Advanced Search
Article Contents
Article Contents

On the strauss index of semilinear tricomi equation

  • * Corresponding author

    * Corresponding author 

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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 $.

    Mathematics Subject Classification: Primary: 35L70, 35L71; Secondary: 35L80, 35L81.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. GeorgievH. 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. HeI. 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. HeI. 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. LupoC. 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. RuanI. 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. RuanI. 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. 
  • 加载中

Article Metrics

HTML views(1565) PDF downloads(368) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint