The invariance of a closed and convex set under a semigroup $S\left(t\right)$ associated with a nonlinear form is investigated. Other properties of increase and domination of the semigroup $S\left(t\right)$ are also derived. Examples are also given to demonstrate the power of the theoretical results.

The existence, uniqueness, regularity and asymptotic behavior of global solutions of semilinear heat equations in Hilbert spaces are studied by developing new results in the theory of one-parameter strongly continuous semigroups of bounded linear operators. Applications to special semilinear heat equations in $L^2\left({ℝ}^n\right)$ governed by pseudo-differential operators are given.

Morse theory for isolated critical points at infinity is used for the existence of multiple critical points for an asymptotically quadratic functional. Applications are also given for the existence of multiple nontrivial periodic solutions of asymptotically Hamiltonian systems.

In this work we study a deformation of the minimal interface of two fluids in a vertical tube under the presence of gravitation. We show that a symmetry of the base of tube let us to apply a method developed earlier by the first author and based on the Crandall-Rabinowitz bifurcation theorem. Using the natural symmetry of the corresponding variational problem defined by a symmetry of region and restricting the functional to spaces of invariant functions we show the existence of bifurcation, and describe its local picture, for interfaces parametrized by the square and disc.

The embedding functions of an intermediate space $A$ into a Banach couple $\left(A_0,A_1\right)$ are defined as its embedding constants into the couples $\left(\frac{1}{\alpha }{A}_0,\frac{1}{\beta }{A}_1\right)$, $\forall \alpha ,\beta >0$. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to obtain new interpolation theorems.

The exponential and asymptotic stability are studied for certain coupled systems involving unbounded linear operators and linear infinitesimal semigroup generators. Examples demonstrating the theory are also given from the field of partial differential equations.