On a semilinear wave equation in anti-de Sitter spacetime: the critical case

In the present paper we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine an ODI result with an iteration argument, by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.


Introduction
In the previous work [14], we studied the following semilinear Cauchy problem associated with a semilinear wave equation in anti-de Sitter spacetime      x ∈ R n , ∂ t v(0, x) = εv 1 (x), x ∈ R n , (1.1) where c, H are positive constants, b, m 2 are nonnegative real parameters satisfying b 2 4m 2 , ε > 0 is a parameter describing the size of initial data, T = T (ε) ∈ (0, ∞] is the lifespan of a classical solution v (i.e., the maximal existence time) and the nonlinear term is given by where p > 1, β 0, and Γ = Γ(t) is a suitable positive function. In cosmology, the constant H is the so-called Hubble constant, m is the mass of a particle and b is taken equal to the space dimension n (cf. (0.6) in [28]). We recall that the assumption b 2 4m 2 guarantees that the damping term b∂ t v is somehow dominant over the mass term m 2 v (cf. [ (1.4) where N = N (H, b, m 2 , p) .
Aim of the present paper is to study the blow-up for local solutions to (1.1) under suitable sign conditions for the Cauchy data in the threshold case ̺ = ̺ crit (n, H, b, m 2 , β, p) for n > N and to derive the corresponding upper bound estimates for the lifespan.
Furthermore, since we consider the limit case ̺ = ̺ crit (n, H, b, m 2 , β, p) we have to prescribe a lower bound for the power of the polynomial factor in (1.3), namely, ς crit (n, H, b, m 2 , p) .
The threshold case that we treat in the present work is somehow a blow-up result for a critical case. Consequently, the approach that we use to prove the blow-up in finite time of the spatial average V = V (t) of a local solution v to (1.1) is inspired by the one in [15] for the derivation of the sharp upper bound estimate for the lifespan of a local solution to the semilinear wave equation in the critical case when n 4. As in [32], when n 2 we use the Radon transform with respect to the spatial variables to handle the problem as it was in one space dimension.
The main difficulty in our argument will be the derivation of a sequence of lower bound estimates for the nonlinear term v(t, ·) p L p (R n ) . In particular, we will derive an iteration frame for v(t, ·) p L p (R n ) combining two estimates involving the Radon transform of v(t, ·). A fundamental tool for this kind of argument is provided by Yagdjian's integral transform approach. Indeed, we will make use of an explicit integral representation formula for the solution to a linear one dimensional wave equation in anti-de Sitter spacetime in order to derive one of the aforementioned inequalities involving the Radon transform of v(t, ·).
After deriving this sequence of lower bound estimates for v(t, ·) p L p (R n ) , we will derive in turn a sequence of lower bound estimates for V with an additional polynomial factor. Combining these lower bound estimates for V with a comparison argument for an ODE with "critical" exponential growth, we will be able to derive the desired upper bound estimates for the lifespan.
We point out that the speed of propagation, namely, the function a(t) .
= c e Ht , is exponentially increasing in the previous semilinear wave equation. Moreover, the amplitude of the forward light-cone is given by . =ˆt This means that, considering smooth solutions, if we assume v 0 and v 1 compactly supported in B R . = {x ∈ R n : |x| R}, given a local solution v to (1.1), we have that supp v(t, ·) ⊂ B R+A(t) for any t ∈ (0, T ). (1.6) For the proof of this support condition one can use the property of finite speed of propagation or, alternatively, the explicit representation formulas from the series of works by Galstian and Yagdjian [27,29,30].
In this second part of the introduction, we provide a short summary of the results in the literature for wave models with a not-flat and time-dependent metric in the spacetime.
In the case of de Sitter spacetime, i.e. for H < 0 in (1.1), the wave equation was considered by several authors. We recall the integral representation formulas (and their applications) established by Yagdjian and Yagdjian-Galstian in [28,21,22,23,24,25,26] and the global existence results for semilinear wave models in [9] and [1]. Concerning blow-up results, we recall the blow-up result with a pure imaginary mass term in (1.1), namely, when we replace m with im, both for de Sitter and anti-de Sitter spacetime in [10,Proposition 1.1]. Moreover, in [20] a blow-up result is proved in a de Sitter-type spacetime when m = 0. Finally, in our recent paper [14], we slightly improved some result from [21] for the semilinear wave equation in de Sitter spacetime with the same nonlinear term as in (1.2), providing further the lifespan estimates (see [ For anti-de Sitter spacetime, we refer to [2,27,29,30], where among other things, L p − L q estimates are derived for the solutions to the corresponding linear Cauchy problem. Moreover, as we have already mentioned above, in [14] some blow-up results for (1.1) have been proved together with the corresponding upper bound estimates for the lifespan.
Finally, for the wave equation in Einstein-de Sitter spacetime (that is, for the d'Alember we cite the papers [3,4] for the linear model, [5,16,11,12,17,19] for the semilinear model with power nonlinearity |v| p and [6,7,18] for the semilinear model with nonlinearity of derivative type |∂ t v| p . It is interesting to compare our approach in this paper to deal with a critical case in comparison to those in [16,11,12] for the treatment of the corresponding critical cases in Einsten-de Sitter spacetime. Indeed, while the critical case in [16] is studied by using the approach from [33], in [11,12] the method from [31] is adapted to the case with time-dependent coefficients. However, in both cases the employment of the techniques from [33,31] to the case with time-dependent coefficients produces some restrictive assumptions: in [16] an upper bound is prescribed for the size of the ball containing the supports of the Cauchy data, whilst in in [11] a restriction on the multiplicative constant in the damping term is prescribed. We emphasize that with the approach of the present work, no restriction of this kind (either on the size of the support for the data or on the range for the multiplicative constants in the lower order terms) appears.

