Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent

The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case of non-effective damping. In this paper we extend his result in two aspects by showing that: (I) the blow-up will happen for bigger exponent, which is closely related to the Strauss exponent, the critical number for non-damped semilinear wave equations; (II) such a blow-up result is established for a wider range of the constant than the known non-effective one in the damping term.


Introduction
In this paper, we consider the following initial value problem.
First, we shall outline a background of (1.1) briefly according to the classifications by Wirth in [20,21,22] for the corresponding linear problem. Let u 0 be a solution of the initial value problem for the following linear damped wave equation.
where µ > 0, β ∈ R, n ∈ N and u 1 , u 2 ∈ C ∞ 0 (R n ). When β ∈ (−∞, −1), we say that the damping term is "overdamping" in which case the solution does not decay to zero when t → ∞. When β ∈ [−1, 1), the solution behaves like that of the heat equation, which means that the term u 0 tt in (1.2) has no influence on the behavior of the solution. In fact, L p -L q decay estimates of the solution which are almost the same as those of the heat equation are established. In this case, we say that the damping term is "effective." In contrast, when β ∈ (1, ∞), it is known that the solution behaves like that of the wave equation, which means that the damping term in (1.2) has no influence on the behavior of the solution. In fact, in this case the solution scatters to that of the free wave equation when t → ∞, and thus we say that we have "scattering." When β = 1, the equation in (1.2) is invariant under the following scaling u 0 (x, t) := u 0 (σx, σ(1 + t) − 1), σ > 0, and hence we say that the damping term is "scale invariant." The remarkable fact in this case is that the behavior of the solution of (1.2) is determined by the value of µ. Actually, for µ ∈ (0, 1), it is known that the asymptotic behavior of the solution is closely related to that of the free wave equation. For this range of µ, we say that the damping term is "non-effective." However, the threshold of µ according to the behavior of the solution is still open. We conjecture that it may be µ = 1 since we have the following L 2 estimates: In this way, we may summarize all the classifications of the damping term in (1.2) in the following table.
Next, we consider the following initial value problem for semilinear damped wave equation.
0 (R n ) and n ∈ N. We assume that ε > 0 is a "small" parameter.
For the constant coefficient case, β = 0, Todorova and Yordanov [15] have shown that the energy solution of (1.3) exists globally-in-time for "small" initial data if p > p F (n), where is the so-called Fujita exponent, the critical exponent for semilinear heat equations. It has been also obtained in [15] that the solution of (1.3) blowsup in finite time for some positive data if 1 < p < p F (n). The critical case p = p F (n) has been studied by Zhang [24] by showing the blow-up result. We note that Li and Zhou [10], or Nishihara [12], have obtained the sharp upper bound of the lifespan which is the maximal existence time of solutions of (1.3) in the case of n = 1, 2, or n = 3, respectively. The sharpness of the upper bound has been studied by Li [11] including the result for more general equations with all n ≥ 1, but for smooth nonlinear terms. The sharp lower bound has been obtained by Ikeda and Ogawa [6] for the critical case. Recently, Lai and Zhou [9] have obtained the sharp upper bound of the lifespan in the critical case for n ≥ 4. For the variable coefficient case of the most part of the effective damping with −1 < β < 1, Lin, Nishihara and Zhai [13] have obtained the blow-up result if 1 < p ≤ p F (n) and the global existence result if p > p F (n). Later, D'Abbicco, Lucente and Reissig [2] have extended the global existence result to more general equations. For the precise estimates of the lifespan in this case, see Introduction in Ikeda and Wakasugi [7]. Recently, similar results on the remaining part of effective damping with β = 1 have been obtained by Wakasugi [19] for the global existence part, and by Fujiwara, Ikeda and Wakasugi [5] for the blow-up part. The sharp estimates of the lifespan are also obtained by [5] except for the upper bound in the critical case. Now, let us turn back to our problem (1.1). Wakasugi [18] has obtained the blow-up result if 1 < p ≤ p F (n) and µ > 1, or 1 < p ≤ 1 + 2/(n + µ − 1) and 0 < µ ≤ 1. He has also shown in [17] that an upper bound of the lifespan is where C is a positive constant independent of ε. We note that the both proofs in [17] and [18] are based on the so-called "test function method" introduced by Zhang [24]. On the other hand, D'Abbicco [1] has obtained the global existence result if p > p F (n), but µ has to satisfy µ ≥    5/3 for n = 1, 3 for n = 2, n + 2 for n ≥ 3 (and p ≤ 1 + 2/(n − 2)) .
It is remarkable that, by the so-called Liouville transform; w(x, t) := (1 + t) µ/2 u(x, t), (1.5) When µ = 2, D'Abbicco, Lucente and Reissig [3] have obtained the following result. Let where is the so-called Strauss exponent, the positive root of the quadratic equation, We note that p 0 (n) is the critical exponent for semilinear wave equations, µ = 0 in (1.1). They have shown in [3] that the problem (1.1) admits a global-in-time solution in the classical sense for "small" ε if p > p c (n) in the case of n = 2, 3 although the radial symmetry is assumed in n = 3, and that the classical solution of (1.1) with positive data blows-up in finite time if 1 < p ≤ p c (n) and n ≥ 1. In the same year, with radial symmetric assumption, D'Abbicco and Lucente [4] extended the global existence result for p 0 (n + 2) < p < 1 + 2/(max{2, (n − 3)/2}) to odd higher dimensions (n ≥ 5). We remark that, in the case of n = 1, Wakasa [16] has studied the estimates of the lifespan and has shown that the critical exponent p c (1) = p F (1) = 3 changes to p 0 (1 + 2) = 1 + √ 2 when the nonlinearity is a signchanging type, |u| p−1 u, and the initial data is of odd functions. Both results in [3] and [16] heavily rely on the special structure of the massless wave equations, µ = 2 in (1.5). In view of them, µ = 2 may be an exceptional case. Recalling Wirth' classification in the linear problem, (1.2), one may regard µ = 1 as a threshold also for the semilinear problem, (1.1). In this sense, the blow-up result in Wakasugi [18] says that the solution may be "heat-like" if µ > 1. Here, "heat-like" means that the critical exponent for (1.1) is Fujita exponent.
In this paper, we claim that the solution of (1.1) is "wave-like" in some case even for µ > 1. Here, "wave-like" means that the critical exponent for (1.1) is bigger than Fujita exponent and is related to Strauss exponent. We also conjecture that such a threshold of µ depends on the space dimension n. The main tool of our result is Kato's lemma in Kato [8] on ordinary differential inequalities which is improved to be applied to semilinear wave equations by Takamura [14]. Together with Yordanov and Zhang's estimate in [23], we can prove a new blow-up result for wave-like solutions by means of some special transform for the time-derivative of the spatial integral of unknown functions. This paper is organized as follows. In the next section, we state our main result. In the section 3, we estimate the spatial integral of unknown functions from below. Making use of such an estimate, we prove the main result for µ ≥ 2 in section 4, and for 0 < µ < 2 in section 5.

