Ricerc@Sapienza

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The main idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form.
The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with the diffusion equation.

We will present some existence and uniqueness theorems for two different two-scale problems ([1]). In this framework, a two-scale problem is a system of PDEs involving two unknowns (u; u_1), the first one just depending on a set of space variables denoted by x (usually called the macroscopic or slow variables) and on the time t, the second one depending on a second set of spatial variables y, beside the old ones (i.e. u_1 depends on (x; y; t)). The second set of space variables y are usually called microscopic or fast variables.

In this paper we prove existence and uniqueness for a parabolic and a parabolic/elliptic two-scale system of partial differential equations typically arising in homogenization theory or in macroscale-microscale applied models.

In this paper we derive a model for heat diffusion in a composite medium in which the different components are separated by thermally active interfaces. The previous result is obtained via a concentrated capacity procedure and leads to a non-stantard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For such a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady state is proved.