Exponentially slow traveling waves on a finite interval for Burgers’ type equation

In this paper we study for small positive the slow motion of the solution for evolution equations of Burgers’ type with small diffusion, ut = uxx + f(u)ux , u(x, 0) = u0(x), u(±1, t) = ±1, (?) on the bounded spatial domain [−1, 1]; f is a smooth function satisfying f(1) > 0, f(−1) < 0 and ∫ 1 −1 f(t)dt = 0. The initial and boundary value problem (?) has a unique asymptotically stable equilibrium solution that attracts all solutions starting with continuous initial data u0. On the infinite spatial domain R the differential equation has slow speed traveling wave solutions generated by profiles that satisfy the boundary conditions of (?). As long as its zero stays inside the interval [−1, 1], such a traveling wave suitably describes the slow long term behaviour of the solution of (?) and its speed characterizes the local velocity of the slow motion with exponential precision. A solution that starts near a traveling wave moves in a small neighborhood of the traveling wave with exponentially slow velocity (measured as the speed of the unique zero) during an exponentially long time interval (0, T ). In this paper we give a unified treatment of the problem, using both Hilbert space and maximum principle methods, and we give rigorous proofs of convergence of the solution and of the asymptotic estimate of the velocity.