Main Result
First we define an energy solution of (1.1).
We note that, employing the integration by parts in (2.2) and letting t → T , we have that This is exactly the definition of a weak solution of (1.1). Our main result is the following theorem.

(2.3)
Assume that both f ∈ H 1 (R n ) and g ∈ L 2 (R n ) are non-negative and do not vanish identically. Suppose that an energy solution u of (1.1) satisfies with some R ≥ 1. Then, there exists a constant ε 0 = ε 0 (f, g, n, p, µ, R) > 0 such that T has to satisfy for 0 < ε ≤ ε 0 , where C is a positive constant independent of ε.
Remark 2.1 Theorem 2.1 can be established also for n = 1 if one define φ 1 (x) = e x + e −x for x ∈ R in Section 3 below. But the result is not new. See the following two remarks.
We note that µ 0 (2) = 1. This means that Theorem 2.1 just covers the noneffective range of µ for n = 2. Since µ 0 (n) is increasing in n, Theorem 2.1 gives us the blow-up result on super-Fujita exponent even for µ in outside of the non-effective range for n ≥ 3. We also note that µ 0 (n) < 2 for n = 2, 3, 4 and µ 0 (n) > 2 for n ≥ 5.

Remark 2.3
One can see also that Therefore we have that for n = 2 and 0 < µ < 1, or n ≥ 3 and µ > 0. This means that Theorem 2.1 includes the blow-up result in Wakasugi [18].

Remark 2.4
If β is in the scattering range, (1, ∞), for the problem, the result will be T ≤ Cε −2p(p−1)/γ(p,n) for 1 < p < p 0 (n) for all µ > 0. This estimate coincides with the one for non-damped equation, u tt − ∆u = |u| p except for the case of R 2 g(x)dx = 0 in n = 2 and 1 < p ≤ 2. See Introduction of Takamura [14] for its summary. The proof of this fact will appear in our forthcoming paper.

Lower bound of the functional
Let u be an energy solution of (1.1) on [0, T ). We estimate which means that All the quantities in this equation except for F ′ 0 (t) are differentiable in t, so that so is F ′ 0 (t). Hence we have Integrating this equation with a multiplication by (1 + t) µ , we obtain It follows from this equation and the assumption on the initial data that 3) From now on, we employ the modified argument of Yordanov and Zhang [23]. Let us define In view of (3.2) and the argument of (2.4)-(2.5) in [23], we know that there is a positive constant C 1 = C 1 (n, p, R) such that In order to get a lower bound of F 1 (t), we turn back to (2.2) and obtain that Multiplying the above equality by(1 + t) µ , we have that Integrating this equality over [0, t], we get

It follows from this equation and
which follows from integration by parts that Hence we obtain that Integrating this inequality over [0, t] with a multiplication by e 2t , we get We note that the assumption on f implies F 1 (0) > 0. Hence we find that there is no zero point of F 1 (t) for t > 0. Because the continuity of F 1 implies F 1 (t) > 0 for small t > 0. If one assumes that there is a nearest zero point t 0 of F 1 to 0, then one has a contradiction in (3.5); The last term in the right-hand side of this inequality is positive by F 1 (t) > 0 for 0 < t < t 0 . Turning back to (3.5), we obtain Here we have used the fact that C f,g > C f,0 .
Plugging this estimate into (3.4), we have Since F ′ 0 (0) > 0 and it follows from t ≥ 1 that we obtain that Here we have used the fact that Integrating this inequality over [1, t] and making use of F 0 (0) > 0 and where where w is the solution of (1.5). We note that (3.1) yields Then it follows from (2.4) and Hölder's inequality that By combining (4.1) and (4.2), and noting the assumption R ≥ 1, we come to where Due to (3.3), we have that which implies that From now on, we focus on the case of µ ≥ 2. Then it follows from (4.3) and (4.4) that We shall employ the following lemma now.
Here we use our lower bound of F in (4.13) again to get where Therefore, taking δ small enough such that Y + 1 > 0, we then have by integrating (5.6) over [T 1 , t], Making use of (4.13) with t = T 1 in this inequality, we obtain that If one sets t = kT 1 with k > 1, then, due to the definition of T 1 , one has 1 > C 7 C γ(p,n+2µ)/4 6 (k Y +1 − 1).
Therefore the conclusion of the Theorem 2.1, is now established, where This completes the proof in the case of 0 < µ < 2. ✷