This work deals with Morse index estimates for a solution $ u\in H_1^p(M)$ of the quasilinear elliptic equation $ -\textrm{div}_g \big ( \big (\alpha +|\nabla u|_g^2 \big )^{(p-2)/2}\nabla u \big )=h(x,u) $, where $(M,g)$ is a compact, Riemannian manifold, $0 < \alpha$, $2 \leq p < n$. The nonlinear right-hand side $h(x,s)$ is allowed to be subcritical or critical.

In this paper, we consider the exact boundary controllability for cubic focusing semilinear wave equations in $1\le n\le 3$ space dimensions. When the initial data and the final data are in the so-called potential well, we find that the sufficient condition for the global existence is also sufficient to ensure the exact boundary controllability of the problem. Moreover, in one space dimension, the control time can be that of the linear wave equation.

In this paper, we discuss the Cauchy problem for Navier-Stokes equations in homogeneous weak Herz spaces $W{\dot{K}}^\alpha_{p,q}({\mathbb{R^{\textit{n}}}})$. More precisely, we construct the solution in the class $L^\infty(0,T; W{\dot{K}}^\alpha_{p,q})$ with the initial data in $W{\dot{K}}^\alpha_{p,q}$. Further, we consider the blow-up phenomena of time-local solutions and the uniqueness of global solutions with large initial data in $W{\dot{K}}^\alpha_{p,q}$. Also, we give several embeddings of weak Herz spaces into homogeneous Besov spaces $B^{-\alpha}_{p,\infty}({\mathbb{R^{\textit{n}}}})\ (\alpha >0),$ or $bmo^{-1}({\mathbb{R^{\textit{n}}}})$.

We prove that the complex-valued modified Benjamin-Ono (mBO) equation is analytically locally well posed if the initial data $\phi$ belongs to $H^s$ for $s\geq 1/2$ with $ \| {\phi} \| _{L^2}$ sufficiently small, without performing a gauge transformation. The key ingredient is that the logarithmic divergence in the high-low frequency interaction can be overcome by a combination of $X^{s,b}$ structure and smoothing effect structure. We also prove that the real-valued $H^\infty$ solutions to the mBO equation satisfy a priori local-in-time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s>1/4$.

We consider problems for quite general second-order abstract elliptic and corresponding parabolic equations on the interval $[0,1]$ and the rectangle $[0,T]\times [0,1]$, respectively. $R$-boundedness estimates of solutions of abstract boundary-value problems for elliptic equations with a parameter are established, in contrast to standard norm-bounded estimates. The results are applied to obtain $L^p$-maximal regularity for corresponding parabolic systems. In applications, the coefficient $A(x)$ of the solution $u$ can be $2m$-order elliptic operators with suitable boundary conditions, while the coefficient $B(x)$ of the first-order derivative of the solution $D_xu$ can be interpreted as an $m$-order differential operator. The corresponding applications to PDEs are presented.