Ricerc@Sapienza

In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely
\begin{equation*}
\begin{cases}
\dys u\geq 0 & \mbox{in } \Omega,\\
\displaystyle - div \,A(x) D u = F(x,u)& \mbox{in} \; \Omega,\\
u = 0 & \mbox{on} \; \partial \Omega,\\
\end{cases}
\end{equation*}
with $F(x,s)$ a Carath\'eodory function such that
$$
0\leq F(x,s)\leq \frac{h(x)}{\Gamma(s)}\,\,\mbox{ a.e. } x\in\Omega,\, \forall s>0,
$$

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.