Traveling Waves with Multiple and Nonconvex Fronts for a Bistable Semilinear Parabolic Equation

We construct new examples of traveling wave solutions to the bistable and balanced semilinear parabolic equation in \input amssym ${\Bbb R}^N+1$ , $N\geq 2$ . Our first example is that of a traveling wave solution with two non planar fronts that move with the same speed. Our second example is a traveling wave solution with a nonconvex moving front. To our knowledge no existence results of traveling fronts with these type of geometric characteristics have been previously known. Our approach explores a connection between solutions of the semilinear parabolic PDE and eternal solutions to the mean curvature flow in \input amssym ${\Bbb R}^N+1$ .