Let $Q_T$ be a cylinder in ${\mathbb{R}}^{n+1}$ and $x=\left(x^{\prime },t\right)\in {\mathbb{R}}^n\times \mathbb{R}$. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator $$\left\{\begin{array}{cc}{u}_t-\stackrel{n}{\sum _{i,j=1}}{a}^{ij}\left(x\right)D_{ij}u=f\left(x\right)\hfill & \text{q.o. in }{Q}_T,\hfill \\ u\left(x\right)=0\hfill & \text{su }\partial Q_T,\hfill \end{array}\right.$$ in the Morrey spaces $W_{p,\lambda }^{2,1}\left(Q_T\right)$, $p\in \left(1,\mathrm{\infty }\right)$, $\lambda \in \left(0,n+2\right)$, supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.

Let $L(z,{\partial }_z)=({\partial }_{z_0}{)}^k-A(z,{\partial }_z)$ be a linear partial differential operator with holomorphic coefficients, where$$A(z,{\partial }_z)=\sum _{j=0}^{k-1}{A}_j(z,{\partial }_{z^{\text{'}}}\left)\right({\partial }_{z_0}{)}^j,\mathrm{ord}.A(z,{\partial }_z)=m\>k$$and$$z=(z_0,z^{\text{'}})\in C^{n+1}.$$We consider Cauchy problem with holomorphic data$$L(z,{\partial }_z)u\left(z\right)=f\left(z\right),\left({\partial }_{z_0}{)}^{i}u\right(0,z^{\text{'}})={\widehat{u}}_i(z^{\text{'}}\left)\right(0\le i\le k-1).$$We can easily get a formal solution $\widehat{u}\left(z\right)={\sum }_{n=0}^{\infty }{\widehat{u}}_n(z^{\text{'}}\left)\right(z_0{)}^n/n!$, bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L(z,{\partial }_z)$, there is a function $u_S\left(z\right)$ holomorphic except on $\{z_0=0\}$ such that $L(z,{\partial }_z)u_S\left(z\right)=f\left(z\right)$ and $u_S\left(z\right)\sim \widehat{u}\left(z\right)$ as $z_0\to 0$ in $S$.

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$u_t=-A\left(t\right)u_{x^4}+B\left(t\right)u_{x^2}+g{\left(u\right)}_{x^2}+f{\left(u\right)}_x+h{\left(u_x\right)}_x+G\left(u\right)$$ with the initial boundary value conditions $$u(-\ell ,t)=u(\ell ,t)=0,u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ or with the initial boundary value conditions $$u_x(-\ell ,t)=u_x(\ell ,t)=0,u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

As a model for elliptic boundary value problems, we consider the Dirichlet problem for an elliptic operator. Solutions have singular expansions near the conical points of the domain. We give formulas for the coefficients in these expansions.

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac-like inequalities. As part of our methods, we give a different approach to a priori estimates of Foiaş, Guillopé and Temam.

In the context of suitable weak solutions to the Navier-Stokes equations we present local conditions of Prodi-Serrin’s type on velocity $𝐯$ and pressure $p$ under which $({𝐱}_0,t_0)\in \Omega \times (0,T)$ is a regular point of $𝐯$. The conditions are imposed exclusively on the outside of a sufficiently narrow space-time paraboloid with the vertex $({𝐱}_0,t_0)$ and the axis parallel with the $t$-axis.

We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.