Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity and geometric character of solution of a degenerate parabolic
  equation   This work studies the regularity and the geometric significance of solution
of the Cauchy problem for a degenerate parabolic equation
$u_{t}=\Delta{}u^{m}$. Our main objective is to improve the H$\ddot{o}$lder
estimate obtained by pioneers and then, to show the geometric characteristic of
free boundary of degenerate parabolic equation. To be exact, the present work
will show that: (1) the weak solution
$u(x,t)\in{}C^{\alpha,\frac{\alpha}{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$,
where $\alpha\in(0,1)$ when $m\geq2$ and $\alpha=1$ when $m\in(1,2)$; (2) the
surface $\phi=(u(x,t))^{\beta}$ is a complete Riemannian manifold, which is
tangent to $\mathbb{R}^{n}$ at the boundary of the positivity set of $u(x,t)$.
(3) the function $(u(x,t))^{\beta}$ is a classical solution to another
degenerate parabolic equation if $ \beta$ is large sufficiently; Moreover, some
explicit expressions about the speed of propagation of $u(x,t)$ and the
continuous dependence on the nonlinearity of the equation are obtained.
Recalling the older H$\ddot{o}$lder estimate
($u(x,t)\in{}C^{\alpha,\frac{\alpha}{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$
with $0<\alpha<1$ for all $m>1$), we see our result (1) improves the older
result and, based on this conclusion, we can obtain (2), which shows the
geometric characteristic of free boundary.

17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int