Main results
Before stating the main theorem for (1.1), we recall the notion of weak solutions to (1.1) that has been employed in [14]. We stress that, although we call them weak solutions, actually for these solutions more regularity than for usual distributional solutions is required with respect to the time variable, in order to handle a space average that is a C 2 function with respect to t. Indeed, in [14] we worked with the larger class of solutions that can be considered when employing the spatial average for proving the blow-up in finite time.
if v fulfills the support condition (1.6) and the integral identitŷ holds for any t ∈ (0, T ) and any test function ϕ ∈ C ∞ 0 ([0, T ) × R n ). We point out explicitly that in the present paper we work with classical solutions to (1.1) since we need to employ an integral representation formula that requires a pointwise evaluation of the Cauchy data and of the nonlinear term. Nonetheless, it is clear that classical solutions to (1.1) are in particular weak solutions according to Definition 1.1. We emphasize that it was necessary to recall Definition 1.1 since in what follows we are going to use some results from [14] (see Section 2) that have been obtained for weak solutions in the aforementioned sense.
Let us assume that (v 0 , v 1 ) ∈ C 2 0 (R) × C 1 0 (R) are nonnegative and nontrivial functions with supports contained in B R for some R > 0. Let v ∈ C 2 ([0, T ) × R n ) be a classical solution to the Cauchy problem (1.1) with lifespan T = T (ε).
Then, there exists a positive constant ε 0 = ε 0 (n, c, H, b, m 2 , β, p, µ, ς, v 0 , v 1 , R) such that for any ε ∈ (0, ε 0 ] the classical solution v blows up in finite time. Furthermore, the following upper bound estimates for the lifespan hold where the positive constant C is independent of ε. The next sections of the paper are organized as follows: in Section 2 we recall briefly some estimates derived in [14]; in Section 3 we prove Theorem 1.2. More precisely, this proof is split in the following intermediate steps: in Subsection 3.1 we derive a comparison argument for an ordinary differential inequality (ODI) with "critical" exponential growth; in Subsection 3.2 we derive an integral representation formula for the solution to the one dimensional linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime; in Subsection 3.3 we derive the crucial iteration frame for v(t, ·) p L p (R n ) and in Subsection 3.4 we use this iteration frame to derive a sequence of lower bound estimates for v(t, ·) p L p (R n ) ; hence, we use in turn such a sequence to derive a sequence of lower bound estimates for the spatial average of the local solution in Subsection 3.5 and complete the proof in the case ς ∈ (ς crit (n, H, b, m 2 , p), 0]; finally, in Subsection 3.6 we conclude the proof also for the case ς > 0.

