Existence of fast traveling waves for some parabolic equations: A dynamical systems approach

We study semilinear elliptic equations Δu + cu_x=f(u,∇ u) and Δ²u + cu_x,= f(u, Δu, ∇²u) in infinite cylinders (x,-y)∈R×Ω⊂Rⁿ⁺¹ using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limit c→∞, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.