Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

In this paper we study several semilinear damped wave equations with"subcritical"nonlinearities, focusing on demonstrating lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. Under different assumptions imposed on the initial data, lifespan estimates from above are clearly showed. The key fact is that we find"transition surfaces", which distinguish lifespan estimates between"wave-like"and"heat-like"behaviours. Moreover we conjecture that the lifespan estimates on the"transition surfaces"can be logarithmically improved. As direct consequences, we reorganize the blow-up results and lifespan estimates for the massless case in which the"transition surfaces"degenerate to"transition curves". Furthermore, we obtain improved lifespan estimates in one space dimension, comparing to the known results. We also study semilinear wave equations with the scattering damping and negative mass term, and find that if the decay rate of the mass term equals to 2, the lifespan estimate is the same as one special case of the equations with the scale-invariant damping and positive mass. The main strategy of the proof consists of a Kato's type lemma in integral form, which is established by iteration argument.


Introduction
The aim of the present work is to study blow-up phenomena and lifespan estimates for solutions of Cauchy problem with small data for several semilinear damped wave models. Indeed we mainly concern about semilinear wave equations with the scale-invariant damping, mass and power-nonlinearity    u tt − ∆u + µ 1 1 + t u t + µ 2 (1 + t) 2 u = |u| p , in R n × (0, T ), u(x, 0) = εf (x), u t (x, 0) = εg(x), x ∈ R n , (1.1) where µ 1 , µ 2 ∈ R, p > 1, n ∈ N, T > 0 and ε > 0 is a "small" parameter. In particular, we are interested in exploring a competition between so-called "heat-like" and "wave-like" behaviour of the solutions, which concerns not only critical exponents, but also lifespan estimates, in a way that we will clarify later.
Let us firstly denote energy and weak solutions of our problem (1.1).
We recall that the critical exponent p crit of (1.1) is the smallest exponent p crit > 1 such that, if p > p crit , there exists a unique global energy solution to the problem, whereas if 1 < p ≤ p crit the solution blows up in finite time. In the latter case, one is also interested in finding estimates for the lifespan T ε , which is the maximal existence time of the solution, depending on the parameter ε.
Our principal model is the one in (1.1), for which we obtain Theorem 2 and Theorem 4, according to the different conditions imposed on the initial data. As straightforward consequences, we also obtain Theorem 1 and Theorem 3 for the massless case, i.e. the model with µ 2 = 0. The lifespan estimate in dimension n = 1 in this case is improved, comparing to the known results. Moreover, we continue the study of semilinear wave equations with scattering damping, negative mass term and power nonlinearity, introduced by the authors in [20,21].
The paper is organized in this way: in the rest of the Introduction, we will sketch the background of the problems under consideration and we will exhibit our results, which will be proved in Section 3, exploiting, as main tool, a Kato's type lemma in integral form presented in Section 2.

