Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data


Author: Xu Runzhang
Journal: Quart. Appl. Math. 68 (2010), 459-468
MSC (2000): Primary 35L05, 35K05
DOI: https://doi.org/10.1090/S0033-569X-2010-01197-0
Published electronically: June 4, 2010
MathSciNet review: 2676971
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the initial boundary value problem of semilinear hyperbolic equations $ u_{tt}-\Delta u=f(u)$ and semilinear parabolic equations $ u_{t}-\Delta u=f(u)$ with critical initial data $ E(0)=d$ (or $ J(u_0)=d$), $ I(u_0)<0$, and prove that there exist non-global solutions under classical conditions on $ f$.


References [Enhancements On Off] (What's this?)

  • 1. J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math., 28 (1977), 473-486. MR 0473484 (57:13150)
  • 2. H. A. Levine, Instability and non-existence of global solutions to nonlinear wave equations of the form $ Pu_{tt} = -Au + F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. MR 0344697 (49:9436)
  • 3. H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146. MR 0399682 (53:3525)
  • 4. H. A. Levine, J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361. MR 1463799 (99b:34110)
  • 5. H. A. Levine, G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc., 129 (2001), 793-805. MR 1792187 (2001k:35212)
  • 6. V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308. MR 1273304 (95b:35141)
  • 7. R. T. Glassey, Blow-up theorems for nonlinear wave equations, Math. Z., 32 (1973), 183-203. MR 0340799 (49:5549)
  • 8. F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207. MR 2201151 (2007c:35118)
  • 9. V. A. Galaktionov, S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Analysis, 53 (2003), 453-466. MR 1964337 (2004b:35230)
  • 10. Jorge A. Esquivel-Avila, The dynamics of a nonlinear wave equation, Journal of Mathematical Analysis and Applications, 279 (2003), 135-150. MR 1970496 (2004c:35278)
  • 11. L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel. J. Math., 22 (1975), 273-303. MR 0402291 (53:6112)
  • 12. L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303. Reviewed by Howard A. Levine. MR 0402291 (53:6112)
  • 13. D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal., 30 (1968), 148-172. MR 0227616 (37:3200)
  • 14. B. Straughan, Further global nonexistence theorems for abstract nonlinear wave equations, Proc. AMS, 48 (1975), 381-390. MR 0365265 (51:1518)
  • 15. M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japan, 17 (1972), 173-193. MR 0355247 (50:7723)
  • 16. G. Todorova, K. Christov, Existence and blow-up of solutions of some nonlinear parabolic problems, C. R. Acad. Bulgare Sci., 42 (1989), 17-20. MR 1027475 (91d:35109)
  • 17. G. Todorova, E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $ \mathbb{R}\sp n$, J. Math. Anal. Appl., 303 (2005), 242-257. MR 2113879 (2005k:35292)
  • 18. Hiroshi Uesaka, Oscillation or nonoscillation property for semilinear wave equations, Journal of Computational and Applied Mathematics, 164-165 (2004), 723-730. MR 2056910 (2005b:35195)
  • 19. E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182. MR 1719145 (2000k:35205)
  • 20. Liu Yacheng, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169. MR 1987088 (2004h:35151)
  • 21. Liu Yacheng, Zhao Junsheng, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Analysis, 64 (2006), 2665-2687. MR 2218541 (2007a:35108)
  • 22. Liu Yacheng, Xu Runzhang, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete and Continuous Dynamical System-Series B, 7 (2007), 171-189. MR 2257457 (2007h:35232)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35L05, 35K05

Retrieve articles in all journals with MSC (2000): 35L05, 35K05


Additional Information

Xu Runzhang
Affiliation: College of Science, Harbin Engineering University, 150001, People’s Republic of China
Email: xurunzh@yahoo.com.cn

DOI: https://doi.org/10.1090/S0033-569X-2010-01197-0
Keywords: Semilinear hyperbolic equations, semilinear parabolic equation, critical initial data, potential wells, global nonexistence
Received by editor(s): November 18, 2008
Published electronically: June 4, 2010
Article copyright: © Copyright 2010 Brown University

American Mathematical Society