Preliminary results
Before beginning with the proof of Theorem 1.2, we recall some estimates that are proved in Section 3 of [14].

Iteration frame for the spatial average
Let v be a local classical solution to (1.1). In particular, by using the property of finite speed of propagation, we have that v satisfies the support condition (1.6). We set In [14,Subsection 3.1], we proved the identity We underline that (2.1) is obtained by choosing a suitable cut-off function in (1.7), that localizes the forward light-cone on the strip [0, t] × R n . Hence, by factorizing the differential operator ∂ 2 t + b∂ t + m 2 in (2.1), we derived then the following inequality for V : for t 0, where α 1 , α 2 are the roots of the quadratic equation α 2 − bα + m 2 = 0. The estimate in (2.2) is very important since it will be used in Subsection 3.5 to derive a sequence of lower bound estimates for V (t) from the sequence of lower bound estimates for v(t, ·) p L p (R n ) derived in Subsection 3.4.

First lower bound estimate for
In Subsection 3.2 of [14], by working with a weighted spatial average of v, with the weight function given by a suitable positive solution of the linear adjoint equation with separable variables, we derived the following lower bound estimates

Proof of Theorem 1.2
The proof of the results in the critical case ̺ = ̺ crit (n, H, b, m 2 , β, p) when we are in the case n > N is more delicate than the ones for n N seen in [14]. Roughly speaking, the functional V is no longer sufficient to show the blow-up in finite time of a local solution. The tools that we are going to use in this section are inspired by the ones for treatment of the critical case for the semilinear wave equation in the flat case. In particular, we are going to combine the approaches from [32] and [15] with some ideas from [13] for the treatment of a semilinear wave equation with time-dependent coefficients.

