On Glassey's conjecture for semilinear wave equations in Friedmann-Lema\^itre-Robertson-Walker spacetime

Consider nonlinear wave equations in the spatially flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes. We show blow-up in finite time of solutions and upper bounds of the lifespan of blow-up solutions to give the FLRW spacetime version of Glassey's conjecture for the time derivative nonlinearity. We also show blow-up results for the space time derivative nonlinearity.


Introduction.
The spatially flat FLRW metric is given by where the speed of light is equal to 1, dσ 2 is the line element of n-dimensional Euclidean space and a(t) is the scale factor, which describes expansion or contraction of the spatial metric. As in our earlier work [16,17,18], we treat the scale factor as a(t) = ct 2 n(1+w) (1.1) where c is a positive constant, and w is the proportionality constant in the range −1 < w ≤ 1.
The constant w appears in the equation of state relating the pressure to the density for the perfect fluid. See [16].
We first consider the following Cauchy problem in order to compare with the related known results including the case of the Minkowski spacetime: with the initial data given at t = 1, where α and µ are nonnegative constants and ε > 0 is a small parameter.
Let T ε be the lifespan of solutions of (1.3) and (1.4), say, T ε is the supremum of T such that (1.3) and (1.4) have a solution for x ∈ R n and 1 ≤ t < T .
Let α = µ = 0 and p G (n) = 1 + 2/(n − 1). The so-called Glassey's conjecture [4] asserts that if p > p G (n), then there exist global solutions in time for small initial data, on the other hand, if 1 < p ≤ p G (n) with n ≥ 2 or if p > 1 for n = 1, then blow-up in finite time occurs. This conjecture is proved to be almost true. Actually, blow-up results in low dimensions (n = 2, 3), or in high dimensions (n ≥ 4) imposing radial symmetry were proved in, e.g., [2,3,10,12,13,14], and Zhou [20] finally gave a simple proof of the blow-up result for 1 < p ≤ p G (n) and n ≥ 2 as well as for p > 1 and n = 1. Global existence of solutions in low dimensions (n = 2, 3) has been proved in, e.g., [7,15,19]. For high dimensions (n ≥ 3), it is proved by [8] that there exist global solutions in the radial case for p > p G (n). They [8] also proved the lifespan of local solutions in time for 1 < p ≤ p G (n).
The present paper treats the case α ≥ 0 and µ ≥ 0. We first show blow-up in a finite time and upper estimates of the lifespan of solutions of (1. 3) and (1.4) in the case 0 ≤ α < 1.
If α = 0, our upper bounds of the lifespan coincide with the results above by [5]. Similar results are independently shown by [6] where energy solutions are treated. In our results, however, another exponent appears as a blow-up condition in some case. This is different from the results by [6]. We emphasize that the generalized exponent of p G (n + µ) cannot always be the critical exponent for the global existence of solutions. Our proofs are based on the test function method with the modified Bessel function of the second kind and on a generalized Kato's lemma. We next treat the case α ≥ 1.
Moreover, we show blow-up results for the problem (1.7) Unlike the above equation ( Assume that u 0 ∈ C 2 (R n ) and u 1 ∈ C 1 (R n ) are nontrivial and satisfy Then, T < ∞ and there exists a constant ε 0 > 0 depending on p, α, µ, R, u 0 , u 1 such that T ε has to satisfy Remark (1) If α = 0, then (2.1) and (2.2) are the same with the upper bounds (1.5) and (1.6).
(2) By the theorem, the exponent p ′ G (n, α, µ) cannot always be the critical exponent for the global existence of solutions. We discuss more details in the end of this subsection.
(3) If p < p 0 (n, α, µ), then the above assumption u 1 (x) ≥ u 0 (x) ≥ 0 can be replaced just by Proof ) Mutiplying (1.3) by a test funtion φ(t, x) and t µ , and integrating over R n , we have Integrating over [1, t], we obtain We remark that the C 2 -solution u of (1.3) and (1.4) has the property of finite speed of propagation, and satisfies provided that supp u 0 , supp u 1 ⊂ {|x| ≤ R}. See [16] for its proof.
We now define a smooth test function by where K ν (t) is the modified Bessel funtion of the second kind which is given by It is well-known that the Bessel function K ν satisfies the following properties (see, e.g., [1]): (2.10) We can verify by (2.8)-(2.10) that there holds (2.11) The following estimate is shown in [16]: See (3.23) in [16]. We also see by [16] that under assumption on the initial data. Moreover, we have (2.14) where we have used (2.10) and ν = (µ − 1)/(2(1 − α)) for the last equality.
) satisfies the following three conditions: where A 0 , A 1 and R are positive constants. Then, T has to satisfy where C is a constant depending on R, A 1 , µ, p, q, r, a, b and c.

