de Groen, P.P.N.; Karadzhov, G.E.

Exponentially slow traveling waves on a finite interval for Burger's type equation

Electron. J. Differ. Equ. 1998, Paper No. 30, 38 p. (1998)

Summary

Summary: In this paper we study for small positive $$u_t=\epsilon u_{xx}+f(u) u_x, u(x,0)=u_0(x), u(\pm 1,t)=\pm 1$ (\star)$ on the bounded spatial domain [-1,1]; $f$ is a smooth function satisfying $\int_{-1}^{1}f(t)dt=0$. The initial and boundary value problem ($\star$) has a unique asymptotically stable equilibrium solution that attracts all solutions starting with continuous initial data $u_0$. On the infinite spatial domain ${\Bbb R}$ the differential equation has slow speed traveling wave solutions generated by profiles that satisfy the boundary conditions of ($\star$). As long as its zero stays inside the interval [-1,1], such a traveling wave suitably describes the slow long term behaviour of the solution of ($\star$) and its speed characterizes the local velocity of the slow motion with exponential precision. A solution that starts near a traveling wave moves in a small neighborhood of the traveling wave with exponentially slow velocity (measured as the speed of the unique zero) during an exponentially long time interval (0,T). In this paper we give a unified treatment of the problem, using both Hilbert space and maximum principle methods, and we give rigorous proofs of convergence of the solution and of the asymptotic estimate of the velocity.$$

Mathematics Subject Classification

35B25, 35K60

Keywords/Phrases

slow motion, singular perturbations, exponential precision, Burgers equation

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