We shall consider weak solutions of nonlinear elliptic boundary value problems and initial-boundary value problems for semilinear and nonlinear parabolic differential equations with certain nonlocal terms.  We shall prove theorems on the number of solutions and find multiple solutions. These statements are based on arguments for fixed points of some real functions and operators, respectively, and existence-uniqueness theorems on partial differential equations (without functional terms).

The logarithmic norm was introduced in 1958 for matrices, and for the purpose of estimating growth rates in initial value problems. Since then, the concept has been extended to nonlinear maps, differential operators and function spaces. There are applications in operator equations in general, including evolution equations as well as boundary value problems. The logarithmic norm is the extremal value of a quadratic form.

In this talk we outline how logarithmic norms of differential operators can be computed, and how they are related to variational calculus and ellipticity. Thus, while one typically seeks the minimizing function in a variational problem, the logarithmic norm is the corresponding extremal value of the functional associated with a particular symmetrized differential operator. There are also connections to eigenvalue problems for selfadjoint and non-selfadjoint operators. This will also be illustrated with an application to classical singular problems, such as the Bessel equation, as well as to the biharmonic operator. 

In this talk we investigate some special qualitative properties of parabolic problems. At the beginning of the talk we review the remarkable qualitative properties of these problems. Then we will turn to two special properties: the first property says that the number of the so-called LL-level points, or specially the number of the zeros, of the solutions must be non-increasing in time. The second property requires a similar property for the number of the local maximizers and minimizers. We show that linear equations and some special nonlinear equations fulfill the above properties in the continuous case. We use the maximum-minimum principles in the proof. Then we generate the numerical solution with the implicit Euler finite difference method and show that the obtained numerical solution satisfies the discrete versions of the above properties without any requirements on the mesh parameters. We show also some numerical test results.