# zbMATH — the first resource for mathematics

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