Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Carleman estimate and application to an inverse source problem for a
  viscoelasticity model in anisotropic case   We consider an anisotropic hyperbolic equation with memory term: $$
\partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) +
\int^t_0 \sum_{| \alpha| \le 2}
b_{\alpha}(x,t,\eta)\partial_x^{\alpha}u(x,\eta) d\eta + F(x,t) $$ for $x \in
\Omega$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for
viscoelasticity. First we establish a Carleman estimate for this equation with
overdetermining boundary data on a suitable lateral subboundary $\Gamma \times
(-T,T)$. Second we apply the Carleman estimate to establish a both-sided
estimate of $| u(\cdot,0)|_{H^3(\Omega)}$ by $\partial_{\nu}u|_{\Gamma\times
(0,T)}$ under the assumption that $\partial_tu(\cdot,0) = 0$ and $T>0$ is
sufficiently large, $\Gamma \subset \partial\Omega$ satisfies some geometric
condition. Such an estimate is a kind of observability inequality and related
to the exact controllability. Finally we apply the Carleman estimate to an
inverse source problem of determining a spatial varying factor in $F(x,t)$ and
we establish a both-sided Lipschitz stability estimate.

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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc