Heteroclinic orbits of semilinear parabolic equations
The authors study attractors of semilinear parabolic problems \[ u_t= u_{xx}+ f(x, u, u_x),\;t> 0,\;0< x< 1,\quad u_x(t, 0)= u_x(t, 1)= 0. \] Here \(f\in C^2\) and appropriate dissipativity conditions are assumed so that the global attractor exists. It is known that the attractor consists of equilibria and heteroclinic orbits connecting them. If all the equilibria are hyperbolic, there are only finitely many of them, say \(v_1,\dots, v_k\), and they can be enumerated such that \(v_1< v_2<\cdots< v_k\) at \(x= 0\). The main concern of the paper is to determine which of these equilibria are connected by heteroclinic solutions and which are not. The main result gives a complete answer to this problem and shows how the presence of a connecting orbit can be read off from the shooting permutation \(\pi\) associated with the ODE \(u_{xx}+ f(x, u, u_x)\), \(0< x< 1\). (\(\pi\) is the permutation of \((1, 2,\dots, k)\) defined by the requirement that at \(x= 1\) the equilibria enumerated as above satisfy \(v_{\pi(1)}< v_{\pi(2)}<\cdots< v_{\pi(k)}\). Recently, in another paper, the authors have proved that the shooting permutation determines the flow on the attractor, up to a flow equivalence, that is, two equations with the same permutation have the flows on their corresponding attractors equivalent.
scientific article; zbMATH DE number 859260