Heat vs. wave
Let us start considering the toy-models of the wave and heat equations: Nowadays the study of these two equations is almost classic: the well-known results include the lifespan estimates and the critical exponents, which are the so-called Fujita exponent p F (n) and the Strauss exponent p S (n), corresponding to the heat and the wave equation respectively. For the purpose of this work, let us define these two exponents for all ν ∈ R: We remark that 1 < p < p F (ν) =⇒ γ F (p, ν) := 2 − ν(p − 1) > 0, 1 < p < p S (ν) =⇒ γ S (p, ν) := 2 + (ν + 1)p − (ν − 1)p 2 > 0.
In particular, if ν > 0, p F (ν) is the solution of the linear equation γ F (p, ν) = 0, whereas if ν > 1, p S (ν) is the positive solution of the quadratic equation γ S (p, ν) = 0. Although the expression γ S (p, ν) is well-known in the literature, the introduction of γ F (p, ν) is justifyed from the fact that γ F plays for the heat equation the same role that γ S plays for the wave equation, as it emerge from the lifespan estimates. Suppose for the simplicity that f, g > 0 are compactly supported (for different conditions on the initial data, we can have different lifespan estimates, see Subsection 1.5). We have that the blow-up results are the ones collected in the following table.
Heat Wave Critical exponent p crit p F (n) p S (n) Subcritical lifespan T ε for 1 < p < p crit ∼ ε −2(p−1)/γF (p,n) ∼ ε −(p−1)/γF (p,n−1) if n = 1 or n = 2, 1 < p < 2 Here and in the following, we use the notation F G (respectively F G) if there exists a constant C > 0 independent of ε such that F ≤ CG (respectively F ≥ CG), and the notation F ∼ G if F G and F G.
For a more detailed story of these results, we refer to the book [7], the doctoral thesis [49], the introductions of [14,40,41,42] and the references therein.
For the comparison between the heat and wave equations, let us introduce an informal but evocative notation to describe the behaviour of the critical exponent and of the lifespan estimates in our models. We will call the critical exponent heat-like if it is related to the Fujita exponent, i.e. p crit = p F (ν) for some ν ∈ R, whereas we will call it wave-like if it is related to the Strauss exponent, i.e. p crit = p S (ν) for some ν ∈ R.
Similarly, we will say that the lifespan estimate is heat-like if it is related in some way to the one of the heat equation, i.e. to the exponent 2(p − 1)/γ F (p, ν) in the subcritical case and to exp(ε −(p−1) ) in the critical one, whereas we will say it wave-like if related to the one of the wave equation, i.e. to the exponent 2p(p−1)/γ S (p, ν) in the subcritical case and to exp(ε −p(p−1) ) in the critical one. However, we also define a mixed-type behaviour when the lifespan estimate is related to 2p(p − 1)/γ F (p, ν) in the subcritical case (as we will see in Theorem 3 & 4), to remark that the lifespan is larger respect to the heat-like one, due to the additional p in the exponent.

Damped wave equation
Let us proceed further by adding the damping term µ/(1 + t) β to the wave equation, with µ ≥ 0 and β ∈ R, hence we consider the Cauchy problem According to the works by Wirth [51,52,53], in the study of the associated homogeneous problem we can classify the damping term accordingly to the different values of β into four cases. When β < 1, the damping term is said to be overdamping and the solution does not decay to zero when t → ∞. If −1 ≤ β < 1, the solution behaves like that of the heat equation and we say that the damping term is effective. Hence, the term u 0 tt in (1.4) has no influence on the behavior of the solution and the L p − L q decay estimates of the solution are almost the same as those of the heat equation. 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 (1.4) 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. Finally, when β = 1, the equation in (1.4) is invariant under the scaling and hence we say that the damping term is scale-invariant. In this case the behaviour of the solution of (1.4) has been observed to be determined by the value of µ. We summarize all the classifications of the damping term in (1.4) in the next table.
Range of β Classification Let us return to problem (1.3), which inherits the above terminology and has very different behaviours from case to case. Indeed, in the overdamping case the solution exist globally for any p > 1. In the effective case, the problem is heat-like, both in the critical exponent and in the lifespan estimates, while in the scattering case the problem seems to be wave-like. Finally, the scale-invariant case has an intermediate behaviour, and a competition between heat-like and wave-like arises. Before moving to the last case, let us collect in the following two tables some global existence and blow-up results for β = 1, at the best of our knowledge.
Global-in-time existence for β = 1

