Abstract :
In this paper, we consider the semilinear wave equation with a power nonlinearity in one space dimension.
We exhibit a universal one-parameter family of functions which stand for the blow-up profile in self-similar
variables at a non-characteristic point, for general initial data. The proof is done in self-similar variables.
We first characterize all the solutions of the associated stationary problem, as a one parameter family. Then,
we use energy arguments coupled with dispersive estimates to show that the solution approaches this family
in the energy norm, in the non-characteristic case, and to a finite decoupled sum of such a solution in the
characteristic case. Finally, in the case where this sum is reduced to one element, which is the case for noncharacteristic
points, we use modulation theory coupled with a nonlinear argument to show the exponential
convergence (in the self-similar time variable) of the various parameters and conclude the proof. This step
provides us with a result of independent interest: the trapping of the solution in self-similar variables near
the set of stationary solutions, valid also for non-characteristic points. The proof of these results is based on
a new analysis in the self-similar variable.
© 2007 Elsevier Inc. All rights reserved.
