Global existence for semilinear damped wave equations in relation with the Strauss conjecture

We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.

In this paper, we are interested in the small data global existence of solutions for the Cauchy problem of the following semilinear wave equations with general lower order term, in relation with the Strauss conjecture, posed on nontrapping asymptotically Euclidean (Riemannian) manifolds (1.3) u tt − ∆ g u + µ(t, x)∂ t u + µ j (t, x)∂ j u + µ 0 (t, x)u = F p (u), u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x).
Here, ∆ g = |g| −1 ∂ i g ij |g|∂ j is the standard Laplace-Beltrami operator, with |g| = det(g ij (x)). The nonlinearity F p is assumed to behave like |u| p , more precisely, we assume Concerning lower order terms, we assume . When g = g 0 and µ = µ j = µ 0 = 0, the problem has been extensively investigated and is known as the Strauss conjecture, which was initiated in the work of John [7]. It is known that, in general (with F p (u) = |u| p ), the problem admits small data global existence only if p > p c (n) (see Yordanov-Zhang [26] and Zhou [27]), where the critical power p c (n) is the positive root of equation The global existence for small initial data when p ∈ (p c (n), 1 + 4/(n − 1)) followed in Georgiev-Lindblad-Sogge [3] (see also Tataru [15]). See Wang-Yu [22], Wang [20] for a complete history and recent works for the problem on various space-time manifolds.
On the other hand, there are many recent works concerning the damped wave equations, g = g 0 and µ j = µ 0 = 0, with typical damping term depending only on time For the case β < 1, the damping term is strong enough to make the problem behaves like heat equations and the problem has been well-understood. There are some interesting critical phenomena happening for the scale-invariant case β = 1 and it appears that the critical power is p c (n + µ) for relatively small µ > 0.
For the remaining case, β > 1 (which is also referred as the scattering case), where the damping term is integrable, it is natural to expect that the problem behaves like the nonlinear wave equations without damping term. In a recent work of Lai-Takamura [8], the authors proved blow up results for 1 < p < p c (n), together with upper bound of the lifespan. In particular, it is shown that for 1 < p < p c (n) with n ≥ 2, we have where T ε denotes the lifespan and ε is the size of the initial data. For the critical case, p = p c (n), with g = g 0 + g 2 (x) and general (nonnegative) damping term µ = µ(t) ∈ L 1 , where |g 2 ij | + |∇g 2 ij | ≤ Ce −α(1+|x|) for some α > 0, at the final stage of preparation of the current manuscript, we learned that Wakasa-Yordanov [17] obtained the expected exponential upper bound of the lifespan, T ε ≤ exp(Cε −p(p−1) ) .
In this paper, we are interested in complementing to the blow up results for the scattering case, by proving global existence results on general nontrapping asymptotically Euclidean manifolds. Moreover, in the process, we find that we could handle more general lower order perturbation terms as in (1.3), under the assumption (1.4). The first main theorem of this paper states as follows: For more precise statement, see Theorem 4.1. Concerning the proof, the idea is to adapt the recent approach of using local energy and weighted Strichartz estimates, which has been very successful in the recent resolution of the Strauss conjecture on various space-times, including Schwarzschild/Kerr black-hole space-times ( [2], [4], [9], [12]). In particular, we revisit the proof of [12,Theorem 4.1] to extract the key weighted Strichartz estimates, Lemma 3.2, which, combined with the local energy estimates ( [1], [14], [18], [10]), is good enough to treat the lower order terms in (1.3) in a perturbative way.
When there are no global solutions, it is also interesting to obtain sharp estimates of the lifespan. On this respect, it turns out that our argument could also be adapted to show some of the sharp results, as long as we could add some favorable terms in the desired space-time estimates, which could be used to absorb the lower order term in (1.3). To illustrate the argument, as an example, we prove the following lower bound estimate of the lifespan for the four-dimensional critical problem, which is sharp in general, comparing with (1.7) of [17].
for any initial data which are sufficiently small (of size ε ≪ 1), decaying and regular.
See Theorem 5.1 for more precise statement.
Remark 1. It is remarkable that, in our statement, the damping coefficient µ(t, x) is not required to be nonnegative, which were assumed in both [8] and [17]. Moreover, the authors believe that the nonnegative assumption there are not necessary.
Remark 2. On the two dimensional Euclidean space, i.e., n = 2 with g 1 = g 2 = 0, the idea in this paper can be exploited to show that the weighted Strichartz estimates of [2] and [4] are strong enough to yield small data global existence for Remark 3. Based on the existence results in previous works, as we have illustrated in our theorems, our argument could be adapted to show the following lower bounds of the lifespan for (1.3) These lower bounds, together with the upper bounds available from [8], show the sharpness of the lifespan estimates for n = 2 with p ∈ (2, p c (2)), n = 3 with p ∈ [2, p c (3)).
1.1. Notation. The vector fields to be used will be labeled as where Ω denotes the generators of spatial rotations For a norm X and a nonnegative integer m, we shall use the shorthand Let L q ω be the standard Lebesgue space on the sphere S n−1 , we will use the following convention for mixed norms L q1 t L q2 r L q3 ω , with r = |x| and ω ∈ S n−1 : , with trivial modification for the case Occasionally, when the meaning is clear, we shall omit the subscripts. As usual, we use · Em to denote the energy norm of order m ≥ 0, We will use · LE to denote the (strong) local energy norm , for a partition of unity subordinate to the (inhomogeneous) dyadic annuli, Σ j≥0 φ 2 j = 1. We denote 2

