Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace

Figure 1.  Propagation of a front supported by a plane/line with a piecewise constant growth rate. A. The general geometry of the model. The $ z $-axis can be absent. In this case, the model is considered on a plane and the growth happens on a line. B. Piecewise constant approximation of a sigmoidal growth rate $ f $ bound by the maximal growth rate $ a $ at infinity

Figure 2.  The front $ w $ of the travelling solution $ u(x,y,t) = w(x- v t,y) $ of (1.2) given by (3.12) for $ a = 2\pi $, $ k = D = 1 $, and $ u_c = 0.3 $ ($ v \approx 6.47 $). A. A contour plot of $ w $ in the $ xy $-plane ($ y \geqslant 0 $). Note that $ w(x,y) = w(x,-y) $. B. The value of $ w $ on the $ x $-axis. Here the production is active on the $ x $-axis with $ x \in (-\infty,0] $ and inactive everywhere else

Figure 3.  Sketches of travelling solution profiles $ w $ of the classical piecewise linear equation in the original units on the phase plane $ ( w, \partial_x w) $. The trajectory that corresponds to a front is depicted as the thick line. A. The regular travelling front that connects the trivial and the nontrivial steady states for the case $ v > 0 $. It is the standard heteroclinic trajectory on the phase plane. Only separatrices of the saddles are shown. See [19] for the details. B. The travelling solution with $ k = 0 $ that connects the trivial steady state and infinity with bound derivative at infinity. The case is degenerated and the whole $ w $-axis consists of steady states for $ w < u_c $. A generic smooth case would correspond to a saddle-node bifurcation at the origin in this situation. C. The homoclinic stationary solution that connects the trivial steady state with itself in the case when the monotone front travels with $ v > 0 $