Regularity and geometric character of solution of a degenerate parabolic
equation

Jiaqing Pan 111Email<EMAIL_ADDRESS>
Institute of Mathematics, Jimei University,
Xiamen, 361021, P.R.China

Abstract: 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 $m\geq 2$ 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.

Keywords : degenerate parabolic equation; regularity; geometric character;
H$\ddot{o}$lder estimate

AMS(2010) Subject Classifications:  35K15, 35K55, 35K65, 53C25

##  1 Introduction

Consider the Cauchy problem of nonlinear parabolic equation

$\displaystyle\left\\{\begin{array}[]{ll}u_{t}=\Delta{}u^{m}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q,\\\
u(x,0)=u_{0}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on}~{}\mathbb{R}^{n},\end{array}\right.$
(1.3)

where $Q=\mathbb{R}^{n}\times\mathbb{R}^{+},n\geq 1,m>1$ and

$\displaystyle 0\leq
u_{0}(x)\leq{}M,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0<\int_{\mathbb{R}^{n}}u_{0}(x)dx<\infty.$
(1.4)

The equation in (1.1) is an example of nonlinear evolution equations and many
interesting results, such as the existence, uniqueness, continuous dependence
on the nonlinearity of the equation and large time behavior are obtained
during the past several decades. By a weak solution of (1.1), (1.2) in $Q$, we
mean a nonnegative function $u(x,t)$ such that, for any given $T>0$,

$\int_{0}^{T}\int_{\mathbb{R}^{n}}\left(u^{2}+|\nabla{}u^{m}|^{2}\right)dxdt<\infty$

and

$\int_{0}^{T}\int_{\mathbb{R}^{n}}\left(\nabla{}u^{m}\cdot\nabla{}f-uf_{t}\right)dxdt=\int_{\mathbb{R}^{n}}u_{0}(x)f(x,0)dx$

for any continuously differentiable function $f(x,t)$ with compact support in
$\mathbb{R}^{n}\times(0,T)$.

We know that (see [3, 4, 5, 6, 8, 12, 13, 14]) the Cauchy problem (1.1),(1.2)
permits a unique weak solution $u(x,t)$ which has the following properties:

$\displaystyle 0\leq{}u(x,t)\leq{}M,$ (1.5)
$\displaystyle\int_{\mathbb{R}^{n}}u(x,t)dx=\int_{\mathbb{R}^{n}}u_{0}(x)dx,$
(1.6) $\displaystyle\frac{\partial{}u}{\partial{}t}\geq\frac{-u}{(m-1)t},$
(1.7)
$\displaystyle\Delta{}\left(\frac{m}{m-1}u^{m-1}\right)\geq\frac{-n}{n(m-1)+2}\cdot\frac{1}{t}$
(1.8)
$\displaystyle\|u-v\|_{L^{2}(Q_{T})}\leq{}C\max_{s\in[0,M]}\left|s^{\frac{1}{j}}-s^{\frac{m}{j}}\right|,$
(1.9)

where $j=1,2,3...$ and $v$ is the solution to the Cauchy problem of linear
heat equation with the same initial value

$\displaystyle\left\\{\begin{array}[]{ll}v_{t}=\Delta{}v~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q,\\\
v(x,0)=u_{0}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on}~{}\mathbb{R}^{n}.\end{array}\right.$
(1.12)

Moreover, the solution $u(x,t)$ can be obtained (see [9, 15]) as a limit of
solutions $u_{\eta}$($\eta\longrightarrow 0^{+}$) of the Cauchy problem

$\displaystyle\left\\{\begin{array}[]{ll}u_{t}=\Delta{}u^{m}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q,\\\
u(x,0)=u_{0}(x)+\eta~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on}~{}\mathbb{R}^{n},\end{array}\right.$
(1.15)

and the solutions $u_{\eta}$ is taken in the classical sense. We know that
D.G.Aronson, Ph.Benilan (see theorem 2, p.104 in [8]) claimed that: if $u$ is
the weak solution to the Cauchy problem (1.1) with the initial value (1.2),
then $u\in{}C(Q)$ and $u\geq 0$; J. L. Vazquez (see Proposition 6 in Ch.2 of
[13]) proved $u\in{}C^{\infty}(Q_{+})$, where

$Q_{+}=\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{+}:~{}u(x,t)>0\\}.$

Before this, the same conclusion was established by A. Friedman (see theorem
11 and corollary 2 in Ch.3 [10] ). Moreover, employing so called
$bootstrap~{}argument$, D.G.Aronson, B.H.Gilding and L. A. Peletier (see [2,
4, 5, 6]) also claimed $u\in{}C^{\infty}(Q_{+})$ with more details. Therefore,
we can divide the space-time $\mathbb{R}^{n}\times\mathbb{R}^{+}$ into two
parts: $Q=Q_{+}\cup{}Q_{0}$, where

$Q_{0}=\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{+}:~{}u(x,t)=0\\}.$

Furthermore, if $Q_{0}$ contains an open set, say, $Q_{1}$, we can also obtain
$u(x,t)\in{}C^{\infty}(Q_{1})$ owing to $u(x,t)\equiv 0$ in $Q_{1}$. Thereby,
we may suspect that the solution of degenerate parabolic equation is actually
smooth in $Q$ except a set of measure 0. In order to improve the regularity of
$u(x,t)$, many authors have made hard effort in this direction. The earliest
contribution to the subject was made, maybe, by D.G. Aronson and B.H.Gilding
and L.A.Peleiter(see [2, 4]). They proved that the solution to the Cauchy
problem

$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial{}u}{\partial{}t}=\frac{\partial^{2}{}u^{m}}{\partial{}x^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}\mathbb{R}^{1}\times\mathbb{R}^{+},~{}m>1,\\\
u(x,0)=u_{0}(x)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on}~{}\mathbb{R}^{1}\end{array}\right.$
(1.18)

is continuous in $\mathbb{R}^{1}\times(0,+\infty)$ if the nonnegative initial
value satisfies a good condition. Moreover, if the initial value
$0\leq{}u_{0}\leq{}M$ and $u_{0}^{m}$ is Lip-continuous, $u(x,t)$ can be
continuous on $\mathbb{R}^{1}\times[0,+\infty)$ (see [2]). As to the case of
$n\geq 1$, L.Caffarelli and A.Friedman (see [15]), proved that the solution
$u(x,t)$ to the Cauchy problem (1.1), (1.2) is H$\ddot{o}$lder continuous in
$Q$ :

