Geometry of heteroclinic cascades in scalar parabolic differential equations (Q5957841)

The author studies the asymptotic behavior of solutions to the semilinear parabolic equation \(u_t = u_{xx} + f(x,u,u_x)\) considered in the interval \((0,1)\) with the homogeneous Neumann boundary condition (\(f\) being a \(C^2\) function). For two equilibria \(v\), \(w\) with a heteroclinic connection, it is shown that the zero-number \(z(v-w)\) determines which strong-unstable manifolds of \(v\) and strong-stable manifolds of \(w\) intersect each other. Also, if \(z(v-w)=h\), it is proved that in the (transversal) intersection of \(W^u_{h+1}(v)\) and \(W^s_h(w)\) there is a unique connecting orbit.