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.
A) The common feature of the proposed models is the periodic microstructure, which calls for a homogenization procedure, in order to provide a simpler macroscopic description. The technical difficulties are due to the intricate microscopic geometry, further complicated by the presence of active interfaces and possibly singular sources.
As pointed out in the previous sections, most of these models are proposed in the literature. However, in many cases they are taken into account for applications with little or no mathematical understanding. We think that a deeper analysis of these models, in which different transmission conditions across the interfaces are considered, could provide a mathematical motivation
of the experimental formula, possibly introducing new and hopefully more efficient macroscopic models.
Clearly, to achieve this goal, the first step is the study of existence and uniqueness of the systems of PDEs that describe the physical models under investigation. They vary from elliptic, parabolic, pseudo-parabolic or abstract parabolic equations to equations involving also memory or tangential operators, due to the presence of active interfaces. The situation can be further complicated by the presence, in some cases, of singular lower order terms. The variety of different situations among these generally non-standard problems needs a further deepening, calling for a delicate mathematical investigation.
B) We aim at a characterization of the macroscopic behavior, from the point of view of diffusion, of media exhibiting a Fokker-Planck microscopic behavior (or a mixed Fokker-Planck and Fick behavior). That is we want to obtain a diffusion equation, whose type is to be ascertained, where the microscopic properties of the medium appear in an averaged way in the coefficients of the equation itself.
C1) The sought-after asymptotic expansion should give not only the stabilization rate (to zero) of the solution, but the limiting shape of the solution itself, when rescaled exactly by the power of time given by that rate. A connected result is a uniqueness theorem under optimal or at least good enough assumptions.
C2) Blow up in graphs has been already studied, mostly in the case of finite graphs and semilinear equations. We look at the setting of nonlinear diffusion equations in infinite graphs, aiming at results connecting sharply the nonlinearities of the equation and the properties of the graph (e.g., the degree function) to the onset of blow up in a finite time.
D) Recently, new viscoelastic materials, such as viscoelastic gels, have been
described by convolution integral with singular kernels (fractional and hypergeometric). When the natural aging [01s] of the viscoelastic material is taken into account, time dependent kernels are used. The behaviour of some new material (ferro-gels and magneto rheological elastomers) can be determined coupling viscoelastic and magnetic effects. The existence of weak solutions and, possibly, their uniqueness is addressed to, both taking into account or not magnetisation effects.
E) The innovative aspect of the present project is represented by the combination of different algebraic properties, such as invariances under Baecklund transformations and/or existence of an admitted recursion operator, are shown to induce the construction of new soliton solutions. The recursion operator admitted by all such equations as well as further non-Abelian equations allows to investigate the multi-soliton manifold which is well known in the Abelian case, but never studied in the non-Abelian setting. The connected Hamiltonian, and, if it is admitted, bi-Hamiltonian structure is also not yet known: we aim to construct it.
F) We will expand the existing model introducing new features, in a multiscale framework. We not only will consider 3D domains with generic boundaries, but also will include in the model the effect of the pH and the temperature of the tissue on the stem cell survival and the proliferation. Moreover, other important processes will be considered, like dynamics of more than one nutrient (oxygen and glucose), angiogenesis (which is the physiological mechanism through which new blood vessels form from pre-existing vessels and which is necessary for the long term nutrient supply and waste chemicals, like carbon dioxide, and necrotic debris elimination), intracellular signal transduction for cell fate decision at the end of the process (proliferation in the early phase and eventually differentiation in cardiomyocytes) supported by miRNA action. Furthermore, the mechanism of chemotaxis will be deepened.
The theoretical and numerical study of models of cardiac tissue regeneration will bring to a better understanding of the relevant properties of the underlying structure and the prediction of their evolution. The results are expected to have a positive impact in the area of Regenerative Medicine, in particular with the aim of improving the stem cell therapy protocols.