ODI comparison argument in the critical case
We state and prove a Kato-type lemma for exponentially growing functions in the "critical case". This result is the counterpart in our framework of Lemma 2.1 in [15].
with T 0 ∈ [0, T ) and for some positive constants B, K. Let us define where ϑ ∈ (0, q−1 2 ) and κ ∈ (0, T 0 ) are arbitrarily chosen and If the multiplicative constant on the right-hand side of (3.4) satisfies K K 0 , then, the lifespan of G is finite and fulfills T 2T 1 . Remark 1. Since q > 1, the two conditions (3.1) and (3.2) imply immediately that We will see that also the condition in (3.7) on k 0 , k 1 , along with (3.1) and (3.2), is fundamental for the proof of Lemma 3.1.
Remark 2. As we are going to see in the proof of Lemma 3.1, we may assume without loss of generality that the coefficient k 1 appearing in the lower bound for G in (3.4) satisfies where the equality −α 2 = −α 1 holds only for b 2 = 4m 2 . Indeed, it is possible to replace the lower bound for G in (3.4) with (3.9) below in order to get k 1 −α 2 . In particular, it makes sense to consider the limit case k 1 + α 1 = 0 (and, consequently, to modify accordingly T 1 and K 0 as in (3.6)) only in the balanced case b 2 = 4m 2 .
We point out explicitly, that the condition (3.2) in our application of Lemma 3.1 will be always satisfied thanks to (1.8).
Proof. By contradiction, we assume that G(t) is defined for any t ∈ [0, 2T 1 ]. We will show that this fact is not compatible with the choice K K 0 .
Let us begin by proving that G is actually nonnegative for any t ∈ [0, T ). By using the factorization straightforward computations lead to the following lower bound estimate for G(t): See [14, Section 2.1] for a detailed derivation of the previous kind of inequality for a function G satisfying the ordinary differential inequality in (3.3). Let us introduce now the further time-dependent function F (t) . = e α1t G(t). By the previous (3.10) By using (3.8), which can be rewritten as we get easily that thanks to the last sign assumption for the initial values of G in (3.5). Therefore, we multiply both sides of the inequality in (3.10) by F ′ (t), arriving at where in the second estimate we used the fact that α 1 α 2 . Integrating both sides of the previous inequality over [0, t], we obtain where in the second step we used (3.7). By straightforward computations, we find (3.14) Next we show that F (t) 2F (0) for t T 0 . For α 1 = α 2 , from (3.13) we have For α 1 = α 2 , the situation is even simpler since Hence, for t T 0 it follows from (3.14) that From the last inequality we get Now we multiply both sides of the previous inequality by ( Integrating both sides of the last inequality over [ The next step will consist in determining a suitable upper (resp. lower) bound estimate for the left-hand (resp. right-hand) side of (3.15). In order to derive both these estimates, we employ (3.4) to establish the following lower bound estimate for F Therefore, employing (3.16), for t T 1 we find, on the one hand, In this last part of the proof, we have to consider separately the case k 1 + α 1 > 0 and the case where in the second step we used the threshold condition (3.1). So, combining (3.15), (3.17) and (3.18) and using the fact that , which contradicts the condition K K 0 for k 1 + α 1 > 0. On the other hand, for k 1 + α 1 = 0 we have Analogously as before, we combine (3.15), (3.17) and (3.19), , which contradicts the condition K K 0 for k 1 + α 1 = 0. This completes the proof.
Remark 3. In the previous proof, we neglected in (3.12) the influence of the term (α 1 −α 2 )(F ′ (t)) 2 in the intermediate steps that lead to (3.14). If we used (3.11) to estimate this term from below and we kept the resulting term till the estimate (3.14), by having ignored the other term which appears on the right-hand side of (3.14), we would obtain (F ′ (t)) 2 e 2(α1−α2)t that would imply in turn F (t) e (α1−α2)t for t sufficiently large. In particular, by comparing this lower bound for F with the one in (3.16), we see that the latter is stronger provided that k 1 −α 2 .
As we pointed out in Remark 2, we may always assume without loss of generality that k 1 −α 2 and, consequently, that (3.16) provides the best lower bound estimate for F .

Integral representation formula for the 1-d linear Cauchy problem
We derive now an integral representation formula for the solution of the following linear inhomogeneous Cauchy problem in one space dimension Elementary computations show that v solves the following Cauchy problem with respect to (τ, y) Applying the transformation v(τ, y) . = e − bτ 2H w(τ, y), we find that v solves the Cauchy problem in (3.22) if and only if w solves = v 0 ( cy H ), w 1 (y) . In particular, with the terminology from [28], w satisfies a Klein-Gordon equation in anti-de Sitter spacetime with complex-valued curved mass ν.
where A(t) = c H (e Ht − 1), the kernel functions are given by . Remark 4. In (3.31) we could replace ν with −ν, thanks to the following property of the hypergeometric function see, for example, [8,Equation (15.8.1)]. However, we preferred the definition provided in (3.31) since in this way we have no singular behavior for the hypergeometric function as ζ → 1 + when ν > 0.
Remark 5. It is clear that the kernel functions E and K 1 are nonnegative on the forward light-cone and on the base of the forward light-cone, respectively. Now we want to show that K 0 (t, x; z; c, H, b, m 2 ) 0 for any z such that |x − z| c H (e Ht − 1). Let us express in a more explicit way the kernel function K 0 in (3.32). For the sake of brevity, we introduce the notation Clearly,

