Blow-Up of Solutions for the Fourth-Order Schrödinger Equation with Combined Power-Type Nonlinearities

Abstract

In this paper, we mainly consider the blow-up solutions of the fourth-order Schrödinger equation with combined power-type nonlinearities

$$\begin{aligned} iu_{t}+\alpha \Delta ^{2}u+\beta \Delta u+\lambda _{1}\left| u \right| ^{\sigma _{1}}u+\lambda _{2}\left| u \right| ^{\sigma _{2}}u=0, \end{aligned}$$

where \(4<n<8,\) \(\beta =\left\{ { 0, 1}\right\} , \alpha ,\,\lambda _{1}\in \mathbb {R}\) and \(\lambda _{2}<0\). Firstly, using Banach’s fixed point theorem, iterative method and nonlinear estimates, we establish the local well-posedness of solutions with the initial data \(u_{0}\in H^{2}(\mathbb {R}^{n})\). Then, based on variational analysis theory for dynamical system, using localized Virial identity, we establish a new Morawetz estimates and upper bound estimates to prove the existence of blow-up solutions in finite time. Finally, applying the local well-posedness above, we demonstrate the blow-up criteria of solutions and prove it by contradiction method.