Wave equations associated to Liouville-type problems: global existence in time and blow up criteria

We are concerned with wave equations associated to some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated to the latter problems and second, to substantially refine the analysis initiated in [11] concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the sub-critical case and we give general blow up criteria for the super-critical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser-Trudinger inequality.


INTRODUCTION
In this paper we are concerned with wave equations associated to some Liouvilletype problems on compact surfaces arising in mathematical physics: sinh-Gordon equation (1.1) and some general Toda systems (1.7). The first wave equation we consider is with u : R + × M → R, where (M, g) is a compact Riemann surface with total area |M| and metric g, ∆ g is the Laplace-Beltrami operator and ρ 1 , ρ 2 are two real parameters. Nonlinear evolution equations have been extensively studied in the literature due to their many applications in physics, biology, chemistry, geometry and so on. In particular, the sinh-Gordon model (1.1) has been applied to a wide class of mathematical physics problems such as quantum field theories, non commutative field theories, fluid dynamics, kink dynamics, solid state physics, nonlinear optics and we refer to [1,7,12,14,37,38,48,52] and the references therein.
The stationary equation related to (1.1) is the following sinh-Gordon equation: In mathematical physics the latter equation describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices, see [29,42]. For more discussions concerning the physical background we refer for example to [13,33,36,39,40] and the references therein. On the other hand, the case ρ 1 = ρ 2 has a close relationship with constant mean curvature surfaces, see [49,50].
Observe that for ρ 2 = 0 equation (1.2) reduces to the following well-know mean field equation: which has been extensively studied in the literature since it is related to the prescribed Gaussian curvature problem [3,44] and Euler flows [8,30]. There are by now many results concerning (1.3) and we refer to the survey [46]. On the other hand, the wave equation associated to (1.3) for M = S 2 , that is was recently considered in [11], where the authors obtained some existence results and a first blow up criterion. Let us focus for a moment on the blow up analysis. They showed that in the critical case ρ = 8π for the finite time blow up solutions to (1.4) there exist a sequence t k → T − 0 < +∞ and a point x 1 ∈ S 2 such that for any ε > 0, lim k→+∞´B (x 1 ,ε) e u(t k ,·) S 2 e u(t k ,·) ≥ 1 − ε, (1.5) i.e. the measure e u(t k ) (after normalization) concentrates around one point on S 2 (i.e. it resembles a one bubble). On the other hand, for the general super-critical case ρ > 8π the blow up analysis is not carried out and we are missing the blow up criteria. One of our aims is to substantially refine the latter analysis and to give general blow up criteria, see Corollary 1.5. As a matter of fact, this will follow by the analysis we will develop for more general problems: the sinh-Gordon equation (1.1) and Toda systems (1.7).
Let us return now to the sinh-Gordon equation (1.2) and its associated wave equation (1.1). In the last decades the analysis concerning (1.3) was generalized for treating the sinh-Gordon equation (1.2) and we refer to [2,23,24,28,40] for blow up analysis, to [17] for uniqueness aspects and to [5,18,19,20] for what concerns existence results. On the other hand, for what concerns the wave equation associated to (1.2), i.e. (1.1), there are few results mainly focusing on traveling wave solutions, see for example [1,16,38,48,52]. One of our aims is to develop the analysis for (1.1) in the spirit of [11] and to refine it with some new arguments. More precisely, by exploiting the variational analysis derived for equation 1.2, see in particular [5], we will prove global existence in time for (1.1) for the sub-critical case and we will give general blow up criteria for the super-critical and critical case. The sub/super-critical case refers to the sharp constant of the associated Moser-Trudinger inequality, as it will be clear in the sequel.
Before stating the results, let us fix some notation. Given T > 0 and a metric space X we will denote C([0, T]; X) by C T (X). C k T (X) and L k T (X), k ≥ 1, are defined in an analogous way. When we are taking time derivative for t ∈ [0, T] we are implicitly taking right (resp. left) derivative at the endpoint t = 0 (resp. t = T). When it will be clear from the context we will simply write H 1 , L 2 to denote H 1 (M), L 2 (M), respectively, and Our first main result is to show that the initial value problem for (1.1) is locally well-posed in H 1 × L 2 . (1.6)