Scale-invariant damped wave equation
We consider now (1.3) for β = 1, hence we consider the Cauchy problem   (1.5) The scale-invariant problem has been studied intensively in the last years. This great interest is motivated by the fact that, differently from the damped wave equation with β = 1, in the scale-invariant case the results depend also on the damping coefficient µ, for determining both the critical exponent and the lifespan estimate. Hence, the situation is a bit more complicated, being the scale-invariant case the threshold between the effective (−1 ≤ β < 1) and non-effective (β > 1) damping, and hence the threshold between a heat-like and a wave-like behaviour. In the following two tables we collect, at the best of our knowledge, results concerning the existence and the blow-up for the scale-invariant damping.
Global-in-time existence for β = 1
The blow-up part of this conjecture has already been proved, combining [48] and [11]. In our next theorem, which is a straightforward corollary of Theorem 2, we reconfirm the blow-up range and we give cleaner estimates for the lifespan in the subcritical case, obtaining improvements mainly in the 1-dimensional case (see Remark 1.2). We refer to Figure 1 for a graphic representation of the results below. Theorem 1. Let µ ≥ 0 and 1 < p < p µ (n), with p µ (n) defined in (1.6). Assume that f ∈ H 1 (R n ), g ∈ L 2 (R n ) and Suppose that u is an energy solution of (1.5) on [0, T ) that satisfies Then, there exists a constant ε 1 = ε 1 (f, g, µ, p, R) > 0 such that the blow-up time T ε of problem (1.5), for 0 < ε ≤ ε 1 , has to satisfy: Moreover, if µ = n = 1 and 1 < p ≤ 2 the estimate for T ε is improved by Remark 1.1. Note that, if n ≥ 3 and 0 ≤ µ < n − 1, we can write the lifespan estimates in Theorem 1 explicitly as Comparing the lifespan estimates in Theorem 1 with the known results summarized in the above table "Blow-up in finite time for β = 1", we remark that the heat-like estimates for n ≥ 1 were already proved by Wakasugi [49], whereas the wave-like ones for n ≥ 2 by Tu and Lin [44]. The wave-like estimates for n = 1 were almost obtained by Ikeda and Sobajima [11] for p F (n) ≤ p < p S (n + µ), with a loss in the exponent given by a constant δ > 0. Hence our improvements are given by the wave-like estimates if n = 1 and by the logarithmic gain T ε φ 0 (ε) if n = µ = 1 and 1 < p ≤ 2. Moreover, about the wave-like estimates for n ≥ 2, in [44] the initial data are supposed to be non-negative, whereas our conditions on the initial data are less restrictive.
Anyway, our approach is different and based on an iteration argument rather than on a test function method. Remark 1.3. We conjecture that the lifespan estimates in Theorem 1 are indeed optimal, except on the " transition curve" (in the (p, µ)-plane) from the wave-like to the heat-like zone, given by On this line, the identity holds true and here we expect a logarithmic gain, as already obtained for the case p = 2, µ = n = 1 in the previous theorem, and for the case n = p = 2, µ = 0 for the wave equation (see Subsection 1.1). As we see from [15,16,17,46] the conjecture holds true if µ = 2 and n ≤ 3.
Remark 1.4. In this work we do not treat the critical case, but, to conclude our prospectus, it is natural to conjecture that for some constant C > 0. We refer to [11,45] for the wave-like lifespan estimate from above in the critical case and to [15,16,17,46] for the proof of the conjecture if µ = 2 and n = 1, 3. However, we expect a different behaviour if µ = µ * and p = p µ * (n), that is when the transition curve from Remark 1.3 intersects the blow-up curve. This expectation is motivated from [15], where the authors prove for n = µ = µ * = p F (2) = p S (4) = 2 that T ε ∼ exp(Cε −1/2 ), which is neither a wave-like critical lifespan, nor a heat-like one. Figure 1: In this figure we collect the results from Theorem 1. If (p, µ) is in the blue area, we have that T ε ε −2p(p−1)/γ S (p,n+µ) and hence the lifespan estimate is wave-like. Otherwise, if (p, µ) is in the red area, then T ε ε −(p−1)/γ F (p,n−[µ−1] − ) and the lifespan estimate is heat-like. In the case n = 1, the dash-dotted line given by µ = 1, 1 < p ≤ 2 highlights the improvement T ε φ 0 (ε).

