Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

In this paper, we investigate the life-span of classical solutions to hyperbolic inversemean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.


Introduction
In this paper, we study hyperbolic inverse mean curvature flow (HIMCF): where k denotes the mean curvature of the curve c and N → and T → are, respectively, the unit inner normal and tangent vectors of the curve c (·, t). c 0 stands for a smooth strictly convex closed curve, c 1 denotes the initial velocity of c 0 , and N → 0 is the unit inner normal vector of c 0 .
Definition 1 (see [1]). A flow is called a normal preserving flow if the normal vector of the curve is independent of time dN → /dt � 0, i.e., c ts , N → � 0 holds for all time.
It is well known that inverse mean curvature flow is an important method to derive the energy estimates in general relativity; for example, Huisken and Ilmanen developed a theory of weak solutions of the inverse mean curvature flow and used it to prove successfully Riemannian Penrose inequality which plays an important role in general relativity (see [2]). In [3], Urbas proved that, for inverse mean curvature flow, the surfaces stay strictly convex and smooth for all time. Furthermore, the surfaces become more and more spherical in the process. Similar results have also been obtained for star-shaped initial data with the positive mean curvature of the surface, see Urbas [3] and Gerhardt [4]. In [2], Huisken and Ilmanen proved a sharp lower bound of mean curvature, from which they proved that if the initial surface is the boundary of a strictly star-shaped domain and has nonnegative mean curvature, a smooth solution of the inverse mean flow will exist for all time and converge to a manifold. e hyperbolic mean curvature flow, i.e., the hyperbolic version of mean curvature flow, has been introduced by some authors, see Gurtin and Podio-Guidugli [5], He et al. [6], Kong et al. [7,8], Lefloch and Smoczyk [9], Notz [10], Rotstein et al. [11], and Wang [12,13]. In fact, Yau in [14] has suggested the following equation related to a vibrating membrane or the motion of a surface: where H is the mean curvature and N → is the unit inner normal vector of the surface and pointed out that very little is known about the global time behavior of the hypersurfaces. Recently, in [1], Chou and Wo proposed a new hyperbolic curvature flow for convex hypersurfaces. is flow is most suited when the Gauss curvature is involved. e equation satisfied by the graph of the hypersurface under this flow gives rise to a new class of fully nonlinear Euclidean invariant hyperbolic equations. Finally, they consider the expanding Gauss curvature flow driving by negative powers − k − β of the Gauss curvature. In the special case β = 1, the expanding flow becomes, once written in terms of the support function S(θ, t), a linear problem: where S(θ, t) � {c(θ, t), (cos θ, sin θ)}, and the normal angle θ is determined modulo 2π. ey get the following result.
Proposition 1 (see [1]). Consider (3) where the initial values are smooth and satisfy ϕ θθ + ϕ > 0 and ψ θθ + ψ > 0. en, the flow remains smooth and expands to infinity like a circle. Motivated by the inverse mean curvature flow and the local solution theory of the Cauchy problem for the quasilinear hyperbolic system (see [15]), we will focus on life span (the maximum existence time of unique local classical solutions) of classical solutions to the Cauchy problem for the hyperbolic inverse mean curvature flow. Our first result is the following local existence theorem for initial value problem (1). Theorem 1 (local existences and uniqueness). Suppose that c 0 is a smooth, strictly convex curve, and c 1 : S 1 ⟶ R 2 is a smooth vector function. en, there exist positive T and a family of strictly convex closed curves c(·, t) with t ∈ [0, T) such that c(·, t) satisfies (1).
The following theorem concerns the life-span of local (in space) smooth solutions of flow (1) that can be written as convex graphs over an interval R ⊂ R in the form Writing the initial conditions as for some f ∈ C 2 (R) and g ∈ C 1 (R), we obtain the following result.

Theorem 2.
Assume that c 0 is a strictly convex closed curve and c 1 : S 1 ⟶ R 2 is a smooth vector function on S 1 . en, a lower bound δ * for the maximum time T of existence of a solution of (1) in the form G is given by where ϕ � (ϕ 1 , . . . , The paper is organized as follows. In Section 2, we derive a hyperbolic Monge-Ampère equation by the hyperbolic inverse mean curvature flow and give the short-time existence theorem, i.e., eorem 1. In Section 3, we reduce the hyperbolic Monge-Ampère equation to a quasilinear hyperbolic system in Riemann invariants. Section 4 is devoted to prove the main result, i.e., eorem 2, by the local solution of theory for the Cauchy problem of the quasilinear hyperbolic system.