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|u(x,t)-u(x_{0},t_{0})|\leq{}C\left(|x-x_{0}|^{\alpha}+|t-t_{0}|^{\frac{\alpha}{2}}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(*)$

for some $0<\alpha<1$ and $C>0$. Moreover, for the general equation
$u_{t}=\nabla\cdot(u\nabla{}p),~{}p=\kappa(u)$, L .Caffarelli and J.L.Vazques
([16]) established the property of finite propagation and the persistence of
positivity, where $\kappa$ may be a general operator. To study this problem
more precisely, D. G. Aronson, S.B. Angenent and J. Graveleau (see [7, 20])
constructed a interesting radially symmetric solution $u(r,t)$ to the focusing
problem for the equation of (1.1). Denoting the porous medium pressure
$V=\frac{m}{m-1}u^{m-1}$, they claimed $V=Cr^{\delta}$ at the fusing time,
where $0<\delta<1,C$ is a positive constant. Moreover,

$\lim_{r\downarrow
0}\frac{V(r,\eta{}r^{\alpha})}{r^{2-\alpha}}=\frac{\varphi(c^{\ast}\eta)}{-\eta}.$

To study the regularity of the weak solution of (1.1), (1.2), the present work
will show the following more precise conclusion: for every $h\in(m-1,m)$,
there exists a $C>0$ such that

$\displaystyle|u(x,t)-u(x_{0},t_{0})|\leq{}C\left(|x-x_{0}|^{\frac{1}{h}}+|t-t_{0}|^{\frac{1}{2h}}\right),$
(1.19)

where,

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h=\left\\{\begin{array}[]{ll}1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}~{}1<m<2,\\\
h\in(m-1,m)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}~{}m\geq
2.\end{array}\right.$ (1.22)

We see that the range of $\frac{1}{h}$ is $(0,1]$ not $(0,1)$, thereby, the
older H$\ddot{o}$lder estimate $(*)$ is improved by (1.11).

Moreover, we will show that the functions
$\frac{\partial{u^{\beta}}}{\partial{}x_{i}}$ are continuous if $\beta$ is
large sufficiently because we can employ (1.11) to obtain

$|u^{\beta}(x,t)-u^{\beta}(x_{\ast},t_{\ast})|\leq{}C\left(|x-x_{\ast}|^{\frac{\beta}{h}}+|t-t_{\ast}|^{\frac{\beta}{2h}}\right)$

for another positive constant $C$. By this inequality, we want to get the
continuous partial derivatives $\frac{\partial{}u^{\beta}}{\partial{}t}$ and
$\frac{\partial{}u^{\beta}}{\partial{}x_{i}}~{}i=1,2,...,n$, that is to say,

$\displaystyle\phi(x,t)\in{}C^{1}(\mathbb{R}^{n})$ (1.23)

for every given $t>0$, where $\phi(x,t)=u^{\beta}(x,t)$. In particular, we
will prove that the function $\phi(x,t)$ satisfies the degenerate parabolic
equation

$\displaystyle\frac{\partial\phi}{\partial{}t}=m\left[\phi^{\frac{m-1}{\beta}}\Delta{}\phi+\frac{m-\beta}{\beta}\phi^{\frac{m-\beta-1}{\beta}}|\nabla\phi|^{2}\right]$

in the classical sense.

For every fixed $t>0$, we define a n-dimensional surface $S(t)$, which floats
in the space $\mathbb{R}^{n+1}$ with the time $t$:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(t):\left\\{\begin{array}[]{ll}x_{i}=x_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,2,3,...,n,{}\\\
x_{n+1}=\phi(x,t),\end{array}\right.$

where the function $\phi(x,t)$ is mentioned above. We will discuss the
geometric character of $S(t)$. We know that Y.Giga and R.V.Kohn studied the
fourth-order total variation flow and the fourth-order surface diffusion law
(see [22]), and proved that the solution becomes identically zero in finite
time. To be exact, the solution surface will coincide with $\mathbb{R}^{n}$ in
finite time. Because this phenomenon will never occur for our surface $S(t)$
owing to (1.2) and (1.4), so we will discuss the relationship between $S(t)$
and $\mathbb{R}^{n}$.

Let

$\displaystyle\left\\{\begin{array}[]{ll}g_{1}=(1,0,...,\frac{\partial{}\phi}{\partial{}x_{1}}),\\\
g_{2}=(0,1,...,\frac{\partial{}\phi}{\partial{}x_{2}}),\\\ ......,\\\
g_{n}=(0,0,...1,\frac{\partial{}\phi}{\partial{}x_{n}}).\end{array}\right.$
(1.30)

Define the Riemannian metric on $S(t)$:

$(ds)^{2}=\sum\limits_{i,j=1}^{n}g_{ij}dx_{i}dx_{j},$

where $g_{ij}={g}_{i}\cdot{g}_{j}$. Clearly,

$\displaystyle(ds)^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(1+\phi^{2}_{x_{i}})(dx_{i})^{2}+\sum_{i\neq{}j,i,j=1}^{n}\phi_{x_{i}}\phi_{x_{j}}dx_{i}dx_{j}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(dx_{i})^{2}+\left(\sum_{i=1}^{n}\phi_{x_{i}}dx_{i}\right)^{2}.$

Recalling $\sum\limits_{i=1}^{n}\phi_{x_{i}}dx_{i}=d\phi=dx_{n+1}$ for fixed
$t>0$, we get

$\displaystyle(ds)^{2}=\sum_{i=1}^{n+1}(dx_{i})^{2}=\sum_{i=1}^{n}(dx_{i})^{2}+(d\phi)^{2}.$

If the derivatives $\frac{\partial{}\phi}{\partial{}x_{i}}$ are bounded for
$i=1,2,...,n$, then we can get a positive constant $C$, such that
$\sum\limits_{i=1}^{n}\left(\phi_{x_{i}}dx_{i}\right)^{2}\leq{}C\sum\limits_{i=1}^{n}\left(dx_{i}\right)^{2}$.
Denoting $(d\rho)^{2}=\sum\limits_{i=1}^{n}(dx_{i})^{2}$, which is the
Euclidean metric on $\mathbb{R}^{n}$, we get

$\displaystyle(d\rho)^{2}\leq(ds)^{2}\leq(1+C)(d\rho)^{2}.$ (1.31)

As a consequence of (1.14), we see that the completeness of $\mathbb{R}^{n}$
yields the completeness of $S(t)$ and therefore, $S(t)$ is a complete
Riemannian manifold. On the other hand, if we can obtain

$\displaystyle\nabla{}\left(\phi(x,t)\right)|_{\partial{}H_{u}(t)}=0$ (1.32)

for every fixed $t>0$, where $H_{u}(t)$ is the positivity set of $u(x,t)$:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{u}(t)=\\{x\in\mathbb{R}^{n}:~{}u(x,t)>0\\}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t>0,$

then (1.15) encourage us to prove that: the manifold $S(t)$ is tangent to
$\mathbb{R}^{n}$ on $\partial{}H_{u}(t)$.

