Abstract
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points $ I_0 $ is open and that the blow-up curve is of class $ C^{1, \mu_0} $ and the phase $ \theta $ is $ C^{\mu_0} $ on this set. In order to prove this result, we introduce a Liouville Theorem for that equation. Our results hold also for the case of solutions with values in $ \mathbb{R}^m $ with $ m\ge 3 $, with the same proof.
Highlights
For other types of nonlinearities, we mention our recent contribution with Masmoudi and Zaag in [5], where we study the semilinear wave equation with exponential nonlinearity
In [4], we proved the existence of the blow-up profile at non-characteristic points
In the real case, relying on the existence of a blow-up profile, together with Liouville type Theorem, Merle and Zaag [18] could prove the openness of the set of non-characteristic points I0 and the C1 regularity of the blow-up curve restricted to I0
Summary
We consider the following complexvalued one-dimensional semilinear wave equation. Where u(t) : x ∈ R → u(x, t) ∈ Rm with m ≥ 2 , u0 ∈ Hl1oc,u and u1 ∈ L2loc,u,with. In order to avoid complicated notations, we will state our results and give our proofs only in the case m = 2. The interested reader may find in [6] the necessary formalism to extend the proof to the case m ≥ 3. The Cauchy problem for equation (1) in the space Hl1oc,u × L2loc,u follows from the finite speed of propagation and the wellposedness in H1 × L2.
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