Iteration frame for v(t, ·) p L p (R n ) via the Radon transform
The idea to apply the Radon transform to reduce somehow the problem to a one-dimensional one when n 2 was introduced in the study of the critical case for the semilinear wave equation in the flat case in [32]. Here we will follow the main ideas from [13, Section 5] to derive an iteration frame for the nonlinear term v(t, ·) p L p (R n ) . In particular, the representation formula obtained via Yagdjian's integral transform approach will have a crucial role in the explicit representation of the Radon transform of v. We emphasize that the case n = 1 can be considered as well; however, rather than working with the Radon transform of v, it is sufficient to work simply with v (cf. Remark 8 below for further details).
We begin with the following remark: without loss of generality we may assume that a local solution is radially symmetric with respect to x. Indeed, when v is not radial it possible to consider insteadv (t, r) Let us clarify the meaning of this statement. By Jensen's inequality we have where ω n denotes the (n − 1)-dimensional measure of the unit sphere of R n . Consequently, combining the previous inequality, from (1.2) we have Since the fundamental solution E defined in (3.31) is nonnegative on the forward light-cone and the averages with respect to the space variables of v andv are equal, the inequality f (t, v) f (t,v), that we just proved, allows us to assume without loss of generality that v is radially symmetric in the proof of the blow-up result. Let us recall the definition of Radon transform of v(t, ·) when n 2. Given ρ ∈ R and ξ ∈ R n , |ξ| = 1, we have where dσ x is the Lebesgue measure on the corresponding hyperplanes. Since v(t, ·) is radially symmetric with respect to x, it turns out that R[v] does not depend actually on ξ and, moreover, Indeed, using polar coordinates x = ωr 1 with r 1 = |x| and ω ∈ S n−1 such that ω · ξ = 0, we obtain Hence, employing the change of variables r = (ρ 2 + r 2 1 ) 1/2 in the last integral, we find (3.36 From Subsection 3.2, it follows the following integral representation formula We point out that R acts only on the factors in the nonlinear term f (t, v) that depend on the space variable, that is, Thus, due to the fact that on the hyperplane where we are integrating it holds and, consequently, the considered hyperplane as empty intersection with the support of v.
In the next step, we shrink the domain of integration with respect to s in (3.37) so that the support of R[|v| p ](s, η) is a subset of the η-domain of integration. In other words, we look for s ∈ [0, t] such that

