LONG TIME EXISTENCE OF HYPERBOLIC RICCI-BOURGUIGNON FLOW ON RIEMANNIAN SURFACES

We consider the hyperbolic Ricci-Bourguignon flow(HRBF ) equation on Riemannian surfaces and we find a sufficient and necessary condition to this flow has global classical solution. Also, we show that the scalar curvature of the solution metric gij convergence to the flat curvature.


Introduction
Let (M, g) be an n-dimensional complete Riemannian manifold with Riemannian metric g ij . The general variation equation was introduced by Kong and Liu ([4]) and called the generalized hyperbolic geometric flow (denoted by HGF). Here F are some smooth functions of the Riemannian metric and its first derivative with respect to t, and we consider R ij as the components of Ricci curvature tensor. Liu and Zhang in ( [8]) have shown that the hyperbolic geometric flow (HGF) has global classical solution on Riemannian surfaces. In this paper, we would like to prove that the global solution of hyperbolic Ricci-Bourguignon flow (HRBF) exists on Riemannian surfaces. The present work investigates the variation of a Riemannian metric g ij on a Riemannian surface M by its Ricci curvature tensor R ij and scalar curvature R under 2020 Mathematics Subject Classification: 58J45, 58J47 Received: 25-06-2019, accepted: 10-05-2020.
the following equation where ρ is a real constant. When ρ = 0, this equation is hyperbolic geometric flow and the global existence and blowup phenomenon of smooth solutions to this flow on Riemannian surface have been investigated in [8]. The Ricci-Bourguignon flow is ∂gij ∂t = −2R ij + 2ρRg ij and the short time existence and uniqueness for solution to the Ricci-Bourguignon flow on [0, T ) were showed by Catino et al ( [1]) for ρ < 1 2(n−1) . This study regards the initial metric as follows on a surface of topological type R 2 , where u 0 (x) is a function from C 2 class with bounded C 2 norm and the following inequality is hold where k and m are positive constants.
Since all the information about curvature is contained in the scalar curvature function R, we can simplify the HRBF equation on this surface. According to our notation, R = 2K, where K denotes Gauss curvature and also the Ricci curvature is given by R ij = 1 2 Rg ij , so the (HRBF) equation simplifies to At least locally the metric for a surface can be written as g ij = u(t, x, y)δ ij , where u(t, x, y) > 0, and δ ij is Kronecker's symbol. Hence, we have as a result, the aforementioned equation (5) reduces to u tt − (1 − 2ρ)∆ ln u = 0. The initial data u 0 (x) depends only on x and not y; thus, we can consider the Cauchy problem as below where u 1 (x) ∈ C 1 with bounded C 1 norm. By using the transformation Kong and Liu in ( [5]) proved a theorem as follows Then, the Cauchy problem (7) admits a unique global solution for all t ∈ R. Moreover, if u 1 (x) ≡ u 0 (x)/ u 0 (x), and there exists a point x 0 ∈ R such that u 0 (x 0 ) < 0, then the Cauchy problem (7) admits a unique classical solution only in [0, T ) × R, where The following theorem will proven without using (6) in our investigate. and Hence, the Cauchy problem (7) has a unique global solution for all t ∈ R. Theorem 1.3. If a point x 0 ∈ R exists, which satisfy or there exists a point x 0 ∈ R, such that thus, the Cauchy problem (7) has a unique classical solution only in [0, T ) × R.
Note. Based on Theorem 1.2, we can conclude the Cauchy problem has a unique smooth solution for all t ∈ R. Besides we can consider the solution metric g ij as below g ij = u(x, t)δ ij for i, j = 1, 2.
(14) We will prove the above mentioned Theorems 1.2 and 1.3, in the subsequent sections (3 and 4, respectively). Moreover, using Theorem 1.2, the following theorem will be proven in Section 5.
Hence, a unique classical solution of (1) is exist as the form (14) for all time. Furthermore, the scalar curvature R(x, t) relates to the solution metric g ij admits where k 1 is a positive constant and independent of t and x.

Preliminaries
In this section we require only to discuss the classical solution on t ≥ 0. The result for t ≤ 0 can be easily obtained.
Suppose that Thus, from the above equations and Cauchy problem (7) we have Eigenvalues of equations (17) can be easily calculated as follows and we have the matrices L(U ) and R(U )(where U = (u, w, v)) of left and right eigenvectors, respectively as below Equation system (17) is a linear degenerate strict hyperbolic system because of Define p and q as follows Lemma 2.1. p and q satisfy the following equations: Proof. By differentiating of the function λ with respect to t and x, λ t and λ x can easily be obtained as, and Therefore, We can prove (22) in the same way as above, and it is obvious that (21) is hold.
For the next lemma, consider λpq.
Lemma 2.2. r and s satisfy Proof. Suppose Hence, by a direct computation we can get Now we can easily prove (23) and (23).
where λ 0 (x) = 1−2ρ u0(x) . Now in following theorem we show that the Cauchy problem (7) has a unique global solution under some conditions.
then, on D(T), Hence, the Cauchy problem (7) has a unique global classical solution on t ≥ 0, by the local existence theorem of the classical solution to quasilinear hyperbolic systems.

Proof of Theorem 1.2
On the basis of the local existence and uniqueness theorems of the classical solutions to the quasilinear hyperbolic systems ( [7]), to prove Theorem (1.2) it suffices to establish uniform a priori estimates of the C 1 norms of p, q and u. We have following lemma from [2,3].
where λ 1 , λ 2 , A and B are continuous functions, and λ 1 ≤ λ 2 . If A and B are both non positive, then For prove Theorem 1.2 we need the following lemma.  (7) and (25), if (10) and (11) hold, then where C > 0 is a constant.
Proof. Along λ 1 characteristics, we can obtain By (25) and (10), p 0 (x) ≥ 0 for all x ∈ R. Therefore, we have p(x, t) ≥ 0. In a similar way we can prove q(x, t) ≥ 0. As a result of these inequalities, we have Hence, by Lemma 3.1 we can easily see that Also, we can get following equality by integrating (21) Thus, we can get to result since p(x, t) ≥ 0 and q(x, t) ≥ 0.
Proof of Theorem 1.2. Now from aforementioned Lemma 3.2 and Theorem 2.3, Theorem 1.2 is obvious.

Proof of Theorem 1.3
The blow-up phenomena of the hyperbolic geometric flow will be discussed in this section. Suppose We have 1 4 By the use of (16), (19) and Lemma 2.1 the following lemma can be proven Observe that at t = 0, set Proof of Theorem 1.3. Without loss of generality, we assume that (12) holds; in the same way we can proceed if (13) holds. As a result of (33) and (34) we have m t − λm x ≤ 0 and n t − λn x ≤ 0. Thus, we can easily see that Notice that u 0 (x) ≥ k > 0, and also from (12) and (35) we have m 0 (x) < 0. Next, the get following equation is obtained from (33) by integrating along λ 1 characteristics. That is, where By (4), (37) and (40), we get We consider three cases.
This imply the system in (7) is meaningless for t ≥ τ 0 , that is, it admits a unique local classical solution.

Proof of Theorem 1.4
In this section, we will study the asymptotic behaviour of the scalar curvature R(x, t).