# Exponentially Slow Traveling Waves on a Finite Interval for Burgers' Type Equation

 dc.contributor.author de Groen, Pieter P. N. dc.contributor.author Karadzhov, G. E. dc.date.accessioned 2019-03-19T16:35:43Z dc.date.available 2019-03-19T16:35:43Z dc.date.issued 1998-11-20 dc.description.abstract 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 + ƒ(u) ux, u(x, 0) = u0(x), u(±1, t) = ±1, (⋆) on the bounded spatial domain [-1, 1]; ƒ is a smooth function satisfying ƒ(1) > 0, ƒ(-1) < 0 and ∫1-1 ƒ(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. dc.description.department Mathematics dc.format Text dc.format.extent 38 pages dc.format.medium 1 file (.pdf) dc.identifier.citation de Groen, P. P. N. & Karadzhov, G. E. (1998). Exponentially slow traveling waves on a finite interval for Burgers' type equation. Electronic Journal of Differential Equations, 1998(30), pp. 1-38. dc.identifier.issn 1072-6691 dc.identifier.uri https://hdl.handle.net/10877/7931 dc.language.iso en dc.publisher Southwest Texas State University, Department of Mathematics dc.rights Attribution 4.0 International dc.rights.uri https://creativecommons.org/licenses/by/4.0/ dc.source Electronic Journal of Differential Equations, 1998, San Marcos, Texas: Southwest Texas State University and University of North Texas. dc.subject Slow motion dc.subject Singular perturbations dc.subject Exponential precision dc.subject Burgers' equation dc.title Exponentially Slow Traveling Waves on a Finite Interval for Burgers' Type Equation dc.type Article

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