R[v](t, ρ)
The next step is to estimate the kernel function E in the right-hand side of the last inequality on the shrunk η-interval of integration. First of all, from the Taylor expansion of the hypergeometric function where (a) 0 . = 1 and (a) k . = a(a+1) · · · (a+k−1) denotes the so-called Pochhammer symbol, we see immediately that we can estimate from below the factor involving the hypergeometric function in E(t, ρ; s, η; c, H, b, m 2 ) by the constant function 1. Furthermore, since the two exponential terms in (3.31) are independent of η, the only factor that we actually have to estimate from below for η ∈ [−(R + A(s)), R + A(s)] is (( c H (e Ht + e Hs )) 2 − (ρ − η) 2 ) − 1 2 +ν . Notice that we have to proceed in a different way in order to get such lower bound estimate depending on whether ν is smaller or greater than 1/2.
Hereafter, for the sake of brevity, we use the notation φ(t) .
= c H e Ht . In particular, we may express the amplitude of the forward light-cone as follows Let us begin with the case ν ∈ [0, 1 2 ]. Let us prove that in this case the following upper bound estimate holds We check the validity of this inequality for s ∈ [0, s 0 ] and η ∈ [−(R + A(s)), R + A(s)] through a chain of inequalities where we used twice the condition A(s) A(s 0 ) and the identity 2A(s 0 ) = A(t) − |ρ| − R.
In a completely analogous way, one proves that for s ∈ [0, s 0 ] and η ∈ [−(R + A(s)), R + A(s)]. Therefore, combining (3.39) and the last inequality, when ν 1 2 we can estimate On the other hand, for ν 1 2 , from the lower bound estimates for s ∈ [0, s 0 ] and η ∈ [−(R + A(s)), R + A(s)], it follows that (3.44) Remark 6. Notice that for R φ(0), in (3.43) we might consider R 1 = 0 even for ν > 1 2 . Remark 7. In the previous considerations we estimate from below the hypergeometric function by a constant. In the limit case b 2 = 4m 2 (that is, for ν = 0), we might think to employ the asymptotic estimate F( 1 2 , 1 2 ; 1; ζ) ∼ − ln(1 − ζ) as ζ → 1 − in order to improve this lower bound estimate. However, for s ∈ [0, s 0 ] and η ∈ [−(R + A(s)), R + A(s)], setting where we used (3.41) and φ(s 0 ) = 1 2 (φ(t) + φ(0) − ρ − R). Therefore, − ln(1 − ζ) does not provide an improvement in the lower bound estimate, since for large t the argument of the logarithmic term on the right-hand side of the last inequality can be only estimated by a constant for ρ ∈ [0, A(t) − R] (this is the actual range that we will consider for ρ at the end of the present subsection). Now we plug the lower bound estimate from (3.43) in (3.38).
where we used the support condition for R[|v| p ] and Fubini's theorem to obtain The inequality in (3.45) is the first crucial estimate to obtain the iteration frame for v(t, ·) p L p (R n ) . Next step is to determine a lower bound for v(t, ·) p L p (R n ) with R[v] appearing in a nonlinear term on the right-hand side.
In order to derive this inequality, we will follow the approach from [13,Section 5]. We introduce the operator for any τ ∈ R and any h ∈ L p (R).
In [13,Section 5] it is proved that T ∈ L(L p (R) → L p (R)) for any p ∈ (1, ∞) and n 2. Even though the function A(t) in [13] is a polynomial function (more precisely, A(t) = 1 ℓ+1 (t ℓ+1 − 1) for some ℓ 0), the proof of this result is actually independent of the explicit expression of A(t) and it can be repeated verbatim in our case with A(t) = c H (e Ht − 1). We consider now the function By the boundedness of the operator T on L p (R), we have that T(h)(t, ·) L p (R) h(t, ·) L p (R) holds uniformly with respect to t ∈ [0, T ). Therefore, We have seen that R[v] is a nonnegative function by using an explicit integral representation. Moreover, by using the monotonicity of R and (3.36), we get where [ 1 p − 1 2 ] ± denote the positive and negative part of 1 p − 1 2 , respectively. Combining (3.46), (3.47) and the above inequality, we have Finally, from (3.45) and (3.48), we obtain for t A −1 (R) the desired iteration frame ]+ is actually present for p = 2. Nevertheless, we will do the computations formally as if both were present in order to consider simultaneously the cases p ∈ (1, 2) and p 2.
Remark 8. Let us underline explicitly that (3.49) is true also for n = 1. First, (3.45) can be obtained exactly as we did for n 2, with the only difference that Yagdjian integral representation formula is applied now directly to v, that is, On the other hand, for n = 1 (3.48) can be replaced by the trivial inequality Hence, combining (3.50) and (3.51), we conclude the validity of (3.49) for n = 1 too.

