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This paper deals with the Cauchy problem associated to the nonlinearfourth-order Schr\"odinger equation with isotropic and anisotropic mixeddispersion. This model is given by the equation $i\partial _{t}u+\epsilon\Delta u+\delta A u+\lambda|u|^\alpha u=0,$ $x\in \mathbb{R}^{n},$ $t\in\mathbb{R},$ where $A$ represents either the operator $\Delta^2$ (isotropicdispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (anisotropicdispersion), and $\alpha, \epsilon, \lambda$ are given real parameters. Weobtain local and global well-posedness results in spaces of initial data withlow regularity, such as weak-$L^p$ spaces. Our analysis also includes thebiharmonic and anisotropic biharmonic equation $(\epsilon=0)$ for which, theexistence of self-similar solutions is obtained as consequence of his scalinginvariance. In a second part, we investigate the vanishing second orderdispersion limit in the framework of weak-$L^p$ spaces. We also analyze theconvergence of the solutions for the nonlinear fourth-order Schr\"odingerequation $i\partial _{t}u+\epsilon \Delta u+\delta \Delta^2 u+\lambda|u|^\alphau=0$, as $\epsilon$ goes to zero, in $H^2$-norm, to the solutions of thecorresponding biharmonic equation $i\partial _{t}u+\delta \Delta^2u+\lambda|u|^\alpha u=0$.

On the Schr\"odinger equations with isotropic and anisotropic fourth-order dispersion