It is well-known that the function

$v(x,t)=\left(\frac{1}{2\sqrt{\pi{}t}}\right)^{-n}\int_{\mathbb{R}^{n}}u_{0}(\xi)e^{-\frac{(x-\xi)^{2}}{4t}}d\xi$

is the solution of the Cauchy problem of the linear heat equation (1.8) and
$v(x,t)>0$ in $Q$ everywhere if only the initial value $u_{0}$ satisfies
(1.2). This fact shows that the speed of propagation of $v(x,t)$ is infinite,
that is to say,

$\displaystyle\sup_{x\in{}H_{v}(t)}|x|=\infty.$ (1.33)

However, the degeneracy of the equation in (1.1) causes an important
phenomenon to occur, i.e. finite speed of propagation of disturbance. We have
observed this phenomenon on the $source-type$ solution $B(x,t;C)$ (see [11]),
where

$\displaystyle
B(x,t,C)=t^{-\lambda}\left(C-\kappa\frac{|x|^{2}}{t^{2\mu}}\right)_{+}^{\frac{1}{m-1}}$
(1.34)

is the equation in (1.1) with a initial mass $M\delta(x)$, and

$\lambda=\frac{n}{n(m-1)+2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu=\frac{\lambda}{n},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\kappa=\frac{\lambda(m-1)}{2mn}.$

We see that the function $B(x,t;C)$ has compact support in space for every
fixed time. More precisely, if $u(x,t)$ is the solution of (1.1), (1.2), then

$\displaystyle\sup_{x\in{}H_{u}(t)}|x|={}O\left(t^{\frac{1}{n(m-1)+2}}\right)$
(1.35)

when $t$ is large enough (see Proposition 17 in [13]). Comparing (1.16) and
(1.18) and recalling the mass conservation
$\int_{\mathbb{R}^{n}}u(x,t)dx=\int_{\mathbb{R}^{n}}u_{0}(x)dx=\int_{\mathbb{R}^{n}}v(x,t)dx$,
we will prove that the solution continuously depends on the nonlinearity of
the equation (1.1):

$\|u(\cdot,t)-v(\cdot,t)\|^{2}_{L^{2}(|x|\leq{}k)}\leq{}C\left[(m-1)+\frac{1}{k}\right].$

We read the main conclusions of the present work as follows:

Theorem 1 Assume $u(x,t)$ be the solution to (1.1), (1.2). Then
$u(x,t)\in{}C(Q)$ and

(1) for every given $\tau>0,K>0$ , there exists a positive $\nu$ such that

$|u(x_{1},t_{1})-u(x_{2},t_{2})|\leq{}\nu\left(|x_{1}-x_{2}|^{\frac{1}{h}}+|t_{1}-t_{2}|^{\frac{1}{2h}}\right)$

where $|x_{i}|\leq{}K,t_{i}\geq\tau,~{}i=1,2$,

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h=\left\\{\begin{array}[]{ll}1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}~{}1<m<2,\\\
h\in(m-1,m)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}~{}m\geq
2;\end{array}\right.$ (1.38)

(2) for every $\beta>h$, the function $\phi=(u(x,t))^{\beta}\in{}C^{1}(Q)$ and
the surface $\phi=\phi(x,t)$ is a complete Riemannian-manifold which is
tangent to $\mathbb{R}^{n}$ on $\partial{}H_{u}(t)$ for every fixed $t>0$, $h$
is defined by (1.19);

(3) if $\beta>2h$, the function $\phi(x,t)$ satisfies the degenerate parabolic
equation

$\displaystyle\frac{\partial\phi}{\partial{}t}=m\left[\phi^{\frac{m-1}{\beta}}\Delta{}\phi+\frac{m-\beta}{\beta}\phi^{\frac{m-\beta-1}{\beta}}|\nabla\phi|^{2}\right]$

in the classical sense in $Q$.

Theorem 2 Assume $u(x,t)$ be the solution to (1.1), (1.2),
$B_{\delta}=\\{x\in\mathbb{R}^{n}:~{}|x|<\delta\\}$ for some $\delta>0$. If
supp $u_{0}\subset{}B_{\delta}$, then for every given $t>0$,

$\displaystyle\sup_{x\in{}H_{u}(t)}|x|\geq\chi(t),$ (1.39)

where,

$\displaystyle\chi(t)=\left[(m-1)\pi^{\frac{(1-m)n}{2}}\Gamma(1+\frac{n}{2})^{m-1}\cdot\left(\int_{\mathbb{R}^{n}}u_{0}dx\right)^{m-1}t\right]^{\frac{1}{2+(m-1)n}}.$

Moreover, for every given $T>0$, there is a positive $C_{\ast}=C_{\ast}(T)$
such that

$\displaystyle\int_{|x|\leq{}k}\left[v(x,t)-u(x,t)\right]^{2}dx\leq{}C_{\ast}\left[(m-1)+\frac{1}{k}\right]$
(1.40)

with respect to $t\in(0,T)$ uniformly, where $v(x,t)$ is the solution of
(1.8).

Let us also note that the manifolds $S(t)$ and $\mathbb{R}^{n}$ are two
surfaces in $\mathbb{R}^{n+1}$ and the Cauchy problem (1.1), (1.2) can be
regarded as a mapping $\Phi(t):~{}\mathbb{R}^{n}\longrightarrow~{}S(t)$. Thus,
besides the theorems mentioned above, we will give an example to show the
intrinsic properties about the manifold $S(t)$.

##  2 The proof of Theorem 1

Lemma 1 If $u(x,t)$ is the weak solution to (1.1), (1.2) in $Q$. Then for
every $h\in(m-1,m)$, there is a $C_{1}=C_{1}(h,m,M)$ such that

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left|\nabla{}u^{h}\right|^{2}\leq{}\frac{1}{{C_{1}t}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q$
(2.1)

in the sense of distributions in $Q$.

Proof: We first prove (2.1) for the classical solutions $u_{\eta}(x,t)$. Set

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}u_{\eta}^{m}=V^{q}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{for}~{}q\in\left(1,\frac{m}{m-1}\right).$

Then

$V_{t}=mV^{q-\frac{q}{m}}\Delta{}V+m(q-1)V^{q-1-\frac{q}{m}}|\nabla{}V|^{2}.$

Differentiating this equation with respect to $x_{j}$ and multiplying though
by $\frac{\partial{}V}{\partial{}x_{j}}$, letting
$h_{j}=\frac{\partial{}V}{\partial{}x_{j}}$, we get

