کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4592068 | 1335072 | 2007 | 79 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension](/preview/png/4592068.png)
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 non-characteristic 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.
Journal: Journal of Functional Analysis - Volume 253, Issue 1, 1 December 2007, Pages 43-121