Remark 1.2.
The assumption on the initial datum u 1 to have zero average guarantees that the average (1.6) of the solution u(t, ·) to (1.1) is preserved in time. A consequence of the latter property is that the energy E(u(t, ·)) given in (3.9) is preserved in time as well, which will be then crucially used in the sequel, see the discussion later on.
The proof is based on a fixed point argument and the standard Moser-Trudinger inequality (2.1), see section 3. Once the local existence is established we address the existence of a global solution to (1.1). Indeed, by exploiting an energy argument jointly with the Moser-Trudinger inequality associated to (1.2), see (2.2), we deduce our second main result. Theorem 1.3. Suppose ρ 1 , ρ 2 < 8π. Then, for any (u 0 , u 1 ) ∈ H 1 (M) × L 2 (M) such that´M u 1 = 0, there exists a unique global solution u ∈ C(R + ; H 1 ) ∩ C 1 (R + ; L 2 ) of (1.1) with initial data (u 0 , u 1 ).
The latter case ρ 1 , ρ 2 < 8π is referred as the sub-critical case in relation to the sharp constant 8π in the Moser-Trudinger inequality (2.2). The critical and supercritical case in which ρ i ≥ 8π for some i is subtler since the solutions to (1.2) might blow up. However, by exploiting the recent analysis concerning (1.2), see in particular [5], based on improved versions of the Moser-Trudinger inequality, see Proposition 2.1, we are able to give quite general blow up criteria for (1.1). Our third main result is the following. Furthermore, if ρ 1 ∈ [8m 1 π, 8(m 1 + 1)π) and ρ 2 ∈ [8m 2 π, 8(m 2 + 1)π) for some m 1 , m 2 ∈ N, then, there exist points {x 1 , . . . , x m 1 } ⊂ M such that for any ε > 0, either or there exist points {y 1 , . . . , y m 2 } ⊂ M such that for any ε > 0, The latter result shows that, once the two parameters ρ 1 , ρ 2 are fixed in a critical or super-critical regime, the finite time blow up of the solutions to (1.1) yields the following alternative: either the measure e u(t k ) (after normalization) concentrates around (at most) m 1 points on M (i.e. it resembles a m 1 -bubble) or e −u(t k ) concentrates around m 2 points on M. We point out this is new for the mean field equation (1.4) as well and generalizes the previous blow up criterion (1.5) obtained in [11] for ρ = 8π. More precisely, the general blow up criteria for the super-critical mean field equation are the following.
Finally, it is worth to point out some possible generalizations of the results so far. Remark 1.6. We may consider the following more general weighted problem It is easy to check that Theorems 1.1, 1.3 and 1.4 extend to this case as well. The same argument applies also to the Toda system (1.7).
On the other hand, motivated by several applications in mathematical physics [1,41,43] we may consider the following asymmetric sinh-Gordon wave equation We next consider the wave equation associated to some general Toda system, where ρ i , i = 1, . . . , n are real parameters and A n = (a ij ) n×n is the following rank n Cartan matrix for SU(n + 1): The stationary equation related to (1.7) is the following Toda system: which has been extensively studied since it has several applications both in mathematical physics and in geometry, for example non-abelian Chern-Simons theory [15,47,51] and holomorphic curves in CP n [6,9,32]. There are by now many results concerning Toda-type systems in particular regarding existence of solutions [5,22,35], blow up analysis [26,31] and classification issues [32].
On the other hand, only partial results concerning the wave equation associated to the Toda system (1.7) were obtained in [11] which we recall here. First, the local well-posedness of (1.7) analogously as in Theorem 1.1 is derived for a general n × n symmetric matrix A n . Second, by assuming A n to be a positive definite symmetric matrix with non-negative entries the authors were able to deduce a global existence result by exploiting a Moser-Trudinger-type inequality suitable for this setting, see [45]. On the other side, no results are available neither for mixed positive and negative entries of the matrix A n (which are relevant in mathematical physics and in geometry, see for example the above Toda system) nor for blow up criteria. Our aim is to complete the latter analysis.
Before stating the results, let us fix some notation for the system setting. We denote the product space as (H 1 (M)) n = H 1 (M) × · · · × H 1 (M). To simplify the notation, to take into account an element (u 1 , . . . , u n ) ∈ (H 1 (M)) n we will rather write H 1 (M) ∋ u : M → (u 1 , . . . , u n ) ∈ R n . With a little abuse of notation we will write´M u when we want to consider the integral of each component u i , i = 1, . . . , n.
Since the local well-posedness of (1.7) is already known from [11], our first result concerns the global existence in time.
The latter result follows by an energy argument and a Moser-Trudinger-type inequality for systems as obtained in [27]. On the other hand, when ρ i ≥ 4π for some i the Moser-Trudinger inequality does not give any control and the solutions of (1.9) might blow up. In the latter case, by exploiting improved versions of the Moser-Trudinger inequality for the system recently derived in [4] we are able to give the following general blow up criteria.  Finally, we have the following possible generalization of the system (1.7). Remark 1.9. We point out that since the improved versions of the Moser-Trudinger inequality in [4] hold for general symmetric, positive definite matrices A n = (a ij ) n×n with non-positive entries outside the diagonal, we can derive similar existence results and blow up criteria as in Theorems 1.7, 1.8, respectively, for this general class of matrices as well. In particular, after some simple transformations (see for example the introduction in [2]) we may treat the following Cartan matrices: which are relevant in mathematical physics, see for example [15]. To simplify the presentation we give the details just for the matrix A n in (1.8).
The paper is organized as follows. In section 2 we collect some useful results and in section 3 we prove the main results of this paper: local well-posedness, global existence and blow up criteria.

