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| Assuming fourth-order semilinear parabolic equations of the Cahn--Hilliard-type
$$ u_t + \Delta^2 u = \gamma u \pm \Delta (|u|^{p-1}u) \quad \hbox{in} \quad \Omega \times \mathbb{R}_+,$$
we will discuss several aspects regarding existence and multiplicity results of classic steady states when $\Omega\subset \mathbb{R}^N$ is a bounded domain under Navier boundary conditions and, also, considering the whole $\mathbb{R}^N$ and in a class of functions properly decaying at infinity,
$$\lim_{|x|\to \infty} u(x) = 0.$$
Moreover, for the different cases presented here we will show global existence of solutions as well as different blow-up patterns.
These discussions will be supported with some numerical results. |
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