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October  2020, 19(10): 4817-4838. doi: 10.3934/cpaa.2020213

On the strauss index of semilinear tricomi equation

Daoyin He 1,, , Ingo Witt 2, and  Huicheng Yin 3,

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

* Corresponding author

Received  September 2019 Revised  May 2020 Published  July 2020

Fund Project: Authors are supported by National Natural Science Foundation of China (grant No. 11901103, No. 11571177, No. 11731007)

In our previous papers, we have given a systematic study on the global existence versus blowup problem for the small-data solution
$ u $
 of the multi-dimensional semilinear Tricomi equation
$ \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*} $
where
$ t>0 $
,
$ x\in \mathbb R^n $
,
$n\geq2$
,
$ p>1 $
, and
$ u_i\in C_0^{\infty}( \mathbb R^n) $
 (
$ i = 0, 1 $
). In this article, we deal with the remaining 1-D problem, for which the stationary phase method for multi-dimensional case fails to work and the large time decay rate of
$\|u(t, \cdot)\|_{L^\infty_x(\mathbb R)}$
 is not enough. The main ingredient of the proof in this paper is to use the structure of the linear equation to get the suitable decay rate of
$u$
 in
$t$
, then the crucial weighted Strichartz estimates are established and the global existence of solution
$ u $
 is proved when
$ p>5 $
.
Keywords: Tricomi equation, weighted Strichartz estimate, global existence, Fourier integral operator.
Mathematics Subject Classification: Primary: 35L70, 35L71; Secondary: 35L80, 35L81.
Citation: Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
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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