Asymptotic expansion for a dissipative Benjamin–Bona–Mahony equation with periodic coefficients. (English) Zbl 1102.35074

Bisognin, E.; Bisognin, V.; Coimbra Charão, R.; Pazoto, A. F.

Port. Math. (N.S.) 60, No. 4, 473-504 (2003).

Summary: In this work we study the asymptotic behavior of solutions of a dissipative BBM equation in \(\mathbb R^N\) with periodic coefficients \[ \rho(x)u_t-{{\partial}\over{\partial x_j}} \Big(a_{jk}(x){{\partial^2 u}\over{\partial x_k\partial t}}\Big)- \nu {{\partial}\over{\partial x_j}} \Big(a_{jk}(x){{\partial u}\over{\partial x_k}}\Big)=0. \tag{\(*\)} \] Here \(\nu\) is a positive constant and Einstein’s convention is used. Moreover, the coefficient \(\rho(x)\) is strictly positive and periodic, and the coefficient matrix \((a_{jk}(x))\) is uniformly positive definite and periodic.
We use Bloch waves decomposition to obtain a complete expansion, as \(t\to+\infty\), and conclude that the solutions behave, in a first approximation, as the homogenized heat kernel.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35C20	Asymptotic expansions of solutions to PDEs