PRELIMINARIES
In this section we collect some useful results concerning the stationary sinh-Gordon equation (1.2), Toda system (1.9) and the solutions of wave equations which will be used in the proof of the main results in the next section.
In the sequel the symbol u will denote the average of u, that is Let us start by recalling the well-known Moser-Trudinger inequality For the sinh-Gordon equation (1.2), a similar sharp inequality was obtained in [40], We recall now some of main features concerning the variational analysis of the sinh-Gordon equation (1.2) recently derived in [5], which will be exploited later on. First of all, letting ρ 1 , ρ 2 ∈ R the associated Euler-Lagrange functional for equation (1.2) is given by J ρ 1 ,ρ 2 : Observe that if ρ 1 , ρ 2 ≤ 8π, by (2.2) we readily have for any u ∈ H 1 (M), where C > 0 is a constant independent of u. On the other hand, as soon as ρ i > 8π for some i = 1, 2 the functional J ρ 1 ,ρ 2 is unbounded from below. To treat the latter super-critical case one needs improved versions of the Moser-Trudinger inequality (2.2) which roughly assert that the more the measures e u , e −u are spread over the surface the bigger is the constant in the left hand side of (2.2). More precisely, we have the following result.
Then, for any ε > 0 there exists From the latter result one can deduce that if the J ρ 1 ,ρ 2 (u) is large negative at least one of the two measures e u , e −u has to concentrate around some points of the surface.

Proposition 2.2. ([5])
Suppose ρ i ∈ (8m i π, 8(m i + 1)π) for some m i ∈ N, i = 1, 2 (m i ≥ 1 for some i = 1, 2). Then, for any ε, r > 0 there exists L = L(ε, r) ≫ 1 such that for any u ∈ H 1 (M) with J ρ 1 ,ρ 2 (u) ≤ −L, there are either some m 1 points or some m 2 points {y 1 , . . . , y m 2 } ⊂ M such that We next briefly recall some variational aspects of the stationary Toda system (1.9). Recall the matrix A n in (1.8) and the notation of u introduced before Theorem 1.7 and write ρ = (ρ 1 , . . . , ρ n ). The associated functional for the system (1.9) is given by J ρ : where (a ij ) n×n is the inverse matrix A −1 n of A n . A Moser-Trudinger inequality for (2.4) was obtained in [27], which asserts that for any u ∈ H 1 (M), where C is a constant independent of u, if and only if ρ i ≤ 4π for any i = 1, . . . , n. In particular, if ρ i > 4π for some i = 1, . . . , n the functional J ρ is unbounded from below. As for the sinh-Gordon equation (1.2) we have improved versions of the Moser-Trudinger inequality (2.5) recently derived in [4] (see also [5]) which yield concentration of the measures e u j whenever J ρ (u) is large negative.
on [0, +∞) × M. Recall the notation of C T (X) and u before Theorems 1.1 and 1.7, respectively. Then, the following Duhamel formula holds true.
The same results hold as well if u 0 , u 1 , and f (t, ·) are replaced by u 0 , u 1 and f(t, ·), respectively.

