| Contents |
| This talk is concerned with the analysis of a spatially-extended model describing
mutual phosphorylation of cytosolic kinases and membrane receptors in immune cells.
A prototype for the systems considered has the form
\begin{eqnarray*}
\frac{\partial K }{\partial t} & = & d \Delta K - {\displaystyle A \frac{H K}{H+K}},
\quad \quad \qquad \quad ~\mbox{in}\;\; {\Omega} \times (0,\infty),\\
\frac{\partial R }{\partial t} & = & (K^w + c_0)(P-R) - bR, \quad \quad \;\; \mbox{in} \;\; \partial \Omega \times (0,\infty),
\end{eqnarray*}
with appropriate initial conditions and a nonlinear Robin-type boundary condition
$$
d \thinspace \nabla K \cdot n = a R (1-K) \quad \textrm{in}\;\; \partial \Omega \times (0,\infty),
$$
that couples the variables $K$ and $R$ on $\partial \Omega$. Here $\Omega \subset
\mathbb{R}^n$ is a bounded domain
with smooth boundary, $K$ represents a concentration of kinase molecules and
$R$ is a concentration of non-diffusing receptors that live on the cell membrane $\partial \Omega$.
The given function $a: \partial \Omega \to [0, \infty)$ is typically zero on part of $\partial \Omega$, giving a zero
Neumann condition, and positive on the remainder of $\partial \Omega$, with the other model
parameters positive constants. We discuss the existence and
stability of stationary spherically-symmetric solutions when $\Omega$ is a spherical
shell via an auxiliary problem in which the Robin boundary condition is replaced
by a uniform Dirichlet boundary condition. |
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