Geometric Optics Expansions for Hyperbolic Corner Problems II: From Weak Stability to Violent Instability

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In this article we are interested in the rigorous construction of geometric optics expansions for weakly well-posed hyperbolic corner problems. More precisely we focus on the case where self-interacting phases occur and where one of them is exactly the phase where the uniform Kreiss--Lopatinskii condition fails. We show that the associated WKB expansion suffers arbitrarily many amplifications before a fixed finite time. As a consequence, we show that such a corner problem cannot be weakly well-posed even at the price of a huge loss of derivatives. The new result, in that framework, is that the violent instability (or Hadamard instability) does not come from the degeneracy of the weak Kreiss--Lopatinskii condition but from the accumulation of arbitrarily many weak instabilities.