We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

A biophysical model describing long-range cell-to-cell communication by a diffusiblesignal mediated by autocrine loops in developing epithelia in the presence of amorphogenetic pre-pattern is introduced. Under a number of approximations, the modelreduces to a particular kind of bistable reaction-diffusion equation with strongheterogeneity. In the case of the heterogeneity in the form of a long strip a detailedanalysis of signal propagation is possible, using a variational approach. It is shown thatunder a number of assumptions which can be easily verified for particular sets of modelparameters, the equation admits a unique (up to translations) variational traveling wavesolution. A global bifurcation structure of these solutions is investigated in a number ofparticular cases. It is demonstrated that the considered setting may provide a robustdevelopmental regulatory mechanism for delivering chemical signals across large distancesin developing epithelia.

We prove existence of minimizing movements for thedislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscositysolutions of the corresponding level-set equation. We also prove theconsistency of this approach, by showing that any minimizing movementcoincides with the smooth evolution as long as the latter exists. Inrelation with this, we finally prove short time existence and uniqueness of a smoothfront evolving according to our law, provided the initial shape issmooth enough.

We investigate the approximationof the evolution of compact hypersurfaces of $\mathbb{R}^N$ depending, not only on terms of curvature of the surface, but alsoon non local terms such as the measure of the set enclosedby the surface.