Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Traveling fronts in space-time periodic media   This paper is concerned with the existence of pulsating traveling fronts for
the equation: $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot
\nabla u = f (t, x, u)$, (1) where the diffusion matrix $A$, the advection term
$q$ and the reaction term $f$ are periodic in $t$ and $x$. We prove that there
exist some speeds $c^*$ and $c^{**}$ such that there exists a pulsating
traveling front of speed $c$ for all $c\ge c^{**}$ and that there exists no
such front of speed $c<c^*$. We also give some spreading properties for
front-like initial data. In the case of a KPP-type reaction term, we prove that
$c^* = c^{**}$ and we characterize this speed with the help of a family of
eigenvalues associated with the equation. If $f$ is concave with respect to
$u$, we prove some Lipschitz continuity for the profile of the pulsating
traveling front.
  Cet articl{\'e} etudie l'existence de fronts pulsatoires pour l'\'equation :
$\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = f (t,
x, u)$, (2) o\`u la matrice de diffusion A, le terme d'advection $q$ et le
terme de r{\'e}action $f$ sont p{\'e}riodiques en $t$ et en $x$. Nous prouvons
l'existence de deux vitesses $c^*$ et $c^{**}$ telles qu'il existe un front
pulsatoire de vitesse $c$ pour tout $c \ge c^{**}$ et qu'il n'existe pas de tel
front de vitesse $c<c^*$. Nous donnons egalement des propri{\'e}t{\'e}s de
spreading pour des donn{\'e}es initiales ressemblant a des fronts. Dans le cas
d'un terme de r{\'e}action de type KPP, nous prouvons que $c^* = c^{**}$ et
nous caract{\'e}risons cette vitess{\`e} a l'aide d'une famille de valeurs
propres associ{\'e}{\`e} a l'\'equation. Si $f$ est concave en $u$, nous
montrons que le profil du front pulsatoire construit est lipschitzien.

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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org