$\displaystyle\frac{1}{2}(h_{j}^{2})_{t}$ $\displaystyle=$ $\displaystyle
mV^{q-\frac{q}{m}}h_{j}\Delta{}h_{j}+m(q-\frac{q}{m})V^{q-1-\frac{q}{m}}h_{j}^{2}\Delta{}V$
$\displaystyle+m(q-1)(q-1-\frac{q}{m})V^{q-2-\frac{q}{m}}h_{j}^{2}|\nabla{}V|^{2}+2m(q-1)V^{q-1-\frac{q}{m}}h_{j}\nabla{}V\cdot\nabla{}h_{j}$

for $j=1,2,...,n$. Setting

$H^{2}=\sum_{j=1}^{n}h_{j}^{2}~{},$

we obtain

$\displaystyle H^{2}_{t}-mV^{q-\frac{q}{m}}\Delta{}H^{2}$ $\displaystyle=$
$\displaystyle
2m(q-\frac{q}{m})V^{q-1-\frac{q}{m}}H^{2}\Delta{}V+2m(q-1)(q-1-\frac{q}{m})V^{q-2-\frac{q}{m}}\sum_{j=1}^{n}h_{j}^{2}|\nabla{}V|^{2}$
$\displaystyle+2m(q-1)V^{q-1-\frac{q}{m}}\nabla{}V\cdot\sum_{j=1}^{n}\nabla{}h^{2}_{j}-2mV^{q-\frac{q}{m}}\sum_{j=1}^{n}|\nabla{}h_{j}|^{2}$
$\displaystyle\leq$ $\displaystyle
2m(q-\frac{q}{m})V^{q-1-\frac{q}{m}}H^{2}\Delta{}V+2m(q-1)(q-1-\frac{q}{m})V^{q-2-\frac{q}{m}}H^{4}$
$\displaystyle+2m(q-1)V^{q-1-\frac{q}{m}}\nabla{}V\cdot\nabla{}H^{2}.$

It follows from $q\in\left(1,\frac{m}{m-1}\right)$ that
$(q-1)(q-1-\frac{q}{m})<0$ and ${q-2-\frac{q}{m}}<0$. Hence,

$\displaystyle 2m(q-1)(q-1-\frac{q}{m})V^{q-2-\frac{q}{m}}$
$\displaystyle\leq$ $\displaystyle{}-\widetilde{C}_{1},$

where
$\widetilde{C}_{1}=2m\left|(q-1)(q-1-\frac{q}{m})\right|(M+\eta)^{m-\frac{2m}{q}-1}.$
Setting

$\displaystyle L({H}^{2})$ $\displaystyle=$ $\displaystyle
mV^{q-\frac{q}{m}}\Delta{}{H}^{2}-\widetilde{C}_{1}{H}^{4}+2m(q-\frac{q}{m})V^{q-1-\frac{q}{m}}{H}^{2}\Delta{}V$
$\displaystyle+2m(q-1)V^{q-1-\frac{q}{m}}\nabla{}V\cdot\nabla{}{H}^{2},$

we get

$H^{2}_{t}\leq{}L(H^{2}).$

Taking
$h^{\ast}_{1}=(\frac{1}{\widetilde{C}_{1}t})^{\frac{1}{2}},h^{\ast}_{i}=0$ for
$i=2,3,...,n$, and setting
$Q_{\ast}^{2}=\sum\limits_{i=1}^{n}h_{i}^{\ast}=\frac{1}{\widetilde{C}_{1}t}$,
we see that the function $Q_{\ast}^{2}$ is a solution to the equation

$\displaystyle\frac{\partial}{\partial{}t}Q_{\ast}^{2}={}L(Q_{\ast}^{2})$

with the initial condition $Q^{2}_{\ast}(0)=+\infty$.

On the other hand, let $u_{\eta}^{m}=K$, then $K$ is the solution to the
Cauchy problem of the linear parabolic equation

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}K_{t}=g(x,t)\Delta{}K~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q$

with the inial value $(u_{0}+\eta)^{m}$, where,
$g(x,t)=m(u_{\eta}(x,t))^{m-1}$ and $u_{\eta}(x,t)$ is a known function, which
is the classical solution to (1.9) and $\eta\leq{}u_{\eta}\leq{}M+\eta$. By
Theorem 5.1 in [18], we get
$K\in{}H^{2+\alpha,1+\frac{\alpha}{2}}(\mathbb{R}^{n}\times(0,T))$ for any
$T>0$ (Even if $u_{0}$ does not have the required smoothness we may
approximate it ( by mollification ) with smooth functions $u_{0\eta}$).
Therefore, $\Delta{}K,\frac{\partial{}K}{\partial{}t}$ and
$\frac{\partial{}K}{\partial{}x_{i}}(i=1,2,...,n)$ are bounded. To be exact,
there is a positive $\mu_{0}$, which may depend on $\eta$, such that

$|\Delta{}K|+|\frac{\partial{}K}{\partial{}t}|+|\nabla{}K|\leq{}\mu_{0}.$

Therefore,

$\displaystyle|\nabla{}V|$ $\displaystyle=$
$\displaystyle|\nabla{}u_{\eta}^{\frac{m}{q}}|$ $\displaystyle=$
$\displaystyle|\frac{1}{q}u_{\eta}^{\frac{m}{q}-m}\nabla{}K|$
$\displaystyle\leq$
$\displaystyle{}\frac{1}{q}\eta^{(\frac{1}{q}-1)m}\mu_{0}.$

Letting $C^{\prime}=\frac{1}{q}\eta^{(\frac{1}{q}-1)m}\mu_{0}$, we obtain
$|\nabla{}V|\leq{}C^{\prime}$. Similarly, we can get positive
$C^{\prime\prime}$, which may depend on $\eta$ also, such that
$|\Delta{}V|\leq{}C^{\prime\prime}$.

Now we can employing the comparison theorem and get

$\displaystyle H^{2}\leq{}\frac{1}{\widetilde{C}_{1}t}.$ (2.2)

Letting $\eta\longrightarrow 0$ in (2.2) gives
$|\nabla{}u^{\frac{m}{q}}|^{2}\leq\frac{1}{{C}_{1}t}$ for
$q\in\left(1,\frac{m}{m-1}\right)$ with

$C_{1}=2m\left|(q-1)(q-1-\frac{q}{m})\right|M^{m-\frac{2m}{q}-1}.$

Setting $h=\frac{m}{q}$ in (2.2) yields $h\in(m-1,m)$, and (2.1) follows.
$\square$

To prove Theorem 1, we need to show an ordinary inequality firstly:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|a-b|^{\beta}\leq|a^{\beta}-b^{\beta}|~{}~{}~{}~{}~{}~{}~{}\mbox{for}~{}a,b\geq
0,~{}\beta{}>1.$ (2.3)

In fact, (2.3) is right for $a=b$. If $a>b$, we can easily get the following
inequalities:

$\left(1-\frac{b}{a}\right)^{\beta}\leq
1-\frac{b}{a}~{}~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}~{}1-\left(\frac{b}{a}\right)^{\beta}\geq
1-\frac{b}{a}$

thanks to $0\leq\frac{b}{a}<1$. Thereby,
$\left(1-\frac{b}{a}\right)^{\beta}\leq 1-\left(\frac{b}{a}\right)^{\beta}.$
This inequality gives

$\displaystyle|a-b|^{\beta}=a^{\beta}\left|1-\frac{b}{a}\right|^{\beta}\leq{}a^{\beta}-b^{\beta}.$

So (2.3) holds for $a>b\geq 0$. Certainly, (2.3) is also right when
$0\leq{}a<b$.

