Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Well-posedness, Global existence and decay estimates for the heat
  equation with general power-exponential nonlinearities   In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\;
u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an
exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in
$\exp L^p_0(\R^N)$ for $f(u)\sim \mbox{e}^{|u|^q},\;0<q\leq p,\; |u|\to
\infty.$ However, if for some $\lambda>0,$ $\displaystyle\liminf_{s\to
\infty}\left(f(s)\,{\rm{e}}^{-\lambda s^p}\right)>0,$ then non-existence occurs
in $\exp L^p(\R^N).$ Under smallness condition on the initial data and for
exponential nonlinearity $f$ such that $|f(u)|\sim |u|^{m}$ as $u\to 0,$
${N(m-1)\over 2}\geq p$, we show that the solution is global. In particular,
$p-1>0$ sufficiently small is allowed. Moreover, we obtain decay estimates in
Lebesgue spaces for large time which depend on $m$.

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The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio