The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions

In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed. The key fact is that, only in two space dimensions, such a special constant in the damping term is a threshold between"wave-like"domain and"heat-like"domain. As a result, we obtain a new type of estimate especially for the critical exponent.


Introduction
We are concerned with the following initial value problem for semilinear wave equations with the scale-invariant damping: where v = v(x, t) is a real valued unknown function, µ > 0, p > 1, n ∈ N, the initial data (f, g) ∈ H 1 (R n ) × L 2 (R n ) has compact support, and ε > 0 is a "small" parameter. It is interesting to look for the critical exponent p c (n) such that p > p c (n) (and may have an upper bound) =⇒ T (ε) = ∞, where T (ε) is the lifespan, the maximal existence time, of the energy solution of (1.1) with an arbitrary fixed non-zero data. Then, we have the following conjecture: =⇒ p c (n) = p F (n) = p S (n + µ) (intermediate), 0 < µ < µ 0 (n) =⇒ p c (n) = p S (n + µ) (wave-like), (1.2) where µ 0 (n) := n 2 + n + 2 n + 2 . is the so-called Strauss exponent which is the critical exponent of the associated semilinear wave equations v tt − ∆v = |v| p . We note that p S (n) (n ≥ 2) is the positive root of γ(p, n) := 2 + (n + 1)p − (n − 1)p 2 = 0 (1.6) and 0 < µ < µ 0 (n) is equivalent to p F (n) < p S (n + µ).
The conjecture (1.2) shows the critical situation of our problem in the following sense. If one replaces the damping term µv t /(1 + t) in (1.1) by µv t /(1+t) β , then one can see that there is no such a p c (n), namely T (ε) = ∞ for any p > 1 when β < −1, the so-called over damping case. Moreover one has p c (n) = p F (n) for any µ > 0 when −1 ≤ β < 1, the so-called effective damping case, and p c (n) = p S (n) for any µ > 0 (it can be any µ ∈ R) when β > 1, the so-called scattering damping case. Therefore one may say that the so-called scale-invariant case, β = 1, is an intermediate situation between wave-like, in which the critical exponent is related to p S (n), and heat-like, in which the critical exponent is p F (n). To see all the references above results, for example, see introductions of related papers to the scattering damping case, Lai and Takamura [15] (the sub-critical case), Wakasa and Yordanov [22] (the critical case), Liu and Wang [17] (partial result of the super-critical case).
In this paper, we consider a special case of µ = 2. The speciality of this value is clarified by setting u(x, t) := (1 + t) µ/2 v(x, t), where v is the solution to (1.1). Then, u satisfies   so that all the technics in the analysis on semilinear wave equations can be employed and we may discussed about not only the energy solution but also the classical solution. In fact, via this reduced problem (1.7), D'Abbicco, Lucente and Reissig [5] have proved the intermediate part of the conjecture (1.2) for n = 2 and the wave-like part for n = 3 when µ = 2. We note that the assumption of the radial symmetry is considered in [5] for the existence part in n = 3. Moreover, D'Abbicco and Lucente [4] have obtained the wave-like existence part of (1.2) for odd n ≥ 5 when µ = 2 also with radial symmetry.
In the case of µ = 2, Lai, Takamura and Wakasa [16] have first studied the wave-like blow-up of the conjecture (1.2) with a loss replacing µ by µ/2 in the sub-critical case. Initiating this result, Ikeda and Sobajima [8] have obtained the blow-up part of (1.2).
In the non-damped case of µ = 0, it is known that (1.9) is true for n ≥ 3, or p > 2 and n = 2, The open part around this fact is p = p S (n) for n ≥ 9. In other cases, (1.9) is still true if R n g(x)dx = 0. On the other hand, we have for n = 2 and p = 2 is a positive number satisfying ε 2 a 2 log(1 + a) = 1. We note that the bounds in (1.10) are smaller than the one of the first line in (1.9) with µ = 0 in each case. For all the references in the case of µ = 0, see Introduction of Imai, Kato, Takamura and Wakasa [10]. The remarkable fact is that even if µ is in the heat-like domain, the lifespan estimate for (1.1) is similar to the one for non-damped case. Indeed, for n = 1 and µ = 2 > µ 0 (1) = 4/3, Kato, Takamura and Wakasa [13] show that the result on (1.8) by Wakasa [24] mentioned above is true only if R {f (x) + g(x)}dx = 0. More precisely, they have obtained that We note that the bounds in (1.11) are larger than those in (1.8) with n = 1 and µ = 2 in each case.
Our aim in this paper is to show the lifespan estimates for (1.1) in two dimensional case, n = 2, with µ = 2 which is similar to one dimensional case as above. We note p c (2) = p F (2) = p S (2 + 2) = 2 and µ 0 (2) = 2. More precisely, we shall show that We note that the critical cases in (1.12) and (1.13) are new in the sense that they are different from (1.8) and (1.9).
(1.12) and (1.13) are announced in Introduction of Lai and Takamura [15], but there are typos in the exponents of ε in the critical case. The strategy of proofs in this paper is based on point-wise estimates of the solution. In the existence part, we employ the classical iteration argument for semilinear wave equations without damping term, which is first introduced by John [11] in three space dimensions, and its variant, which is developed by Imai, Kato, Takamura and Wakasa [10] in two space dimensions. In the blow-up part, we also employ an improved version of Kato's lemma on ordinary differential inequality by Takamura [18] for the sub-critical cases. We note that, till now, the so-called test function method such as in Ikeda, Sobajima and Wakasa [9] cannot be applicable to delicate analysis to catch the logarithmic growth of the solution in the case of p = 2 in (1.10), (1.11), (1.12) and (1.13). Therefore we employ the so-called slicing method of the blow-up domain for the critical case, which is introduced by Agemi, Kurokawa and Takamura [1] to handle weakly coupled systems of non-damped semilinear wave equations with critical exponents. This paper is organized as follows. In the next section, our goals, (1.12) and (1.13), are described in four theorems, and we introduce the linear decay estimate and basic lemmas for a-priori estimates. Section 3, or Section 4, is devoted to the proof of the lower bound, or upper bound, of the lifespan respectively.