Hyperbolic Monge-Ampère Equation
Suppose for z ∈ (a, b) and t ∈ (t 0 , t 1 ), the curve c(z, t) can be expressed in the form of a graph (x, u(x, t)), and we have Taking the inner product with the choice It would be convenient to write the expression of the curvature k for graphs: Flow (1) is a normal preserving flow. First, we need to compute z t c tz , N → . Using the identities 2 Discrete Dynamics in Nature and Society one computes is is an ODE of the form dy/dt � by. Clearly, (1) is a normal preserving flow. en, having observed that also the tangent vector is a constant, by using it, we can easily to show the identity By x t � − (u xt /u xx ), the normal preserving flow is reducible, and the curvature k for graphs u xx /(1 + u 2 x ) 3/2 , graph form of flow (1), satisfies the following equation: is is a fully nonlinear hyperbolic Monge-Ampère equation as long as the curve is uniformly convex, i.e., flow (1) reduces to .
Remark 1. We have expressed flow (1) as the equation of support function in (3). When the curve c is represented as a graph and described by the support function simultaneously, the following relations hold: For an unknown function h � h(x, t) defined on (x, t) ∈ R 2 , Monge-Ampère equation is where the coefficients A, B, C, D, and E depend on t, x, h, h t , h x . We say that (18) is a hyperbolic Monge-Ampère equation if Initial conditions for the Cauchy problem on the Ox axis are where h 0 ∈ C 3 (R 1 ) and h 1 ∈ C 2 R 1 . Suppose h 0 and h 1 satisfy the next two conditions. First, the axis Ox is free, i.e., Secondly, on the axis Ox, the hyperbolic condition holds.
It is easy to prove that (15) is a hyperbolic Monge-Ampère equation, where In fact, Furthermore, we suppose that g(x) is the third continuous differential function and f(x) is second continuous differential function; then, initial conditions satisfy Hence, by the standard theory of hyperbolic equations (see [16]), we have local existences and uniqueness theorem (i.e., eorem 1).

Remark 2.
In fact, the derivation of (16) and eorem 1 can also be obtained by a direct application of the arguments therein (cf. Section 1 in [1]). When establishing the local Discrete Dynamics in Nature and Society solvability for the normal preserving flow, the authors in [1] suppose the initial curve c 0 ∈ H k (S 1 ) and c t (0) ∈ H k− 1 (S 1 ), k > 5/2; however, we just provide that c 0 and c t (0) are smooth.

Systems in Riemann Invariants
is section is concerned with the reduction of (15). Let u(x, t) be a C 3 solution of equation (15) in some domain D. Suppose Condition (26) means that vertical lines t � const are free. By definition, put e functions r and s are tangents of angles of inclinations of characteristics of equation (15), and we always call them Riemann invariants of (15). Let p � u t and q � u x ; then, (15). Suppose (26) and (15) Let D be a domain in which this solution is defined. Suppose holds. en, u(x, t) be a C 3 solution of (15) in the domain D, and furthermore, u t � p, u x � q, and (26) holds.
Then, we reduce equation (28) to the following Cauchy problem for a quasilinear hyperbolic system in terms of Riemann invariants: Throughout this paper, we shall use the following notation: the absolute value of any vector u � (u 1 , . . ., u 5 ) is defined as and the C 0 norm of a vector function u � u(t, x) � (u 1 (t, x), . . ., u n (t, x)) on a bounded domain R is defined as Firstly, suppose ϕ � (ϕ 1 , . . . , ϕ 5 ) T is bounded on R and ϕ ∈ C 1 (R). Take a suitable positive constant Ω such that For the continuous differentiable vector function u(t, x), we may define Furthermore, for any finite set of functions set Γ ⊂ C 1 (R) or Γ ⊂ C 1 (R × [0, δ]), and we can similarly define its norm ‖Γ‖, ‖Γ‖ 1 . Denote where R is a bounded domain, and define its exterior domain 4 Discrete Dynamics in Nature and Society x) ∈ R(δ 0 ), |u| ≤ Ω}. Furthermore, suppose λ i , (zλ i /zx), (zλ i /zu j ), μ i , (zμ i /zx), (zμ i /zu j ) are C 0 functions defined on E(δ 0 ), by the standard hyperbolic theory, there exists a constant 0 < δ * ≤ δ 0 such that the Cauchy problem of systems (28) and (29) has an unique C 1 solution on R(δ * ).

