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.


Introduction
Consider the Cauchy problem of nonlinear parabolic equation where Q = R n × R + , n ≥ 1, m > 1 and 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, for any continuously differentiable function f (x, t) with compact support in R n × (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: where j = 1, 2, 3... and v is the solution to the Cauchy problem of linear heat equation with the same initial value v t = ∆v in Q, v(x, 0) = u 0 (x) on R n . (1.8) Moreover, the solution u(x, t) can be obtained (see [9,15]) as a limit of solutions u η (η −→ 0 + ) of the Cauchy problem u t = ∆u m in Q, u(x, 0) = u 0 (x) + η on R n , (1.9) and the solutions u η 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 ∈ C(Q) and u ≥ 0; J. L. Vazquez (see Proposition 6 in Ch.2 of [13]) proved u ∈ C ∞ (Q + ), where 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 ∈ C ∞ (Q + ) with more details. Therefore, we can divide the space-time R n × R + into two parts: Furthermore, if Q 0 contains an open set, say, Q 1 , we can also obtain u(x, t) ∈ C ∞ (Q 1 ) owing to u(x, t) ≡ 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 is continuous in R 1 ×(0, +∞) if the nonnegative initial value satisfies a good condition. Moreover, if the initial value 0 ≤ u 0 ≤ M and u m 0 is Lip-continuous, u(x, t) can be continuous on R 1 × [0, +∞) (see [2]). As to the case of n ≥ 1, L.Caffarelli and A.Friedman (see [15]), proved that the solution u(x, t) to the Cauchy problem (1.1), (1.2) is Hölder continuous in Q : for some 0 < α < 1 and C > 0. Moreover, for the general equation u t = ∇ · (u∇p), p = κ(u), L .Caffarelli and J.L. Vazques ([16]) established the property of finite propagation and the persistence of positivity, where κ 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 = m m−1 u m−1 , they claimed V = Cr δ at the fusing time, where 0 < δ < 1, C is a positive constant. Moreover, 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 ∈ (m − 1, m), there exists a C > 0 such that where, We see that the range of 1 h is (0, 1] not (0, 1), thereby, the older Hölder estimate ( * ) is improved by (1.11).
Moreover, we will show that the functions ∂u β ∂x i are continuous if β is large sufficiently because we can employ (1.11) to obtain for another positive constant C. By this inequality, we want to get the continuous partial derivatives ∂u β ∂t and ∂u β ∂x i i = 1, 2, ..., n, that is to say, for every given t > 0, where φ(x, t) = u β (x, t). In particular, we will prove that the function φ(x, t) satisfies the degenerate parabolic equation in the classical sense.
For every fixed t > 0, we define a n-dimensional surface S(t), which floats in the space R n+1 with the time t: where the function φ(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 fourthorder 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 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 R n .
Let          g 1 = (1, 0, ..., ∂φ ∂x 1 ), g 2 = (0, 1, ..., ∂φ ∂x 2 ), ......, g n = (0, 0, ...1, ∂φ ∂xn ). (1.13) Define the Riemannian metric on S(t): where g ij = g i · g j . Clearly, If the derivatives ∂φ ∂x i are bounded for i = 1, 2, ..., n, then we can get a positive constant C, such that (1.14) As a consequence of (1.14), we see that the completeness of 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 It is well-known that the function dξ 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, 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 is the equation in (1.1) with a initial mass M δ(x), and We see that the function B(x, t; C) has compact support in space for every fixed time. More when t is large enough (see Proposition 17 in [13]). Comparing (1.16) and (1.18) and recalling the mass conservation R n u(x, t)dx = R n u 0 (x)dx = R n v(x, t)dx, we will prove that the solution continuously depends on the nonlinearity of the equation (1.1): We read the main conclusions of the present work as follows: (1) for every given τ > 0, K > 0 , there exists a positive ν such that (2) for every β > h, the function φ = (u(x, t)) β ∈ C 1 (Q) and the surface φ = φ(x, t) is a complete Riemannian-manifold which is tangent to R n on ∂H u (t) for every fixed t > 0, h is defined by (1.19); (3) if β > 2h, the function φ(x, t) satisfies the degenerate parabolic equation in the classical sense in Q. where, Moreover, for every given T > 0, there is a positive Let us also note that the manifolds S(t) and R n are two surfaces in R n+1 and the Cauchy problem (1.1), (1.2) can be regarded as a mapping Φ(t) : R n −→ 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 in the sense of distributions in Q.
Proof: We first prove (2.1) for the classical solutions u η (x, t). Set Differentiating this equation with respect to x j and multiplying though by ∂V ∂x j , letting h j = ∂V ∂x j , we get for j = 1, 2, ..., n. Setting we obtain we get Taking h * 1 = ( 1 2 , h * i = 0 for i = 2, 3, ..., n, and setting Q 2 , we see that the function Q 2 * is a solution to the equation with the initial condition Q 2 * (0) = +∞. On the other hand, let u m η = K, then K is the solution to the Cauchy problem of the linear parabolic equation ) is a known function, which is the classical solution to (1.9) and η ≤ u η ≤ M + η. By Theorem 5.1 in [18], we get K ∈ H 2+α,1+ α 2 (R n × (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 0η ). Therefore, ∆K, ∂K ∂t and ∂K ∂x i (i = 1, 2, ..., n) are bounded. To be exact, there is a positive µ 0 , which may depend on η, such that |∆K| + | ∂K ∂t | + |∇K| ≤ µ 0 . Therefore, Similarly, we can get positive C ′′ , which may depend on η also, such that |∆V | ≤ C ′′ . Now we can employing the comparison theorem and get Setting h = m q in (2.2) yields h ∈ (m − 1, m), and (2.1) follows.
To prove Theorem 1, we need to show an ordinary inequality firstly: In fact, (2.3) is right for a = b. If a > b, we can easily get the following inequalities:

So (2.3) holds for a > b ≥ 0. Certainly, (2.3) is also right when 0 ≤ a < b.
We are now in a position to establish our Theorem 1.
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 = n i,j=1 g ij dx i dx j . By (2.10), i is just the first fundamental form on R n . Thus, when t is large sufficiently.

The proof of Theorem 2
To prove Theorem 2, we need to establish a more precise Poincaré inequality. It is wellknown that if λ 1 is the minimum positive eigenvalue and ψ 1 is the corresponding eigenfunction of the Dirichlet problem ∆u = −λu in Ω, u = 0 on ∂Ω, where Ω is a boundary domain in R n . Moreover, if ψ ∈ H 1 0 (Ω), then Poincaré inequality claims that there exists a positive constant k such that k ψ 2 L 2 (Ω) ≤ ∇ψ 2 L 2 (Ω) . According to Qiu (see p.98 in [21]), λ 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 In order to prove Theorem 2 we need to show that such choice is also right if Ω is a sphere of R n .
for x 0 ∈ R n and ρ > 0. We have the following result.

Lemma 2
If Ω ⊂ B, u ∈ H 1 (Ω) and u(x) = 0 for x ∈ ∂Ω. Then Proof We first suppose u ∈ C ∞ 0 (Ω). For every x ∈ Ω, there is a x * ∈ ∂Ω, such that the three points x 0 , x and x * lie on a radius x 0 x * . Denote the vector from x * to x by r. We have Using the Hölder inequality gets Thus, The general case is done by approximation.