Ricerc@Sapienza

Physical properties of composite materials and biological tissues, and their interactions are relevant topics in applications. Some examples are the study of electrical conduction in the heart; reconstruction of infarcted cardiac tissues by means of stem cell implantation; encapsulation of electronic devices, where the increasing miniaturization poses a big challenge in attaining an efficient heat dissipation; diffusion in a spatially inhomogeneous medium, being a not trivial extension of the case with constant diffusivity.
While the previous argument deals essentially with the effects of local inhomogeneities, we are interested also in global effects of locally smooth coefficients depending on space.

More precisely, the problems we are interested in are essentially the following:

i) well-posedness and homogenization of multiphase systems described by PDEs coupled through suitable interface conditions and, eventually, involving singular sources;
ii) homogenization of Fokker-Planck equations, when inhomogeneities are finely sparsed in the medium;
iii) existence or nonexistence of global in time solutions in problems with nonlinear sources and asymptotic behavior for large times of solutions to parabolic equations containing terms sharply depending on space at infinity;
iv) study of nonlinear evolution operator equations (Banach spaces): determination of admitted special solutions both finite and infinite dimensional ones (which are relevant, for instance, in quantum mechanics);
v) investigation on memory effects in evolution phenomena in heat conductors, viscoelastic and magneto-viscoelastic solids;
vi) stem cell transplantation into the damaged myocardium as a potential strategy to restore the heart function by gradually reducing and repairing the necrotic tissue. Our aim is to further develop a model able to reproduce cell implantation, migration and proliferation, in order to include angiogenesis and the role of miRNAs in stem cell differentiation.