Summary: This work is the continuation of our previous paper [Discrete Contin. Dyn. Syst. 21, No. 1, 41–67 (2008; Zbl 1173.35542)]. There, we dealt with the reaction-diffusion equation
\[ \partial_t u=\Delta u+f(x-cte,u),\qquad t>0,\quad x\in\mathbb R^N, \]
where \(e\in S^{N-1}\) and \(c>0\) are given and \(f(x,s)\) satisfies some usual assumptions in population dynamics, together with \(f_s(x,0)<0\) for \(|x|\) large. In [loc. cit.], we proved that existence and uniqueness of travelling wave solutions of the type \(u(x,t)=U(x-cte)\) and the large time behaviour of solutions with arbitrary nonnegative bounded initial datum depend on the sign of the generalized principal in \(\mathbb R^N\) of an associated linear operator. Here, we establish analogous results for the Neumann problem in domains which are asymptotically cylindrical, as well as for the problem in the whole space with \(f\) periodic in some space variables, orthogonal to the direction of the shift \(e\).
The \(L^1\) convergence of solution \(u(t,x)\) as \(t\to\infty\) is established next. In this paper, we also show that a bifurcation from the zero solution takes place as the principal crosses 0. We are able to describe the shape of solutions close to extinction thus answering a question raised by M. Mimura. These two results are new even in the framework considered in [loc. cit.].
Another type of problem is obtained by adding to the previous one a term \(g(x-c'te,u)\) periodic in \(x\) in the direction \(e\). Such a model arises when considering environmental change on two different scales. Lastly, we also solve the case of an equation
\[ \partial_t u=\Delta u+f(t,x-cte,u), \]
when \(f(t,x,s)\) is periodic in \(t\). This for instance represents the seasonal dependence of \(f\). In both cases, we obtain a necessary and sufficient condition for the existence, uniqueness and stability of pulsating travelling waves, which are solutions with a profile which is periodic in time.