We are now in a position to establish our Theorem 1.

To prove (1) of Theorem 1  It follows from (2.1) that

$\displaystyle\left|u^{h}(x_{1},t)-u^{h}(x_{2},t)\right|\leq{}(C_{1}t)^{-\frac{1}{2}}|x_{1}-x_{2}|$
(2.4)

for every $(x_{1},t),(x_{2},t)\in{}Q,h\in(m-1,m)$. If $1<m<2$, we take $h=1$,
and therefore, (2.4) yields

$\displaystyle\left|u(x_{1},t)-u(x_{2},t)\right|\leq{}(C_{1}t)^{-\frac{1}{2}}|x_{1}-x_{2}|.$

If $m\geq 2$, we take $h>1$ for every $h\in(m-1,m)$. In this case, we use
(2.3) in (2.4) and obtain

$\displaystyle\left|u(x_{1},t)-u(x_{2},t)\right|\leq{}(C_{1}t)^{-\frac{1}{2h}}|x_{1}-x_{2}|^{\frac{1}{h}}.$

Therefore, for every given $m>1$, there always exists a suitable positive
number $h\geq 1$ such that

$\displaystyle\left|u(x_{1},t)-u(x_{2},t)\right|\leq(C_{1}\tau)^{-\frac{1}{2h}}|x_{1}-x_{2}|^{\frac{1}{h}}$
(2.5)

for every $(x_{1},t),(x_{2},t)\in\mathbb{R}^{n}\times[\tau,\infty)$ with given
$\tau>0$, where ${h}=1~{}\mbox{for}~{}1<m<2;~{}h\in(m-1,m)$ for $m\geq 2$.
Employing the well-known theorem on the H$\ddot{o}$lder continuity with
respect to the time variable (see [1]), we obtain

$\displaystyle\left|u(x,t_{1})-u(x,t_{2})\right|\leq{}\mu{}|t_{1}-t_{2}|^{\frac{1}{2h}}$
(2.6)

for $|x|\leq{}K,~{}t_{1},t_{2}>\tau$ and $|t_{1}-t_{2}|$ small sufficiently,
where $\mu$ depends on $(C_{1}\tau)^{-\frac{1}{2h}},K$ is any fixed positive
constant. Combining (2.5) and (2.6) we get another positive constant $\nu$
such that

$\displaystyle\left|u(x_{1},t_{1})-u(x_{2},t_{2})\right|\leq{}\nu\left[|x_{1}-x_{2}|^{\frac{1}{h}}+|t_{1}-t_{2}|^{\frac{1}{2h}}\right]$
(2.7)

for all $|x_{i}|\leq{}K,t_{i}\geq\tau,~{}i=1,2$, where

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}h=\left\\{\begin{array}[]{ll}1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}~{}1<m<2,\\\
h\in(m-1,m)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}~{}m\geq
2.\end{array}\right.$ (2.10)

Certainly, this gives $u\in{}C(Q).$ $\square$

To prove (2) of Theorem 1 Denote $\beta=h+\varepsilon$ and set

$\displaystyle\phi(x,t)=u^{\beta}$

for every $\varepsilon>0$. It follows from (2.1) that

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|\nabla{}\phi(x,t)|\leq{}C_{2}u^{\varepsilon}(x,t)t^{-\frac{1}{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q$
(2.11)

for some $C_{2}>0$. To prove the function $\phi(x,t)\in{}C^{1}(Q)$, we first
see $u\in{}C^{\infty}(Q_{+})$; second, (2.8) implies
$\nabla{}\phi(x_{\ast},t_{\ast})=0$ for every $(x_{\ast},t_{\ast})\in{}Q_{0}$.
Combining (2.7) and (2.8) gives

$\displaystyle\left|\nabla{}\phi(x,t)-\nabla{}\phi(x_{\ast},t_{\ast})\right|$
$\displaystyle=$ $\displaystyle\left|\nabla{}\phi(x,t)\right|$ (2.12)
$\displaystyle\leq$ $\displaystyle
C_{2}{}t^{-\frac{1}{2}}\nu^{\varepsilon}\left[|x-x_{\ast}|^{\frac{1}{h}}+|t-t_{\ast}|^{\frac{1}{2h}}\right]^{\varepsilon}.$

This inequality tells us $\nabla\phi(x,t)\in{}C(Q)$ and therefore,
$\phi(x,t)\in{}C^{1}(Q)$. Furthermore, for every fixed $t>0$, the continuity
of $u(x,t)$ implies $H_{u}(t)$ is a open set. Thus
$\phi(x_{\ast},t)=\nabla\phi(x_{\ast},t)=0$ for
$(x_{\ast},t)\in{\partial{}H_{u}(t)}$. This fact claims that the surface
$\phi=\phi(x,t)$ touches $\mathbb{R}^{n}$ at $\partial{}H_{u}(t)$. In other
words, $\mathbb{R}^{n}$ is just the tangent plane of $\phi(x,t)$ at
$\partial{}H_{u}(t)$.

Now we can define a n-dimensional surface $S(t)$ for every fixed $t>0$ as
before:

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(t):\left\\{\begin{array}[]{ll}x_{i}=x_{i},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,2,3,...,n,{}\\\
x_{n+1}=\phi(x,t).\end{array}\right.$

Defining the Riemannian metric (1.13) on $S(t)$, we get (1.14), more
precisely,

$\displaystyle(d\rho)^{2}\leq(ds)^{2}$ $\displaystyle\leq$
$\displaystyle\left(1+\max_{i=1,2,...,n}\left|\frac{\partial\phi}{\partial{}x_{i}}\right|^{2}\right)(d\rho)^{2}.$

It follows from (2.8) that
$\max\limits_{i=1,2,...,n}\left|\frac{\partial\phi}{\partial{}x_{i}}\right|^{2}\leq{}\left(C_{2}u^{\varepsilon}t^{-\frac{1}{2}}\right)^{2}$.
Moreover, as a result of theorem 9 in Chap III of [13], we get
$u(x,t)\leq{}C_{3}t^{-\frac{n}{n(m-1)+2}}$ for some $C_{3}>0$. Hence,

$\displaystyle\max\limits_{i=1,2,...,n}\left|\frac{\partial\phi}{\partial{}x_{i}}\right|^{2}\leq{}C_{4}t^{\frac{-2n\varepsilon}{n(m-1)+2}-1}$
(2.15)

for another positive $C_{4}$. Thus,

$\displaystyle(d\rho)^{2}\leq(ds)^{2}$ $\displaystyle\leq$
$\displaystyle\left(1+C_{4}t^{\frac{-2n\varepsilon}{n(m-1)+2}-1}\right)(d\rho)^{2}.~{}$
(2.16)

