Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounds for the extremal parameter of nonlinear eigenvalue problems and
  application to the explosion problem in a flow   We consider the nonlinear eigenvalue problem $ L u = \lambda f(u) $, posed in
a smooth bounded domain $ \Omega \subseteq \Bbb{R}^{N} $ with Dirichlet
boundary condition, where $ L $ is a uniformly elliptic second-order linear
differential operator, $ \lambda > 0 $ and $ f:[0,a_{f}) \rightarrow
\Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing and
convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $.
First we present some upper and lower bounds for the extremal parameter $
\lambda^{*} $ and the extremal solution $ u^{*} $. Then we apply the results to
the operator $ L_A = - \Delta + A c(x) $ with $ A>0 $ and $ c(x) $ is a
divergence-free flow in $ \Omega $. We show that, if $\psi_{A,\Omega}$ is the
maximum of the solution $\psi_{A}(x)$ of the equation $ L_A u = 1 $ in $\Omega$
with Dirichlet boundary condition, then for any incompressible flow $ c(x) $ we
have, $\psi_{A,\Omega} \longrightarrow 0$ as $A \longrightarrow \infty$ if and
only if $c(x)$ has no non-zero first integrals in $H_{0}^{1}(\Omega)$. Also,
taking $ c(x)=-x\rho(|x|) $ where $\rho$ is a smooth real function on $[0,1]$
then $c(x)$ is never divergence-free in unit ball $ B\subset \Bbb{R}^{N} $, but
our results completely determine the behaviour of the extremal parameter $
\lambda^{*}_{A} $ as $ A \longrightarrow \infty $.

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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio