi: 10.1016/j.jde.2004.07.028. |
[33] |
K. Yagdjian,
Global existence in the Cauchy problem for nonlinear wave equations with variable speed of propagation, Oper. Theory Adv. Appl., 159 (2005), 301-385.
doi: 10.1007/3-7643-7386-5_4. |
[34] |
K. Yagdjian,
Global existence for the $n$-dimensional semilinear Tricomi-type equations, Commun. Partial Differ. Equ., 31 (2006), 907-944.
doi: 10.1080/03605300500361511. |
[35] |
K. Yagdjian,
The self-similar solutions of the Tricomi-type equations, Z. angew. Math. Phys., 58 (2007), 612-645.
doi: 10.1007/s00033-006-5099-2. |
[36] |
B. Yordanov and Q. S. Zhang,
Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.
doi: 10.1016/j.jfa.2005.03.012. |
[37] |
Y. Zhou,
Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Differ. Equ., 8 (1995), 135-144.
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