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

In this paper, we mainly consider the blow-up solutions of the fourth-order Schrödinger equation with combined power-type nonlinearities \(\begin{aligned} iu_{t}+\alpha \Delta ^{2}u+\beta \Delta u+\lambda _{1}\left| u \right| ^{\sigma _{1}}u+\lambda _{2}\left| u \right| ^{\sigma _{2}}u=0, \end{aligned}\) $$\begin{array}{c}\hfill i{u}_t+\alpha {\mathrm{\Delta }}^{2}u+\beta \mathrm{\Delta }u+{\lambda }_1{\left|u\right|}^{{\sigma }_1}u+{\lambda }_2{\left|u\right|}^{{\sigma }_2}u=0,\end{array}$$ where \(4<n<8,\) $4<n<8,$ \(\beta =\left\{ { 0, 1}\right\} , \alpha ,\,\lambda _{1}\in \mathbb {R}\) $\beta =\left\{0,1\right\},\alpha ,{\lambda }_1\in \mathbb{R}$ and \(\lambda _{2}<0\) ${\lambda }_2<0$ . Firstly, using Banach’s fixed point theorem, iterative method and nonlinear estimates, we establish the local well-posedness of solutions with the initial data \(u_{0}\in H^{2}(\mathbb {R}^{n})\) $u_0\in H^2({\mathbb{R}}^n)$ . 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.