Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Propagation acceleration in reaction diffusion equations with anomalous
  diffusions   In this paper, we are interested in the properties of solution of the
nonlocal equation $$\begin{cases}u_t+(-\Delta)^su=f(u),\quad t>0, \
x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R}\end{cases}$$ where $0\le
u_0<1$ is a Heaviside type function, $\Delta^s$ stands for the fractional
Laplacian with $s\in (0,1)$, and $f\in C([0,1],\mathbb{R}^+)$ is a non negative
nonlinearity such that $f(0)=f(1)=0$ and $f'(1)<0$. In this context, it is
known that the solution $u(t,s)$ converges locally uniformly to 1 and our aim
here is to understand how fast this invasion process occur. When $f$ is a
Fisher-KPP type nonlinearity and $s \in (0,1)$, it is known that the level set
of the solution $u(t,x)$ moves at an exponential speed whereas when $f$ is of
ignition type and $s\in \left(\frac{1}{2},1\right)$ then the level set of the
solution moves at a constant speed. In this article, for general monostable
nonlinearities $f$ and any $s\in (0,1)$ we derive generic estimates on the
position of the level sets of the solution $u(t,x)$ which then enable us to
describe more precisely the behaviour of this invasion process. In particular,
we obtain a algebraic generic upper bound on the "speed" of level set
highlighting the delicate interplay of $s$ and $f$ in the existence of an
exponential acceleration process. When $s\in\left (0,\frac{1}{2}\right]$ and
$f$ is of ignition type, we also complete the known description of the
behaviour of $u$ and give a precise asymptotic of the speed of the level set in
this context. Notably, we prove that the level sets accelerate when
$s\in\left(0,\frac{1}{2}\right)$ and that in the critical case $s=\frac{1}{2}$
although no travelling front can exist, the level sets still move
asymptotically at a constant speed. These new results are in sharp contrast
with the bistable situation where no such acceleration may occur, highlighting
therefore the qualitative difference between the two type of nonlinearities.

21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: eess
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an