Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nonexistence of global solutions of
a nonlinear hyperbolic system


Author: Keng Deng
Journal: Trans. Amer. Math. Soc. 349 (1997), 1685-1696
MSC (1991): Primary 35L15, 35L55, 35L70
DOI: https://doi.org/10.1090/S0002-9947-97-01841-2
MathSciNet review: 1401767
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the initial value problem

\begin{equation*}\begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0, \\ [2\jot ] u(x,0)=f(x),&v(x,0)=h(x),&{}&{} \\ [2\jot ] u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), &{}&{} \end {array} \end{equation*}

with $1\le n\le 3$ and $p,q>0$. We show that there exists a bound $B(n)\ (\le \infty )$ such that if $1<pq<B(n)$ all nontrivial solutions with compact support blow up in finite time.


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

  • 1. M. Escobedo and M.A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations 89 (1991), 176-202. MR 91j:35040
  • 2. H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 16 (1966), 105-113. MR 35:5761
  • 3. R.T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177 (1981), 323-340. MR 82i:35120
  • 4. R.T. Glassey, Existence in the large for $\square u=F(u)$ in two space dimensions, Math. Z. 178 (1981), 233-261. MR 84h:35106
  • 5. J.W. Jerome, Approximation of nonlinear evolution systems, Math. in Sci. Engineering 164, Academic Press, New York, 1983. MR 85g:35064
  • 6. F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268. MR 80i:35114
  • 7. T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 32 (1980), 501-505. MR 82f:35128
  • 8. S. Klainerman, The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math. 23 (1986), 293-326. MR 87h:35217
  • 9. H.A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_{tt}=-Au+{\mathcal {F}}(u)$, Trans. Amer. Math. Soc. 192 (1974), 1-21. MR 49:9436
  • 10. H.A. Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, Z. Angew Math. Phys. 42 (1991), 408-430. MR 92g:35097
  • 11. J. Schaeffer, The equation $u_{tt}-\Delta u=\vert u\vert ^{p}$ for the critical value $p$, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 31-44. MR 87g:35159
  • 12. T. Sideris, Ph.D. thesis, Indiana University, Bloomington, 1981.
  • 13. T. Sideris, Nonexistence of global solutions of semilinear wave equations in high dimensions, J. Differential Equations 52 (1984), 378-406. MR 86d:35090
  • 14. F.B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29-40. MR 82g:35059

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35L15, 35L55, 35L70

Retrieve articles in all journals with MSC (1991): 35L15, 35L55, 35L70


Additional Information

Keng Deng
Affiliation: Department of Mathematics, University of Southwestern Louisiana, Lafayette, Louisiana 70504
Email: kxd5858@usl.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01841-2
Received by editor(s): March 16, 1995
Received by editor(s) in revised form: December 1, 1995
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society