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The equation utt − Δu = |u|p for the critical value of p

  • Jack Schaeffer (a1)

The equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.

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1Fujita H.. On the blowing-up of solutions of the Cauchy problem for ut = Δu + u1+α J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109124.
2Glassey R.. Finite-time blow-up for solutions of nonlinear wave equations. Math. Z. 177 (1981), 323340.
3Glassey R.. Existence in the large for u = F (u) in two space dimensions. Math. Z. 178 (1981), 233261.
4John F.. Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28 (1979), 235268.
5Sideris T.. Nonexistence of global solutions to semilinear wave equations in high dimensions. Brown University L.C.D.S. Report #81–23 (October 1981).
6Weissler F. B.. Existence and nonexistence of global solutions for a semilinear heat equation. Israel J. Math. 38 (1981), 2940.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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