## Asymptotic expansion for the generalized Benjamin-Bona-Mahony-Burgers equation.(English)Zbl 1021.35072

This paper is devoted to the asymptotic expansion as $$t\to\infty$$ of the solutions to the $$n$$-dimensional Benjamin-Bona-Makowy-Burgers equation: $\begin{cases} u_t-\Delta u_t-\Delta u+(b\cdot\nabla u)=\nabla\cdot F(u),\quad x\in\mathbb{R}^n,\;t>0\\ u(x,0)=u_0(x),\end{cases}\tag{1}$ where $$b\in\mathbb{R}^n$$ and $$F\in C^1(\mathbb{R},\mathbb{R}^n)$$ are a fixed vector and a vector field, respectively, and $$\nabla$$ stands for the divergence operator. The authors compute also the second term in the asymptotic expansion with quadratic nonlinear term.

### MSC:

 35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators 35C20 Asymptotic expansions of solutions to PDEs 35L05 Wave equation 35B40 Asymptotic behavior of solutions to PDEs