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| We consider fourth order parabolic problems in $\mathbb{R}^N$ where the main operator is the bi-Laplacian. The equations we will consider have perturbations of the form $D^a(h(x)D^b u)$ with $a+b$, and $h$ in the locally uniform spaces $\dot L_U^p(\mathbb{R}^N)$ which will be described in detail.
We want to find the range of $\gamma$ such that for $u_0\in H^{4\gamma,q}(\mathbb{R}^N)$, $q\in (1,\infty)$, the problem has a unique solution. We prove that $\gamma\in I$, where $I$ is an interval depending only on $a$, $b$, $p$, $q$ and $N$.
Furthermore the solution is given by an analytic semigroup $S(t)u_0$ and satisfies smoothing estimates between the space of initial data an the space of the solutions.
Finally, we will briefly explain how to extend these results to a larger spaces such as the uniform Bessel spaces, and also how similar techniques can be used to consider non-linear terms. |
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