We study the blow-up behaviour of a class of solutions of the incompressible Navier-Stokes equations in $\mathbf{R}^2$ which are described by a parabolic free-boundary problem with a nonlocal term. We prove that the solutions blow up in finite time globally in space and calculate the corresponding rates and profiles. The asymptotic profile and the free boundary are obtained as a nonviscous limit, solution of a nonlocal first-order equation of the Hamilton-Jacobi type. This means that the final highly nonstationary stage of such flows is essentially a convection-type singularity. The viscosity term, though asymptotically negligible, provides the eventual regularity of the limit profiles. Other types of unstable blow-up and global patterns with a non-generic behaviour are discussed.

We classify all complete noncompact embedded convex hypersurfaces in $\mathbf{R}^{n+1}$ which move homothetically under flow by some negative power of their Gauss curvature.

We investigate the existence of solutions for a large class of semilinear elliptic problems with nonlinear Neumann boundary conditions. We study separately the one-dimensional case and show how to count the solutions. In higher dimension we give existence results under some compactness assumption.

The dual variational method is applied to establish results on the multiplicity of solutions for nonlocal semilinear Tricomi problems of the type introduced in [9]. For classes of odd sublinear nonlinearities, the existence of infinitely many solutions as preimages of critical points of the dual functional at minimax values is established with the aid of the Krasnoselski genus. For classes of asymptotically linear nonlinearities the existence of at least one nontrivial solution is captured as the preimage of a mountain pass or a linking critical point of the dual functional. In all the cases, information on the eigenvalues of the inverse of the linear operator is needed and demonstrated. An analogous problem involving a second order ordinary differential operator and with homogeneous Cauchy conditions on one endpoint of an interval is resolved in the same way and the role of the nonlocal effect is studied in terms of the question of the uniqueness of the trivial solution.

Let $X$ be a real separable reflexive Banach space with dual space $X^*.$ Assume that the operator $A:X\supset\mathcal{D}(A)\to X^*$ is such that $L\subset \mathcal{D}(A)$ and $L=X,$ where $L$ is a subspace of $X.$ It is shown that it is possible to define a topological degree for such operators $A$ that satisfy, mainly, Condition $(S_+)_{0,L}.$ It is also shown that a topological degree can be defined for operators of the type $M+A,$ where $M+A: X\supset\mathcal{D}(M+A)\to X^*, L\subset\mathcal{D}(M+A),$ and $ L=X.$ Here, $X$ is not necessarily separable and $M$ satisfies a variant of the maximal monotonicity condition (with respect to the space $L$) as well as an approximation condition. The operator $A$ satisfies, mainly, analogues of the quasi-boundedness condition and the $(S_+)$ condition (with respect to the operator $M).$ Properties of these degrees are studied and applications are given for nonlinear Dirichlet elliptic problems of the type $$ \sum_{i=1}^n\frac{\partial}{\partial x_i}\left\{\rho^2(u)\frac{\partial u}{\partial x_i}+ a_i\left(x,u,\frac{\partial u}{\partial x}\right)\right\}=\sum_{i=1}^n\frac{\partial}{\partial x_i}f_i(x), $$ as well as Cauchy-Dirichlet parabolic problems of the type $$ \frac{\partial u}{\partial t}-\sum_{i=1}^n\frac{\partial}{\partial x_i}a_i\left(x,t,u,\frac{\partial u}{\partial x}\right) +\rho(x,t,u)=\sum_{i=1}^n\frac{\partial}{\partial x_i}f_i(x,t). $$