Theorems and preliminaries
In this section, we state our results (1.12) and (1.13) in four theorems. After them, we list useful point-wise estimates of linear wave equations. For the sake of the simplicity, we assume that throughout this paper. The existence parts of our goals in (1.12) and (1.13) are guaranteed by the following two theorems. Recall the definitions of µ 0 (n), p F (n), p S (n) and γ(p, n) respectively in (1.3), (1.4), (1.5) and (1.6).
Theorem 2.2 Suppose that the assumptions in Theorem 2.1 are fulfilled. Assume additionally that Then, there exists a positive constant ε 0 = ε 0 (f, g, p, k) such that (1.7) admits a unique solution u ∈ C 2 (R 2 × [0, T )) if p = 2, or the integral equation associated with (1.7) admits a unique solution u ∈ C 1 (R 2 × [0, T )) otherwise, as far as T satisfies for 0 < ε ≤ ε 0 , where c is a positive constant independent of ε.
On the other hand, the blow-up parts of our goals in (1.12) and (1.13) are guaranteed by the following two theorems.
Then, there exists a positive constant ε 1 = ε 1 (f, g, p, k) such that the solution cannot exist whenever T satisfies Theorem 2.4 Suppose that the assumptions in Theorem 2.3 are fulfilled.
Assume additionally that Then, there exists a positive constant ε 1 = ε 1 (f, g, p, k) such that the solution cannot exist whenever T satisfies From now on, we introduce some definitions and useful lemmas.
in the classical sense, and it holds We introduce the decay estimates for the solutions of (2.2) which will be used in the proof of Theorem 2.1 and Theorem 2.2. For the proof, see Lemma 2.1 in [10]. Lemma 2.1 (Imai, Kato, Takamura and Wakasa [10]) Let u L be the one in (2.2). Then, there exist positive constants C 0 = C 0 (k) and Next, we prepare the following decay estimate which will be employed in the proof of Theorem 2.3 and Theorem 2.4.
Proof. First we shall prove (2.4). Denote r := |x|. For t − r ≥ 2k and t ≥ 4k, we shall split the domain into the interior domain t ≥ 2r and the exterior domain t ≤ 2r. We set This expression gives us Using the Taylor expansion with respect to y at the origin, we get and From (2.6), (2.7) and (2.8), it follows that Therefore, combining (2.5), (2.9) and Next, we prove (2.4) in D ext . Here, we employ the following different representation formula from (2.2) which is established by (6.24) in Hörmander [7]: where ω = x/r ∈ S 1 , ρ = r − t, z = 1/r and For (x, t) ∈ D ext and |y| ≤ k, we have it follows from (2.10) and (2.11) that Hence, we obtain Making use of the Taylor expansion with respect to s at the origin, we have from (2.13) (2.14) Since ρ = r − t and z = 1/r, for |s| ≤ 5k/4, we obtain Hence, for (x, t) ∈ D ext , it follows from (2.12), (2.14), (2.15) and (2.16) that Therefore, we obtain (2.4) in D ext .
Finally, we show (2.3). It follows from the proof of Lemma 2.1 in [10] that In what follows, we consider the following integral equation: where we set and . We note that (2.21) solves . Then, the following lemma is one of the basic tools. [2]) Let L be the linear integral operator defined by (2.21) and

