Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of a Stable Blow-up profile for the nonlinear heat equation
  with a critical power nonlinear gradient term   We consider the nonlinear heat equation with a nonlinear gradient term:
$\partial_t u =\Delta u+\mu|\nabla u|^q+|u|^{p-1}u,\; \mu>0,\; q=2p/(p+1),\;
p>3,\; t\in (0,T),\; x\in \R^N.$ We construct a solution which blows up in
finite time $T>0.$ We also give a sharp description of its blow-up profile and
show that it is stable with respect to perturbations in initial data. The proof
relies on the reduction of the problem to a finite dimensional one, and uses
the index theory to conclude. The blow-up profile does not scale as
$(T-t)^{1/2}|\log(T-t)|^{1/2},$ like in the standard nonlinear heat equation,
i.e. $\mu=0,$ but as $(T-t)^{1/2}|\log(T-t)|^{\beta}$ with
$\beta=(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $\nabla u$ blow up
simultaneously and at a single point, and give the final profile. In
particular, the final profile is more singular than the case of the standard
nonlinear heat equation.

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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con