Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

We consider the fourth-order Schrödinger equation $i\partial_tu+Δ^2 u+μΔu+λ|u|^αu=0,$ where $α>0,μ=\pm1$ or $0$ and $λ\in\mathbb{C}$. Firstly, we prove local well-posedness in $H^4\left(\R^N\right)$ in both $H^4$ subcritical and critical case: $α>0$, $(N-8)α\leq8$. Then, for any given compact set $K\subset\mathbb{R}^N$, we construct $H^4(\R^N)$ solutions that are defined on $(-T, 0)$ for some $T>0$, and blow up exactly on $K$ at $t=0$.

No reviews yet.

No comments yet.