As a consequence of (2.11), the completeness of $\mathbb{R}^{n}$ yields the
completeness of $S(t)$ and therefore, $S(t)$ is a complete Riemannian
manifold. $\square$

To prove (3) of Theorem 1  Recalling $u(x,t)=\lim\limits_{\eta\longrightarrow
0^{+}}u_{\eta}$ and $u_{\eta}\geq\eta$, $u_{\eta}$ are the classical solutions
to the Cauchy problem (1.9), we can make

$\phi_{\eta}=u_{\eta}^{\beta}$

for $\beta=h+\varepsilon>2h$, and then $\phi_{\eta}$ satisfies the degenerate
parabolic equation

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial\phi_{\eta}}{\partial{}t}=m\left[\phi_{\eta}^{\frac{m-1}{\beta}}\Delta{}\phi_{\eta}+\frac{m-\beta}{\beta}\phi_{\eta}^{\frac{m-\beta-1}{\beta}}|\nabla\phi_{\eta}|^{2}\right]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}Q.$

Recalling $\lim\limits_{\eta\longrightarrow 0^{+}}\phi_{\eta}=\phi$ and
$u\in{}C^{\infty}(Q_{+})$, we see that $\phi(x,t)\in{}C^{\infty}(Q_{+})$ and
$\phi$ satisfies the equation

$\displaystyle\frac{\partial\phi}{\partial{}t}=m\left[\phi^{\frac{m-1}{\beta}}\Delta{}\phi+\frac{m-\beta}{\beta}\phi^{\frac{m-\beta-1}{\beta}}|\nabla\phi|^{2}\right]~{}$
(2.17)

in $Q_{+}$ . We next prove that (2.12) is also right in $Q$. To do this, we
first see that (2.7) yields

$\displaystyle\left|u(x,t)-u(x,t_{\ast})\right|=u(x,t)\leq{}\nu\left|t-t_{\ast}\right|^{\frac{1}{2h}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{for}~{}(x,t_{\ast})\in{}Q_{0}.$

Thus,
$|\phi(x,t)-\phi(x,t_{\ast})|=\phi(x,t)\leq{}\nu^{\beta}|t-t_{\ast}|^{\frac{\beta}{2h}}$.
Thus,

$\displaystyle\left|\frac{\phi(x,t)-\phi(x,t_{\ast})}{t-t_{\ast}}\right|\leq\nu^{\beta}\left|t-t_{\ast}\right|^{\frac{\beta{}-2h}{2h}}.$
(2.18)

(2.13) shows the continuity of the function
$\frac{\partial\phi}{\partial{}t}$, specially,
$\frac{\partial{\phi}}{\partial{}t}|_{(x,t_{\ast})}=0$ owing to $\beta>2h$. On
the other hand, similar to (2.5) and (2.8), we can get

$|u(x_{1},...,x_{j},...,x_{n},t)-u(x_{1},...,x_{j\ast},...,x_{n},t)|\leq(C_{1}t)^{-\frac{1}{2h}}|x_{j}-x_{j\ast}|^{\frac{1}{h}}$

for every
$(x_{1},...,x_{j},...,x_{n},t),(x_{1},...,x_{j\ast},...,x_{n},t)\in{}Q$ and
$\left|\frac{\partial{}\phi}{\partial{}x_{i}}\right|\leq{}C_{2}u^{\varepsilon}t^{-\frac{1}{2}}~{}\mbox{for}~{}i=1,2,...,n.$
Thereby, if $(x_{1},...,x_{j\ast},...,x_{n},t)\in{}Q_{0}$, we have

$\displaystyle\left|\frac{\partial{}\phi(x_{1},...,x_{j},...,x_{n},t)}{\partial{}x_{i}}-\frac{\partial{}\phi(x_{1},...,x_{j_{\ast}},...,x_{n},t)}{\partial{}x_{i}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\partial{}\phi(x_{1},...,x_{j},...,x_{n}t)}{\partial{}x_{i}}\right|$
$\displaystyle\leq$ $\displaystyle
C_{2}t^{-\frac{1}{2}}(C_{1}t)^{-\frac{\varepsilon}{2h}}|x_{j}-x_{j\ast}|^{\frac{\varepsilon}{h}}.$

This yields

$\displaystyle\frac{1}{|x_{j}-x_{j\ast}|}\left|\frac{\partial{}\phi(x_{1},...,x_{j},...,x_{n},t)}{\partial{}x_{i}}-\frac{\partial{}\phi(x_{1},...,x_{j_{\ast}},...,x_{n},t)}{\partial{}x_{i}}\right|\leq{}C_{2}(C_{1}t)^{-\frac{\varepsilon}{2h}}|x_{j}-x_{j\ast}|^{\frac{\varepsilon}{h}-1}$

for $i,j=1,2,...,n$. This gives the continuity of the function
$\frac{\partial^{2}{\phi}}{\partial{}x_{i}x_{j}}$ thanks to $\varepsilon>h$.
In particular,

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial^{2}{\phi}}{\partial{}x_{i}x_{j}}=0~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on}~{}Q_{0}$

for $i,j=1,2,...,n$. It follows from $m>1$ that
$\phi^{\frac{m-1}{\beta}}\Delta{}\phi=0$ on $Q_{0}$. Similarly,
$\phi^{\frac{m-\beta-1}{\beta}}|\nabla\phi|^{2}=0$ on $Q_{0}$.

Finally, as previously mentioned above, we deduce that the function
$\phi(x,t)$ satisfies (2.12) on $Q_{0}$. $\square$

As an applications of our Theorem 1, here we give an example to show the large
time behavior on the intrinsic properties of the manifold $S(t)$.

Example ( the first fundamental form on $S(t)$)  In fact, the first
fundamental form on the manifold $S(t)$ is
$(ds)^{2}=\sum\limits_{i,j=1}^{n}g_{ij}dx_{i}dx_{j}$. By (2.10),

$\displaystyle\left|(ds)^{2}-(d\rho)^{2}\right|=(d\phi)^{2}$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{n}{\phi_{x_{i}}}dx_{i}\right)^{2}$
$\displaystyle\leq$ $\displaystyle
C_{4}t^{-\frac{2n\varepsilon}{n(m-1)+2}-1}(d\rho)^{2}.$

where $(d\rho)^{2}=\sum\limits_{i,j=1}^{n}dx_{i}^{2}$ is just the first
fundamental form on $\mathbb{R}^{n}$. Thus,

$\displaystyle\frac{(ds)^{2}}{(d\rho)^{2}}=1+O\left(t^{t^{-\frac{2n\varepsilon}{n(m-1)+2}-1}}\right)$
(2.19)

when $t$ is large sufficiently.

