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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:132

Article

de Groen, PPN ; Karadzhov, GE

关键词: slow motion;    singular perturbations;    Burgers' equation;   

DOI  :  10.1016/S0377-0427(00)00597-5

来源: Elsevier

【 摘 要 】

In this paper we study for small positive E the slow motion of the solution for evolution equations of Burgers' type with small diffusion, u(t) = epsilonu(xx) + F(u)(x), u(x,0) = u0(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 (star) has a unique asymptotically stable equilibrium solution that attracts all solutions starting with continuous initial data u(0). On an interval [ -1 - c(0)epsilon ,1 + c(0)epsilon], c(0) > 0 the differential equation has slow speed travelling wave solutions generated by profiles that satisfy the boundary conditions of(star). During a long but finite time interval, such travelling waves suitably describe the slow long-term behaviour of the solution of evolution problem (star). Their speed characterizes the local velocity of the slow motion with algebraic precision (w.r.t. epsilon) 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 rime interval (0, T). If F has zeros of order higher than 1, the equilibrium and the travelling wave are multi-shock solutions of (star). 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 (star). Moreover, we improve some results of the previous paper, allowing a larger ball of initial data. (C) 2001 Elsevier Science B.V. All rights reserved.

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