We consider the spectral function, $\rho_{\alpha}(\lambda)$, associated with the linear, second-order differential equation $$ y{''} + (\lambda - q) y = 0 \quad \mbox{on} \ [0,\infty) $$ with the initial condition $$ y(0) \cos (\alpha) + y'(0) \sin (\alpha) = 0 \quad \mbox{for some} \ \ \alpha \in [0,\pi). $$ It is shown that if $(1+x)^n q(x) \in L^1[0,\infty)$, then $(n+1)$ derivatives of $\rho_0(\lambda)$ exist and are continuous for $\lambda>0$. Under similar conditions, the derivatives are explicitly computed for $\lambda \ge \Lambda_0$ where $\Lambda_0$ is computable.

We show a smoothing effect of near full-derivative for low-regularity global-in-time solutions of the periodic Korteweg--de Vries (KdV) equation. The smoothing is given by slightly shifting the space-time Fourier support of the nonlinear solution, which we call resonant phase-shift. More precisely, we show that $\mathcal{S}[u](t) - e^{-t{\partial}_x^3} u(0) \in H^{-s+1-}$, where $u(0) \in H^{-s}$ for $0\leq s < 1/2$ where $\mathcal{S}$ is the resonant phase-shift operator described below. We use the normal form method to obtain the result.

Initial--boundary-value problems in a strip with different types of boundary conditions for two-dimensional generalized Zakharov--Kuznetsov equations are considered. Results on global existence and uniqueness of weak solutions in certain weighted spaces are established.

The existence of global and almost-global solutions of heat equations with derivative nonlinear terms is considered for small initial data in the Besov or Triebel--Lizorkin spaces. As an application, the Navier--Stokes equation and the Keller--Segel system of parabolic elliptic type are considered.

By introducing a suitable setting, we study the behavior of finite Morse-index solutions of the equation \begin{equation} -\operatorname{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \mbox{in $\Omega \subset \mathbb{R}^N \; (N \geq 2)$}, \tag{1} \end{equation} where $p>1$, $\theta, l\in\mathbb{R}^1$ with $N+\theta>2$, $l-\theta>-2$, and $\Omega$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^\sigma u(x)$, equation (1) can be rewritten as a nonlinear Schrödinger equation with Hardy potential \begin{equation} -\Delta u=|x|^\alpha |u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \mbox{in $\Omega \subset \mathbb{R}^N \;\; (N \geq 2)$}, \tag{2} \end{equation} where $p>1$, $\alpha \in (-\infty, \infty)$, and $\ell \in (-\infty, (N-2)^2/4)$. We show that under our chosen setting for the finite Morse-index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent $p$ in (1) that divide the behavior of finite Morse-index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2).

We study the Cauchy problem associated with nonlinear Schrödinger-type equations with a nonlocal term in $\mathbb{R}^n$. Existence and uniqueness of local and global solutions are established in spaces which allow singular initial data. Scattering, asymptotic stability, and decay rates are also proved.