Slow travelling waves on a finite interval for Burgers’-type equations

Abstract

In this paper we study for small positive ε the slow motion of the solution for evolution equations of Burgers’ type with small diffusion, () u t=εu xx+F(u) x, u(x,0)=u 0(x), u(±1,t)=u ± on the bounded spatial domain [−1,1]; F is a smooth nonpositive function having only a finite number of zeros (at least two) between u − and u +, all of finite order. The initial and boundary value problem (★) has a unique asymptotically stable equilibrium solution that attracts all solutions starting with continuous initial data u 0. On an interval [−1−c 0ε,1+c 0ε], c 0>0 the differential equation has slow speed travelling wave solutions generated by profiles that satisfy the boundary conditions of (★). During a long but finite time interval, such travelling waves suitably describe the slow long-term behaviour of the solution of evolution problem (★). Their speed characterizes the local velocity of the slow motion with algebraic precision (w.r.t. ε) in general, and with exponential precision, if F has only two zeros of first order located at u + and u −. A solution that starts near a travelling wave, moves in a small neighbourhood of such a travelling wave during a long time interval (0, T). If F has zeros of order higher than 1, the equilibrium and the travelling wave are multi-shock solutions of (★). This situation differs strongly from the case where F has only a first-order zero at both u ±, studied by the authors in a previous paper. In this paper we consider multi-shock solutions of (★). Moreover, we improve some results of the previous paper, allowing a larger ball of initial data.