PROOF OF THE MAIN RESULTS
In this section we derive the main results of the paper, that is local well-posedness, global existence and blow up criteria for the wave sinh-Gordon equation ( Proof of Theorem 1.1. Let (u 0 , u 1 ) ∈ H 1 (M) × L 2 (M) be such that´M u 1 = 0. Take T > 0 to be fixed later on. We set and we introduce the space B T given by For u ∈ B T we consider the initial value problem Step 1. We aim to show that v ∈ B T if T is taken sufficiently small. Indeed, still by for some C > 0, where we used u * ≤ R. Similarly, we have Hence, recalling the definition of R in (3.1), by (3.3) and the above estimates we conclude Therefore, If T > 0 is taken sufficiently small, Moreover, observe that if we integrate both sides of (3.2) on M we get Thus, for this choice of T we conclude that v ∈ B T . Therefore, we can define a map

It follows that
Step 2. We next prove that by taking a smaller T if necessary, F is a contraction.
(3.6) Reasoning as before, the first term in the righ hand side of the latter estimate is bounded by for some C > 0, where we used the Hölder inequality. Moreover, we have e u i (s,·) 4 L 4 =ˆM e 4(u i (s,·)−u 1 (s)) e 4I ≤ Ce 4I exp for some C > 0. Using the latter estimate for the second term in (3.7) and the Sobolev inequality for the first term, we can bound (3.7) by On the other hand, by using (3.4), the second term in (3.6) is bounded by where in the last step we used the Sobolev inequality.
The same arguments with suitable adaptations show that the initial value problem (1.1) is locally well-posed so we omit the details. The proof is completed.
We next prove that if the two parameters in (1.1) are taken in a sub-critical regime, then there exists a global solution to the initial value problem associated to (1.1). To this end we will exploit an energy argument jointly with the Moser-Trudinger inequality related to (1.2), see (2.2). For a solution u(t, x) to (1.1) we define its energy as for t ∈ [0, T]. We point out that where J ρ 1 ,ρ 2 is the functional introduced in (2.3). We first show that the latter energy is conserved in time along the solution u.
Proof. We will show that We have (3.10) After integration by parts, the first two terms in the right hand side of the latter equation givê where we have used the fact that u satisfies (1.1). Plugging the latter equation into (3.10) we readily have since´M u(t, ·) =´M u 0 for all t ∈ [0, T], see Theorem 1.1. This concludes the proof.
We can now prove the global existence result for (1.1) in the sub-critical regime ρ 1 , ρ 2 < 8π.
Proof of Theorem 1.3. Suppose ρ 1 , ρ 2 < 8π. Let (u 0 , u 1 ) ∈ H 1 (M) × L 2 (M) be such that´M u 1 = 0 and let u be the solution to (1.1) with initial data (u 0 , u 1 ) obtained in Theorem 1.1. Suppose that u exists in [0, T 0 ). With a little abuse of notation C([0, T 0 ); H 1 ) will be denoted here still by C T 0 (H 1 ). Analogously we will use the notation We claim that for some C > 0 depending only on ρ 1 , ρ 2 and (u 0 , u 1 ). Once the claim is proven we can extend the solution u for a fixed amount of time starting at any t ∈ [0, T 0 ), which in particular implies that the solution u can be extended beyond time T 0 . Repeating the argument we can extend u for any time and obtain a global solution as desired.
On the other hand, to estimate u(t, ·) L 2 we recall that´M u(t, ·) =´M u 0 for all t ∈ [0, T 0 ), see Theorem 1.1, and use the Poincaré inequality to get where C > 0 is independent of u(t, ·). By the latter estimate and (3.14) we readily have (3.11).
Suppose now one of ρ 1 , ρ 2 's is not positive. Suppose without loss of generality ρ 1 ≤ 0. Then, recalling (3.13) and by using the standard Moser-Trudinger inequality (2.1) we have Reasoning as before one can get (3.11).
where (a ij ) n×n is the inverse matrix A −1 n of A n . Analogous computations as in Lemma 3.1 show that the latter energy is conserved in time, i.e. E(u(t, ·)) = E(u(0, ·)) for all t ∈ [0, T].
To prove the global existence in Theorem 1.7 for ρ i < 4π, i = 1, . . . , n, one can then follow the argument of Theorem 1.3 jointly with the Moser-Trudinger inequality associated to the Toda system (1.9), see (2.5).
3.2. Blow up criteria. We next consider the critical/super-critical case in which ρ i ≥ 8π for some i. The fact that the solutions to (1.2) might blow up makes the problem more delicate. By exploiting the analysis introduced in [5], in particular the improved version of the Moser-Trudinger inequality in Proposition 2.1 and the concentration property in Proposition 2.2, we derive the following general blow up criteria for (1.1). We stress this is new for the wave mean field equation (1.4) as well.