Proof )
Mutiplying assumption (ii) by t µ , we have Integrating the above inequality over [T 0 , t] yields By assumption (iii), we see that F (t) > 0 for t ≥ T 0 . Hence, by assumption (i), we have Based on the fact above, we define the sequences a j , b j , c j , D j for j = 0, 1, 2, · · · by Solving (2.26) and (2.27), we obtain and thus by assumption, choosing t large enough, we can find a positive δ such that It then follows from (2.29) that F (t) −→ ∞ as j → ∞ for sufficiently large t. We therefore see that the lifespan T of F (t) has to satisfy and c. This completes the proof of the proposition.
This completes the proof of Theorem 2.1.
If we compare the two upper bounds (2.1) and (2.3), then in the region (G) in Fig. 1 and

Case α ≥ 1
We next consider the same problem for the case α ≥ 1.
and u 1 ∈ C 1 (R n ) are nontrivial and satisfy Then, T < ∞ and there exists a constant ε 0 > 0 depending on p, α, µ, R, u 0 , u 1 such that T ε has to satisfy Proof ) We first remark that the C 2 -solution u of (1.3) and (1.4) has the property of finite speed of propagation, and satisfies where Proceeding as in the proof of Theorem 2.1 for the case 0 ≤ α < 1, we have by (2.34), Therefore, we obtain Proceeding as before, by (2.34), we have Therefore, we obtain 3 Space derivative nonlinearity.
In this section we consider the problem (1.7). Let u 0 and u 1 be nonnegative and satisfy supp We prepare several basic inequalities which will be used repeatedly. Let Then integrating equation (1.7) over R n and using Poincaré's and Hölder's inequalities imply that On the other hand, mutiplying (3.1) by t µ and integrating imply Since F ′ (1) > 0 by assumption, Integrating again, we have from F (1) > 0 by assumption, We call wavelike and heatlike cases if a blow-up condition is concerned with exponents similar to the Strauss and Fujita ones, respectively.
Let T ≥ T 1 > T 0 ≥ 1. Assume that F ∈ C 2 ([T 0 , T )) satisfies the following three conditions: where A 0 , A 1 and R are positive constants. Then T has to satisfy where C is a constant depending on R, A 1 , µ, p, q, a and b.
Remark One cannot apply Corollary 1 to the original equation (1.2) for |∇ x u| p in the FLRW spacetime since n + 1 + (µ − α)/(1 − α) > 0. This is covered later in Section 5 in more detail.

Wavelike and critical case
We next consider the critical case p = p ′ c (n, α, µ).

