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| In [{\it J. Math. Anal. Appl.} 411 (2014), 853-872] we study Cauchy problem in $R^N$ for the Korteweg-de Vries-Burgers system
\begin{equation}\label{1}
\begin{split}
&u_t + \sum_{i=1}^N \frac{\partial}{\partial x_i} \nabla \Phi(u) + \sum_{j=1}^N \frac{\partial}{\partial x_j} \sum_{i=1}^N \frac{\partial^{2p}}{\partial x_i^{2p}} u = \alpha \Delta u + g(x,u), \\
&u(0,x) = u_0(x), \ t>0, \ x \in R^N,
\end{split}
\end{equation}
where $\alpha > 0$, $2p > N \geq 1$ and $\Phi$ is a scalar function of the vector $u(t,x) = (u_1(t,x),...,u_m(t,x))$; $\nabla$ denotes gradient with respect to $u$.
Parabolic regularization technique is used to prove global in time solvability of \eqref{1}; we study first its {\it parabolic regularization}:
\begin{equation}\label{2}
\begin{split}
u^\epsilon_t + \sum_{i=1}^N \frac{\partial}{\partial x_i} \nabla \Phi(u^\epsilon) &+ \sum_{j=1}^N \frac{\partial}{\partial x_j} \sum_{i=1}^N \frac{\partial^{2p}}{\partial x_i^{2p}} u^\epsilon = \alpha \Delta u^\epsilon + \epsilon (-1)^{p} (\Delta)^{p+1} u^\epsilon \\
&+ g(x, u^\epsilon), \; u^\epsilon(0,x) = u_0(x), \ \ t>0, \ x \in R^N,
\end{split}
\end{equation}
where $\epsilon > 0$ is the {\it viscosity coefficient}, that will later tend to $0^+$. Certain estimates for solutions $u^\epsilon$ to \eqref{2} are extended next to the limit problem \eqref{1}. The regularization effect of the Laplacian term is observed for the {\it viscous solutions} of \eqref{1} constructed in the paper. Asymptotic behavior, as $t \to \infty$, is finally discussed in the language of the global attractors.
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\end{itemize} |
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