Lemma 2.3 (Agemi and Takamura
where a + := max{a, 0} and Moreover, the following estimates hold in [0, ∞) 2 : In order to construct our solution in the weighted L ∞ space, we define the following weighted functions: where we set and We remark that w 2 can be described as For these weighted functions, we denote the weighted L ∞ norms of V by Finally, we shall introduce some useful representations for L. It is trivial Changing the variables by and extending the domain of (α, β)-integration, we obtain from Lemma 2.3 and (2.28) and Similarly, we get and 3 Proof of Theorem 2.1 and Theorem 2.2 In this section, we prove Theorem 2.1 and Theorem 2.2. The proof is based on the classical iteration method in John [11]. Lemma 3.3 will be used to prove Theorem 2.1, whereas we prove Theorem 2.2 by using Lemma 3.4. First, we prepare the elementary inequalities in Lemma 3.1 and Lemma 3.2.
Proof. For 0 ≤ r ≤ t + k, the integration by parts yields This completes the proof. ✷
Let t − r ≥ k which implies t − r ≥ (t − r + 2k)/4. Then, breaking the integral up into two pieces, we get It is easy to see that We obtain The following lemma contains one of the most essential estimates.
Lemma 3.3 Let 1 < p ≤ 2 and L be the linear integral operator defined by Then, there exists a positive constant C 1 independent of k and T such that
Proof. In order to show the a-priori estimate (3.5), it is enough to prove where L j are defined in Lemma 2.3. By (2.22), (2.25) and (2.29), we have We shall prove (3.7) in the following two cases.

Proof of Theorem 2.3
We divide the proof of Theorem 2.3 into two cases, 1 < p < 2 and p = 2. First, we shall handle the sub-critical case.
Proof of Theorem 2.3 with 1 < p < 2. We shall follow the arguments in Section 4 of Takamura [18]. In order to obtain the estimates in Theorem 2.3, we shall take a look on the ordinary differential inequality for (1.7) with µ = 2 and (2.1) imply that Hence, the Hölder's inequality and (2.1) yield that Due to the assumption on the initial data in Theorem 2.3, f (x) ≡ 0, g(x) ≥ 0 ( ≡ 0), we have It follows from (4.3) in [18] that From (4.1), it follows that Plugging (4.4) into the right-hand side of this inequality, we have that We evaluate the integral of the last term in (4.5). For t ≥ 3k, we obtain From (4.5) and (4.6), we obtain 3 · 2 3p−1 π p−1 ε p t 3−2p for t ≥ 3k. Integrating this inequality in [3k, t], we get from (4.3) Hence, we obtain from (4.3) where In the sub-critical case, the following basic lemma is useful.
Lemma 4.1 (Takamura [18]) Let p > 1, a > 0 and q > 0 satisfy where A, B, k, T 0 are positive constants. Then, there exists a positive constant D 0 = D 0 (p, a, q, B) such that This is exactly Lemma 2.1 in [18], so that we shall omit the proof here. According to (4.2), (4.3) and (4.7), we are in a position to apply our situation to Lemma 4.1 with A = D 1 ε p , B = π 1−p , a = 5 − 2p, q = 3(p − 1) which imply that (4.8) yields If we set we find that there is an ε 0 = ε 0 (g, p, k) such that This means that T 1 = T 0 in (4.10). Therefore, from (4.11), the conclusion of Lemma 4.1 implies The proof of Theorem 2.3 with 1 < p < 2 is now completed. ✷ Proof of Theorem 2.3 with p = 2.

Proof of Theorem 2.4
We divide the proof of Theorem 2.4 into two cases, 1 < p < 2 and p = 2. First, we shall handle the sub-critical case.
For the key inequality, we employ the following proposition.