Proof ) Let
where K ν (t) is the modified Bessel function given by (2.7). We now define a test function and It is proved in [16] that the function φ q (t, x) satisfies the following properties: Lemma 3.4. Let φ q (t, x) be defined by (3.16). Assume that q satisfies (3.17) and (3.18).
(i) Then, there exists a T 0 > 0 such that φ q satisfies We now prove the following key lemma to prove the theorem.
and let p satisfy (3.14). Define Then, G(t) satisfies with some T 2 sufficiently large, where C is a constant independent of ε.
Proof ) We first verify that q satisfies the required conditions (3.17) and (3.18) to use Lemma 3.4. We claim that there holds We note that ∂ t φ q (1, x) ≤ 0, which is shown in [16]. Hence, by positivity assumption on u 0 and u 1 , it holds that C data (t) ≥ 0 for t ≥ 1. Thus, the right-hand side of (3.24) becomes We now estimate the left-hand side of (3.24). By Poincaré's and Hölder's inequalities, the first integral is estimated by We remark that q satisfies (3.19). Applying Lemma 3.4 to the last integral above, we have Thus, we obtain The second integral on the left-hand side of (3.24) can be estimated as before by Using Lemma 3.4, 0 by (3.26), we obtain We finally estimate the third integral on the left-hand side of (3.24), Set T 2 = max{T 0 , T 1 } to apply Lemma 3.4 (i) and (ii). Proceeding in a similar way as before, we have We remark here that q satisfies (3.21). By Lemma 3.4, Hence, where we note that since (n − 1)(1 − α)/2 = q + (µ + α)/2 + (1 − α)/p and (3.26), This completes the proof of Lemma 3.5.

Thus, we obtain
Then the rest of the proof is the same as that of Theorem 2.3 in [16].
satisfies the following three conditions: where A 0 , A 1 and R are positive constants. Then, T has to satisfy where C is a constant depending on R, A 1 , µ, p and b.
In the end of this subsection, we discuss the blow-up condtions and the estimates of the lifespan in the subcritical cases in Theorems 3.1 and 3.6.
By (2.38) and (3.3), Hence, from (3.5), We here use another Kato's lemma. We can combine Lemmas 2.3 and 3.3 in [18] to obtain the following lemma. Let T ≥ T 1 > T 0 ≥ 1. Assume that F ∈ C 1 ([T 0 , T )) satisfies the following three conditions: where A 0 , A 1 and R are positive constants. Then, T has to satisfy where C is a constant depending on R, A 1 , µ, p, q, a, b and c.

Wave Equations in FLRW
Consider the equation with the time derivative nonlinear term |u t | p . Denote p ′ G (n, 2/(n(1+ w)), 2/(1 + w)) by p ′ G (n, w). Fig. 6 below shows the range of blow-up conditions in terms of w and p in the case n = 3. For 2/n − 1 < w ≤ 1 and n ≥ 2, applying Theorem 2.1 to (1.2) for |u t | p , we obtain the following upper bounds of the lifespan: where C > 0 is a constant independent of ε. We note that the estimate (2.3) in Theorem 2.1 is not applied to (1.2) since µ ≥ 1 in our case. See Region (G) in Fig. 6.
See Region (A) in Fig. 6.
From these results, we see that the blow-up range of p in the flat FLRW spacetime is smaller than that in the Minkowski spacetime because p ′ G (n, w) < p G (n) = 1 + 2/(n − 1). Moreover, in the subcritical case p < p ′ G (n, w), the lifespan of the blow-up solutions in the We next consider the equation with the space derivative nonlinear term |∇ x u| p . We define here γ ′ 0 (n, p, w) corresponding to γ ′ (n, p, α, µ) in (3.6) by γ ′ 0 (n, p, w) = 1 − 2 n(1 + w) γ ′ n, p, 2 n(1 + w) , 2 1 + w .
For −1 < w ≤ 2/n − 1 and n ≥ 2, say an accelerated expanding universe, we obtain from Theorem 3.8 T 2 ε (ln T ε ) −(n(p+1)−n) ≤ Cε −(p−1) if p > 1 and w = 2 n − 1, We see that blow-up in finite time can happen to occur for all p > 1. This is in contrast to the case of decelerated expansion above. Finally, let us compare the results for the term |u t | p with those for |∇ x u| p , especially in the decelerated expanding universe, say (G), (F) and (C). We observe that which has a larger blow-up range depends on the value of w for each n. If n = 2, then max{p ′ F (2, w), p ′ c (2, w)} > p ′ G (2, w) for 0 = 2/n − 1 < w ≤ 1. The higher the dimension n becomes, however, the larger the w-interval such that p ′ G (n, w) > max{p ′ F (n, w), p ′ c (n, w)} becomes. We will treat the remaining case w = −1 in future papers.