| Abstract: |
| We study the equation $ u_t = u_{xx} + \beta(x; L) g(u) - a u$, $x \in \mathbb{R}$, where the Borel-measure coefficient $\beta(x;L)$ represents the local density of calcium release sites with $L$ being the separation between release sites and $g(u)$ satisfies a weak Allee effect condition. The model describes intracellular waves in a continuum excitable media with discrete release sites.
In this paper, we first establish the well-posedness of the Cauchy problem for general Borel-measure $\beta(x;L)$ within a rigorous mathematical framework, using an approximation procedure. Next, we investigate the propagation dynamics of the model where $\beta(x;L)$ is an $L^1$ perturbation of the periodic Dirac sources $\sum_{n=-\infty}^{+\infty} \delta(x - nL)$ and $g(u) = u^p (1 - u)$ with the integer $p \ge 2$. We focus on the bistable regime of the model, which is determined by the decay strength parameter $a$ and the separation distance $L$ between release sites, in which the model admits exactly three $L$-periodic steady states. We establish the existence of pulsating waves that spatially connect stable steady states by a dynamical systems approach. Furthermore, we prove that any such traveling wave with nonzero speed is both unique and globally exponentially stable via the squeezing method. |
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