Small Data Blowup for Semilinear Klein-Gordon Equations

                             by

                 Markus Keel, UCLA and MSRI

         JWB 208, 4:15pm  Friday, December 12, 1997

                          Abstract


This is joint work with T. Tao.

We consider nonlinear Klein-Gordon equations,

$$
(- \partial_t^2 + \Delta -1) u = F(u, \partial_t u, \nabla u),
$$
$$
u(0,\cdot) = \epsilon f,\quad  \partial_t u(0,\cdot) = \epsilon g
$$
where $F$ behaves like a power nonlinearity of order $p$.

Energy heuristics predict global existence from small data
for nonlinear Klein-Gordon equations when the non-linearity
has order $p > 1 + \frac{2}{n}$. We complement these
heuristics and previous rigorous existence results by
demonstrating blow-up from small data for $p \leq 1 +
\frac{2}{n}$. Our blow-up results also yield lifespan
estimates.

As an example of how the lifespan estimates complement
previous results: when $p=3$, our nonlinearity is smooth,
and in dimension $n = 1$ we prove the solution blows up
before time $T_{\epsilon} =
\epsilon^{\frac{C}{\epsilon^2}}$. This answers a conjecture
made by Hormander, who proved the same order lower bound on
the lifespan.

REFERENCES:

L. Hormander,
Lectures on nonlinear hyperbolic differential equations,
Mathematics Applications, 26. Springer-Verlag, 1997.

S. Klainerman,  Global existence of small amplitude
solutions to nonlinear Klein-Gordon equations in four space-time
dimensions, Comm. Pure Appl. Math., 38,(1985),631--641.

H. Lindblad, C.D. Sogge , Restriction theorems and semilinear
Klein-Gordon equations in $(1+3)$ dimensions,
Duke Math J., 85, 227--252, 1996.

T. Ozawa, K. Tsutaya, Y. Tsutsumi,
Global existence and asymptotic behavior of solutions for the
Klein-Gordon equations with quadratic nonlinearity in two space
dimensions, Math Z., 222, 1996, 341-362.

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon
equations, Comm. Pure. Appl. Math., 38, 1985, 685--696.

From toner300@hotmail.com Thu Dec 11 12:10:48 1997
Flags: 000000000000
Received: from bmd.clis.com (bmd.clis.com [207.22.129.1])
        by csc-sun.math.utah.edu (8.8.5/8.8.5) with SMTP id MAA04628
        for <gustafso@math.utah.edu>; Thu, 11 Dec 1997 12:10:46 -0700 (MST)
                                Abstract


  The talk will present two global existence and uniqueness theorems
  for the critical power Yang-Mills-Higgs equations in RxR^3.

  These theorems extend the known results by allowing a critical power
  Higgs nonlinearity for both smooth and finite energy initial data.
  In particular, we allow Higgs nonlinearities in the Lagrangian which
  behave like $V(\phi) = \norm{\phi}^6$ for large $\norm{\phi}$.  This
  corresponds to a degree 5 power nonlinearity in the equations
  themselves.

  Some of the ideas for the results are easily seen in the context of
  the wave equation.  I'll first show a simple extension of known
  non-concentration results for the critical power wave equation
  (CPWE)

                 \square  \phi  =  \norm{\phi}^4 \phi

  for (t,x) in RxR^3 which will be necessary for the Yang - Mills -
  Higgs results. I'll then give a simple proof of global existence for
  the CPWE.

  As for the Yang-Mills-Higgs results: I first indicate why the Higgs
  field enjoys the same local non-concentration as the CPWE. Finally,
  I'll describe how to take advantage of this by using the local
  estimates of Klainerman-Machedon [Comm. Pure and Applied Math, 1993,
  1221-1268] and their application to the Yang-Mills Equations [K-M,
  Annals of Math, 1995, 39-119].

Requests for preprints and reprints to: rmm@math.utah.edu Attn: Mark Keel
This source can be found at http://www.math.utah.edu/research/