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Let $M$ be a smooth manifold with boundary $partial M$ and bounded geometry, $partial_D M subset partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $partial M smallsetminus partial_D M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f mbox{ in } M, u = 0 mbox{ on } partial_D M, partial^P_ u u + bu = 0 mbox{ on } partial M setminus partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s geq 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincar{e} inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a emph{uniform Agmon condition} and show that it is equivalent to the Gaa rding inequality. This extends an important result of Agmon (1958).
The diffusion system with time-fractional order derivative is of great importance mathematically due to the nonlocal property of the fractional order derivative, which can be applied to model the physical phenomena with memory effects. We consider an
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