. Preliminary
In this section, we collect some inequalities we shall use later.
Proof. When s = 1, by Hölder's inequality When s = 0, by duality, we need only to show Since f g Ḣ1 ≤ ∇f g L 2 + f ∇g L 2 , by Hölder's inequality and Sobolev embedding By interpolation, (2.1) follows.
Proof. When s = 1, it is the classical Hardy's inequality By duality, we get which is the case s = 0. Then by interpolation, (2.2) follows.

Space-time estimates
In this section, we collect various space-time estimates for linear wave equation Proof. It is proven in Wang [18,Lemma 3.5]. The result with g 1 = 0 has been proven in Bony-Häfner [1] and Sogge-Wang [14]. See also Metcalfe-Sterbenz-Tataru [10].
Proof. It is essentially proved in [ Proof. This is essentially proved in [19,Lemma 5.6], which is based on the local energy estimates on nontrapping asymptotically Euclidean (Riemannian) manifolds, Lemma 3.1, as well as a sharper version of local energy estimates for small metric perturbation, due to Metcalfe-Tataru [11].

Global Existence
In this section, we prove the global existence results, Theorem 1.1.
) and q ∈ (p c , 1 + 4/(n − 1)) is any fixed choice when p ≥ 1 + 4/(n − 1). Then there exist ε 0 sufficiently small and a R sufficiently large, so that if 0 < ε < ε 0 and Proof. Without loss of generality, we may assume p ∈ (p c , 1 + 4/(n − 1)). If not, one need only fix any q ∈ (p c , 1 + 4/(n − 1)) and apply the proof below while noting that Sobolev embeddings provide u L ∞ t,x u X2 , which suffices to handle the p − q extra copies of the solution in the nonlinearity.

Almost Global Existence
In this section, we prove the almost global existence for the four-dimensional critical problem, Theorem 1.2. We set, for T ∈ (0, ∞), Proof. Basically, the proof follows the similar way in Theorem 4.1. For convenience of statement, we introduce Then by Lemma 3.3 with n = 4, for linear equation (3.1), there exists a constant C 5 > 0 such that for any 0 ≤ m ≤ 3, we have We set u (0) = 0 and recursively define u (k+1) be the solution to the linear equation To complete the proof, we need to show that u (k) is well defined, bounded inX 3 Tε and convergent inX 0 Tε . Well defined : It is easy to see u (1) by Hölder's inequality and Sobolev embedding, we have, for any t ∈ [0, T ] with fixed T < ∞, Thus (u (k) ) 2 ∈ L 1 t ([0, T ]; H 3 ) for any T < ∞. By standard existence theorem of linear wave equations, we have u (k+1) ∈ C([0, T ]; H 4 )∩C 1 ([0, T ]; H 3 ) for any T < ∞ and so u (k+1) ∈ C([0, ∞); H 4 ) ∩ C 1 ([0, ∞); H 3 ). Hence the iteration sequence is well defined.