Iteration argument for
Our next goal is to derive a sequence of lower bound estimates for v(t, ·) p L p (R n ) through the iteration argument (3.49).
The starting point of our iteration procedure is given by (2.3).
Let us derive now a first lower bound estimate for v(t, ·) p L p (R n ) with an additional polynomial growing factor. Plugging (2.3) where, henceforth, q . = (β + 1)p and Hq s ds.
We recall that Γ(s) = µ(1 + s) ς e ̺crits , being ̺ crit defined by (1.4). Therefore, using the actual value of Γ(s), we have where we used for the coefficient in the exponential term.
Hereafter, we consider only the case ς 0. The complementary case ς > 0 will be discuss at the end of Section 3 (cf. Subsection 3.6). Therefore, where we used the analytic expression of the inverse function of A given by Plugging the last lower bound for I 0 (t, ρ) in (3.52), we get The next step consists in estimating from below the integral J 0 (t). First, we consider the factor ((φ(t) + R 1 ) 2 − ρ 2 ) (− 1 2 +ν)p . Recalling that the value of R 1 depends on the range for ν (cf. (3.44) for the definition of R 1 ), we derive a lower bound for this factor separately in the case ν 1 2 and in the case ν > 1 2 . For ν 1 2 , since the power (− 1 2 + ν)p is nonpositive, we consider an upper bound for (φ(t) + R 1 ) 2 − ρ 2 . For ρ ∈ [0, A(t) − R], since R 1 = 0 in this case, we have and Thus, for ν 1 2 and ρ ∈ [0, A(t) − R] we obtained We consider now ν > 1 2 . In this case we determine a lower bound for (φ(t) H . We emphasize that the lower bound for v(t, ·) p L p (R n ) that we are going to prove will be valid for t A −1 (a 0 R + b 0 c H ) and for suitable a 0 2 and b 0 > 0. In particular, we will use the inequality φ(t) + R 1 + ρ 1 2 φ(t) when R > c H without specifying the further condition on t. Hence, for ν > 1 2 and ρ ∈ [0, A(t) − R] we proved that c, H, b, m 2 , p, µ, v 0 , v 1 , R, a 0 , b 0 , δ) . = δ (n−1)[1− p 2 ]+ K B q B pB . We recall that the only assumptions on the parameters a 0 and b 0 that we did in order to obtain (3.61) are the following a 0 2 and b 0 > 0. (3.62) In the next subsection, however, we will require further conditions on a 0 , cf. (3.78) and (3.84). We stress that, since the amplitude function A for the light-cone grows exponentially, the logarithmic term in (3.61) provides, together with (1 + t) ςp , a polynomially increasing factor. This factor constitutes the improvement with respect to the estimate in (2.3) provided that ς ∈ (− 1 p , 0]. The next step is to prove that v(t, ·) p L p (R n ) satisfies the following sequence of lower bound estimates where {B j } j∈N is a suitable sequence of positive real numbers that we will determine iteratively during the proof and Notice that in (3.67) we shrank the ρ-domain of integration in order to have a nonempty s-domain of integration when using (3.63) to obtain I j+1 (t, ρ). We begin by deriving a lower bound estimate for I j+1 (t, ρ). By using (3.53), we can rewrite the first three factors in I j+1 (t, ρ) as follows: are satisfied if and only if the inequalities in (3.77) hold. Therefore, thanks to (3.78), we can continue the lower bound estimate for J j+1 (t), obtaining t A −1 ((8a j + 1)R + 4(2b j + 1) c H ) , with D = D(n, c, H, b, m 2 , β, p, a 0 , δ) . = KB C p δ (n−1)[1− p 2 ]+ (q − 1)q −1 . Hence we proved (3.63) for j + 1 which is exactly (3.80) with B j+1 given by (3.81). Finally, it is convenient for our future considerations to derive an explicit representation of ln B j . Applying the logarithmic function to both sides of (3.81) and using the resulting identity in an iterative way, we find ln B j = q ln B j−1 − j ln q + ln D = q 2 ln B j−2 − (j + (j − 1)q) ln q + (1 + q) ln D = · · · = q j ln B 0 − where E . = B 0 q −q/(q−1) 2 D 1/(q−1) .

Improved lower bound estimates for the spatial average of the solution
In the previous subsection we derived the sequence of lower bound estimates for v(t, ·) p L p (R n ) in (3.63). Now, we are going to use (3.63) to derive a sequence of lower bound estimates for the spatial average V (t).

Upper bound estimates for the lifespan when ς > 0
In the previous proof we showed the validity of (1.10) when ς ∈ (− 1 p , 0]. Of course, we can repeat the same argument as before when ς > 0 obtaining the same upper bound estimate for the lifespan as in the case ς = 0, that is T (ε) ε −p(q−1) . On the other hand, when ς > 0 and ̺ = ̺ crit we may use the same approach as in the proof of [14,Theorem 1.9] to prove the blow-up in finite time of V and the lifespan estimate T (ε) ε − q−1 ς . Comparing the last two lifespan estimates when ς > 0, we conclude the validity of (1.10).

Final remarks and open problems
We point out explicitly that we expect that the lifespan estimate in (1.10) is not sharp when ς > 0.
We stress also that when n > N in the double limit case ̺ = ̺ crit (n, H, b, m 2 , β, p) and ς = ς crit (n, H, b, m 2 , p) we are not able to prove the blow-up in finite time of V . This is due to the fact that the function L(t, ε) does no longer depend on t as the power for t is 0 in (3.91) when ς = ς crit (n, H, b, m 2 , p).