##  3 The proof of Theorem 2

To prove Theorem 2, we need to establish a more precise Poincar$\acute{e}$
inequality. It is well-known that if $\lambda_{1}$ is the minimum positive
eigenvalue and $\psi_{1}$ is the corresponding eigenfunction of the Dirichlet
problem

$\displaystyle\left\\{\begin{array}[]{ll}\Delta{}u=-\lambda{}u~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{in}~{}\Omega,\\\
u=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{on}~{}\partial\Omega,\end{array}\right.$
(3.3)

then
${\lambda_{1}}\|\psi_{1}\|^{2}_{L^{2}(\Omega)}=\|\nabla{}\psi_{1}\|^{2}_{L^{2}(\Omega)}$,
where $\Omega$ is a boundary domain in $\mathbb{R}^{n}$. Moreover, if
$\psi\in{}H_{0}^{1}(\Omega)$, then Poincar$\acute{e}$ inequality claims that
there exists a positive constant $k$ such that
$k\|\psi\|^{2}_{L^{2}(\Omega)}\leq{}\|\nabla{}\psi\|^{2}_{L^{2}(\Omega)}$.
According to Qiu (see p.98 in [21]), $\lambda_{1}$ is the maximum of all such
$k$. We know that there are many kinds of choice for such $k$. For example, Wu
(see p.13 in [23]) proved that

$\displaystyle k\leq{}\rho^{-2}$

if
$\Omega=\\{(x_{1},x_{2},...,x_{N})\in\mathbb{R}^{N}:~{}a_{i}<x_{i}<a_{i}+\rho~{}\\}$.
In order to prove Theorem 2 we need to show that such choice is also right if
$\Omega$ is a sphere of $\mathbb{R}^{n}$.

Denote

$B=\\{x\in\mathbb{R}^{n}:~{}|x-x_{0}|<\rho\\}$

for $x_{0}\in\mathbb{R}^{n}$ and $\rho>0$. We have the following result.

Lemma 2  If $\Omega\subset{}B$, $u\in{}H^{1}(\Omega)$ and $u(x)=0$ for
$x\in\partial{\Omega}$. Then

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\|u\|_{L^{2}(\Omega)}\leq{}\rho\|\nabla{}u\|_{L^{2}(\Omega)}~{}.$
(3.4)

Proof  We first suppose $u\in{}C_{0}^{\infty}(\Omega)$. For every
$x\in{}\Omega$, there is a $x_{{}_{\ast}}\in{}\partial{}\Omega$, such that the
three points $x_{0},~{}x$ and $x_{\ast}$ lie on a radius
$\overline{x_{0}~{}x_{\ast}}$. Denote the vector from $x_{\ast}$ to $x$ by
$r$. We have

$\displaystyle u(x)$ $\displaystyle=$ $\displaystyle u(x)-u(x_{\ast})$
$\displaystyle=$
$\displaystyle\int_{x_{\star}}^{x}\frac{\partial{}u(x)}{\partial{}r}dr.~{}~{}$

Using the H$\ddot{o}$lder inequality gets

$\displaystyle|u(x)|^{2}$ $\displaystyle\leq$
$\displaystyle\rho\int_{x_{\ast}}^{x}\left|\frac{\partial{}u(x)}{\partial{}r}\right|^{2}dr.$

Thus,

$\displaystyle\int_{\Omega}|u(x)|^{2}dx$ $\displaystyle\leq$
$\displaystyle\rho\int_{\Omega}\int_{x_{\ast}}^{x}\left|\frac{\partial{}u(x)}{\partial{}r}\right|^{2}dr{}dx$
$\displaystyle\leq$
$\displaystyle\rho^{2}\int_{\Omega}|\nabla{}u(x)|^{2}dx.~{}$

The general case is done by approximation. $\square$

To prove (1.20)  Assume $u(x,t)$ be the solution of (1.1), (1.2). Integrating
(1.5) from $t_{1}$ to $t_{2}$ yields
$\ln{}u(x,t_{2})-\ln{}u(x,t_{1})\geq-\frac{1}{m-1}(\ln{}t_{2}-\ln{}t_{1})$ for
$t_{2}>t_{1}$. This means
$u(x,t_{2})\cdot{}t_{2}^{\frac{1}{m-1}}\geq{}{u(x,t_{1})}\cdot{}t_{1}^{\frac{1}{m-1}}$
for $~{}t_{2}>t_{1}\geq 0$. Therefore,

$\displaystyle~{}\left\\{\begin{array}[]{ll}if~{}u(x_{0},t_{0})=0,~{}then~{}u(x_{0},t)=0~{}~{}for~{}every~{}~{}0\leq{}t<t_{0};\\\
if~{}u(x_{0},t_{0})>0,~{}then~{}u(x_{0},t)>0~{}~{}for~{}every~{}~{}t>t_{0}.\end{array}\right.$
(3.7)

By (3.2), we see that

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}H_{u}(t)\supset{}B_{\delta}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t>0.$

Absolutely, the proof is finished if $\sup\limits_{x\in{}H_{u}(t)}|x|=\infty$,
otherwise, we set

$K^{\prime}=\gamma+\sup\limits_{x\in{}H_{u}(t)}|x|$

for $\gamma>0$. Thus,

$\displaystyle
u(x,t)=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{for}~{}x\in\mathbb{R}^{n}-B_{K^{\prime}}.$
(3.8)

It follows from (1.5) that

$\displaystyle\int_{B_{K^{\prime}}}u^{m}\Delta{}u^{m}dx\geq{}-\frac{1}{(m-1)t}\int_{B_{K^{\prime}}}u^{1+m}dx,$

so that

$\displaystyle\int_{B_{K^{\prime}}}|\nabla{}u^{m}|^{2}dx\leq{}\frac{1}{(m-1)t}\int_{B_{K^{\prime}}}u^{1+m}dx.$

Using (3.1) in this inequality, we obtain

$\displaystyle\int_{B_{K^{\prime}}}u^{2m}dx\leq{}\frac{K^{\prime
2}}{(m-1)t}\int_{B_{K^{\prime}}}u^{1+m}dx.$

Employing the H$\ddot{o}$lder inverse inequality, we have

$\displaystyle\int_{B_{K^{\prime}}}u^{2m}dx\geq\left(\int_{B_{K^{\prime}}}u^{1+m}dx\right)^{\frac{2m}{1+m}}|B_{K^{\prime}}|^{\frac{1-m}{1+m}},$

where $|B_{K^{\prime}}|$ is the volume of $B_{K^{\prime}}$ and
$|B_{K^{\prime}}|=\pi^{\frac{n}{2}}\Gamma(1+\frac{n}{2})^{-1}K^{\prime n}$. So
we get

$\displaystyle\left(\int_{B_{K^{\prime}}}u^{1+m}dx\right)^{\frac{m-1}{1+m}}|B_{K^{\prime}}|^{\frac{1-m}{1+m}}\leq\frac{K^{\prime
2}}{(m-1)t}.$ (3.9)

