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

Abstract

We consider the fourth-order Schrödinger equation $$i{\partial }_{t}u+{\Delta }^{2}u+\mu \Delta u+\lambda |u{|}^{\alpha }u=0,$$ where $\alpha >0,\mu =\pm 1$ or $0$ and $\lambda \in \mathbb{C}$. Firstly, we prove local well-posedness in $H^4\left({\mathbb{R}}^N\right)$ in both $H^4$ subcritical and critical case: $\alpha >0$, $(N-8)\alpha \le 8$. Then, for any given compact set $K\subset {\mathbb{R}}^N$, we construct $H^4({\mathbb{R}}^N)$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up exactly on $K$ at $t=0$.

arXiv comment36 pages, 3 figures

References in corpus (6)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations
• Construction of a blow-up solution for the Complex Ginzburg-Landau equation in some critical case
• Singular standing-ring solutions of nonlinear partial differential equations
• Finite-time blowup for a Schrödinger equation with nonlinear source term
• Solutions blowing up on any given compact set for the energy subcritical wave equation

Cited by in corpus (1)

• Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schrödinger equation