The Proof of Theorem 2
In this section, we consider the life-span of the solution by the construct method of the local solution for the Cauchy problem of the quasilinear hyperbolic system. To prove this theorem, we first consider the initial value problem of the equation for the graph (x, u(x, t)) of the curve. Since the initial curve c 0 is a smooth curve, we can find open intervals J α , α � 1, . . ., N, in different coordinates and smooth functions u α 0 defined on J α such that c 0 can be described as the union of the graphs x, u α 0 (x): x ∈ J α , α � 1, . . ., N. Assume u α is the solution with initial value u α 0 for each α. Note that u(x, t) ≡ u α satisfies (16).
e following is about the proof of eorem 2. First, we consider quasilinear hyperbolic system (28) to be linear. Take a suitable positive constant Ω 1 such that Here, Ω 1 is a positive constant derived from second a priori estimate, _ ϕ � dϕ/dx. Suppose the following function set and for each ] ∈ (δ|Ω, Ω 1 ), by the linear hyperbolic system, and the initial conditions Defining the function set Γ * � λ l , (zλ l /zx), (zλ l /z u j ), μ l , (zμ l /zx), (zμ l /zu j ), l, j � 1, . . . , 5}, we get the C 0 norm estimates of the set Γ * : max l�1,...,5 Hence, the C 0 norm of the set Γ * can be bounded as follows: we have Discrete Dynamics in Nature and Society erefore, we can bound the C 0 norm of the set Γ 2 []] as follows: In order to prove eorem 2, we will get some priori estimates for the solutions of Cauchy problems for linear hyperbolic systems. ese estimates will be useful in proving the existence and uniqueness of solutions for the initial value problem of quasilinear hyperbolic systems.
Firstly, by the first associated integral relations of initial value problem (39), Hereafter, the argument of each function in the l-th integrand is (τ, ξ) � (τ, ω l (τ; t, x)), and d/d l τ means the directional derivative along the l-th characteristic curve, i.e., Moreover, where (δ kl (x, t)) is the inverse matrix of (δ lj (t, x)), and each δ kl (x, t) is also a C 1 function of (t, x). By max k�1,...,5 is is first a priori estimate for the solution of Cauchy problem.
We are going to construct second a priori estimate for the first derivatives of the solutions of Cauchy problem. Define the function set (53) By the second associated integral, (57) Let obviously; we have (60) By Gronwall inequality, we have By (62), we have 6 Discrete Dynamics in Nature and Society i.e., By the definition of ‖u‖ 1 , Without loss generality, we assume δ 0 ≤ (1/Ω 1 ), and we restrict to δ ∈ (0, δ 0 ); then, By Taylor's expansion, Plugging (68) into (66) and simplifying the second order term of δ, is is second estimate for the first derivatives of the solution of the Cauchy problem.
Next, we are going to estimate the modulus of continuity of the first-order derivatives of the solution. Let ψ(t, x) be a function on the bounded domain R. e module of continuity of ψ(t, x) is defined by the following nonnegative function: Similarly, we can define the module of continuity of a function with any number of independent variables. Also, the modulus of continuity of a vector function ψ � (ψ i (t, x))(i � 1, . . ., n) or a set of functions Γ � {ψ i } is defined as follows: It is easy to see that the modulus of continuity possesses the properties stated in Lemmas 1 and 2.
Lemma 1 (see [15]). It holds that (72) f is a continuous function on [t 1 , f is Hölder continuous on[t 1 , t 2 ] ⊂ R with exponent α(0 < α ≤ 1) iff there exists a constant L such that for any natural number N and any positive constant C, we have where [C] denotes the integer part of C.
Discrete Dynamics in Nature and Society