Using the H$\ddot{o}$lder inverse inequality again, we have

$\displaystyle\int_{B_{K^{\prime}}}u^{1+m}dx$ $\displaystyle\geq$
$\displaystyle\left(\int_{B_{K^{\prime}}}udx\right)^{1+m}|B_{K^{\prime}}|^{-m}.$
(3.10)

It follows from (1.4) and (3.3) that
$\int_{\mathbb{R}^{n}}u(x,t)dx=\int_{B_{K^{\prime}}}u(x,t)dx=\int_{\mathbb{R}^{n}}u_{0}(x)dx$.
Combining (3.4) and (3.5) we get

$\displaystyle\left(\int_{\mathbb{R}^{n}}u_{0}dx\right)^{m-1}|B_{K^{\prime}}|^{1-m}\leq\frac{K^{\prime
2}}{(m-1)t}.$ (3.11)

Now we get

$\displaystyle(m-1)\left(\int_{\mathbb{R}^{n}}u_{0}dx\right)^{m-1}t\leq
K^{\prime
2+(m-1)n}\cdot\pi^{\frac{(m-1)n}{2}}\cdot\left(\Gamma(1+\frac{n}{2})\right)^{1-m}.$

Letting $\gamma\longrightarrow 0$ gives

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sup_{x\in{}H_{u}(t)}|x|\geq{}\chi(t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}t>0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$

To prove (1.21)  Assume $u(x,t)$ and $v(x,t)$ be the solutions to (1.1) and
(1.8) respectively. Employing the well-known result (see [13]), we have

$\displaystyle\frac{1}{1+m}\int_{\mathbb{R}^{n}}u^{1+m}(x,T)dx+\int_{Q_{T}}|\nabla{}u^{m}|^{2}dxdt\leq\frac{1}{1+m}\int_{\mathbb{R}^{n}}u_{0}^{1+m}dx$
(3.12)

and

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{n}}v^{2}(x,T)dx+\int_{Q_{T}}|\nabla{}v|^{2}dxdt\leq\frac{1}{2}\int_{\mathbb{R}^{n}}u_{0}^{2}dx$
(3.13)

for every given $T>0$ and $Q_{T}=\mathbb{R}^{n}\times(0,T)$. Let

$\displaystyle G$ $\displaystyle=$ $\displaystyle v-u_{\eta}^{m},$
$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\int_{T}^{t}Gd\tau~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0<t<T,$

where $u_{\eta}$ are the solutions of (1.9). Let $\\{\zeta_{k}\\}_{k>1}$ be a
smooth cutoff sequence with the following properties:
$\zeta_{k}(x)\in{}C_{0}^{\infty}(\mathbb{R}^{n})$ and

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\zeta_{k}(x)=\left\\{\begin{array}[]{ll}1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|x|\leq{}k,\\\
0<\zeta_{k}(x)<1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}k<|x|<2k,\\\
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|x|\geq{}2k.\end{array}\right.$
(3.17)

Clearly, there is a positive constant $\gamma$ such that

$\displaystyle|\nabla{}\zeta_{k}|\leq\frac{\gamma}{k}~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}|\Delta{}\zeta_{k}|\leq\frac{\gamma}{k^{2}}.$
(3.18)

Recalling $(v-u_{\eta})_{t}=\Delta{}G$ in $Q_{T}$, we multiply the equation by
$\psi\zeta_{k}$ and integrate by parts in $Q_{T}$, we obtain

$\displaystyle\int_{Q_{T}}(\zeta_{k}\nabla{}G\cdot\nabla{}\psi{}+\psi\nabla{}G\cdot\nabla{}\zeta_{k})dxdt=\int_{Q_{T}}(v-u_{\eta})G\zeta_{k}dxdt.$
(3.19)

Differentiating (3.10) with respect to $T$, we get

$\displaystyle\int_{\mathbb{R}^{n}}(v-u_{\eta})G\zeta_{k}dx$ $\displaystyle=$
$\displaystyle-\int_{Q_{T}}\left(\zeta_{k}|\nabla{}G|^{2}+G\nabla{}G\cdot\nabla{}\zeta_{k}\right)dxdt$
(3.20) $\displaystyle\leq$
$\displaystyle-\frac{1}{2}\int_{Q_{T}}\nabla{}G^{2}\cdot\nabla{}\zeta_{k}dxdt.$

Letting $\eta\longrightarrow 0$ in (3.11), we get

$\displaystyle\int_{\mathbb{R}^{n}}(v-u)(v-u^{m})\zeta_{k}dx$
$\displaystyle\leq$ $\displaystyle\frac{\varphi}{k}$ (3.21)

for some positive $\varphi=\varphi(T)$ thanks to (3.7), (3.8) and (3.9). On
the other hand,

$\displaystyle\int_{\mathbb{R}^{n}}(v-u)G\zeta_{k}dx=\int_{\mathbb{R}^{n}}(v-u)^{2}\zeta_{k}dx+\int_{\mathbb{R}^{n}}(v-u)(u-u^{m})\zeta_{k}dxdt.$

Now we conclude that

$\displaystyle\int_{\mathbb{R}^{n}}(v-u)^{2}\zeta_{k}dx$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{n}}|v-u|\cdot|u-u^{m}|\zeta_{k}dx+\frac{\varphi}{k}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{n}}\left[(v-u)^{2}+(u-u^{m})^{2}\right]\zeta_{k}dx+\frac{\varphi}{k}.$

This implies

$\displaystyle\int_{\mathbb{R}^{n}}(v-u)^{2}\zeta_{k}dx$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{n}}(u-u^{m})^{2}\zeta_{k}dx+\frac{2\varphi}{k}$
$\displaystyle\leq$
$\displaystyle(m-1)\int_{\mathbb{R}^{n}}\xi^{m-1}|u-u^{m}|\zeta_{k}dx+\frac{2\varphi}{k}$
$\displaystyle\leq$
$\displaystyle(m-1)M^{m-1}\int_{\mathbb{R}^{n}}|u-u^{m}|\zeta_{k}dx+\frac{2\varphi}{k}.$

Recalling the definition of $\zeta_{k}$, we see
$\int_{\mathbb{R}^{n}}(v-u)^{2}\zeta_{k}dx\geq\int_{|x|\leq{}k}(v-u)^{2}dx$.
Moreover, $\int_{\mathbb{R}^{n}}|u-u^{m}|\zeta_{k}dx$ is bounded thanks to
(1.3) and (1.4). Hence, we can get a positive constant $C_{\ast}=C_{\ast}(T)$
such that

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\int_{|x|\leq{}k}\left[v(x,t)-u(x,t)\right]^{2}dx$
$\displaystyle\leq$ $\displaystyle C_{\ast}\left[(m-1)+\frac{1}{k}\right]$

with respect to $t\in(0,T)$ uniformly. $\square$

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