Wave equation with scale-invariant damping and mass
Finally, we return to our main problem (1.1). The scale-invariant damped and massive wave equation was studied by A. Palmieri as object of his doctoral dissertation [33], under the supervision of M. Reissig. However, as far as we know, the research of the lifespan estimates in case of blow-up is still underdeveloped.
A key value for the study of this problem is which, roughly speaking, quantify the interaction between the damping and the mass term. Indeed, if δ ≥ 0, the damping term is predominant and we observe again a competition between the wave-like and heat-like behaviours. In particular, the critical exponent seems to be wave-like for small positive values of δ, while it is heat-like for large ones. If on the contrary δ < 0, the mass term has more influence and the equation becomes of Klein-Gordon type. To see this, apply again the Liouville transform v(x, t) : (1.7) In the following, we will consider only the case δ ≥ 0. Let us start by collecting some known results. From [32,34,37], we know that for µ 1 , µ 2 > 0 and δ ≥ (n + 1) 2 the critical exponent for problem (1.1) is the shifted Fujita exponent On the contrary, from [35,36], in the special case δ = 1 and under radial symmetric assumptions for n ≥ 3, Palmieri proved that the critical exponent is The case δ = 1 is clearly the analogous of the case µ = 2 for the scale-invariant damped wave equation without mass: under this assumption we see from (1.7) that the equation can be transformed into a wave equation without damping and mass and with a suitable nonlinearity. In [38], Palmieri and Reissig proved, by using the Kato's lemma and Yagdjian integral transform, a blow-up result for δ ∈ (0, 1], and showed a competition between the shifted Fujita and Strauss exponents. Indeed, they obtained the blow-up result for except for the critical case p = p S (n + µ 1 ) in dimension n = 1. Finally, Palmieri and Tu in [39], under suitable sign assumption on the initial data and for µ 1 , µ 2 , δ non-negative, established a blow-up result for 1 < p ≤ p S (n + µ 1 ) and furthermore the following lifespan estimates: They used an iteration argument based on the technique of double multiplier for the subcritical case and a version of test function method developed by Ikeda and Sobajima [11] for the critical case. Of course, we refer to the works by Palmieri and to his doctoral thesis for a more detailed background.
We present now our main result, concerning the blow-up of (1.1) for µ 1 , µ 2 ∈ R and δ ≥ 0 and the upper bound for the lifespan estimates.
Firstly, let us introduce the value and set for the simplicity Observe that, if n + µ 1 > 1, then (1.10) The following result holds.
Suppose that u is an energy solution of (1.1) on [0, T ) that satisfies with some R ≥ 1.
Then, there exists a constant ε 2 = ε 2 (f, g, µ 1 , µ 2 , n, p, R) > 0 such that the blow-up time T ε of problem (1.1), for 0 < ε ≤ ε 2 , has to satisfy: If in particular δ > 0, then Here and in the following, the sign function is defined as sgn Remark 1.5. We can write the exponent in (1.11) explicitly as Remark 1.6. Note that, setting the mass coefficient µ 2 = 0 and the damping coefficient It is straightforward to check that, by imposing µ 2 = 0, the results in Theorem 2 coincide with those in Theorem 1.
Remark 1.7. Analogously as in Remark 1.3, we conjecture that p µ 1 ,δ (n) defined in (1.11) is indeed the critical exponent and that the lifespan estimates presented in Theorem 2 are optimal, except on the " transition surface"(in the (p, µ 1 , δ)-space) defined by on which we expect a logarithmic gain.
The exponent p = 2 n− √ δ already emerged in Palmieri and Tu [39], but as a technical condition. We underline that this exponent comes out to be the solution of the equation for some constant C > 0. See [39] for the proof of the wave-like upper bound of the lifespan estimate in the critical case. Moroever, if √ δ = n − d * (n + µ 1 ) and p = p µ 1 ,δ (n), we expect a different lifespan estimate, as in the massless case.

Different lifespans for different initial conditions
In Theorems 1 & 2 we impose the condition on the initial data One could ask if this is only a technical condition, but it turns out that this is not the case: if we impose h = 0, the lifespan estimates change drastically. This phenomenon was recently taken in consideration also in the works by Imai, Kato, Takamura and Wakasa [14,15,17]. Let us return to the wave equation Since µ 1 = µ 2 = 0, in this case the condition h = 0 is equivalent to g = 0. Indeed, under the assumption collecting the results from the works [14,24,28,29,40,41,56,57,58], we have that, for n ≥ 1, the following lifespan estimates holds: excluding the critical case p = p S (n) for n ≥ 9 and without radial symmetry assumptions. We refer to the Introduction by Imai, Kato, Takamura and Wakasa [14] for a detailed background on these results. What is interesting is the fact that now we observe always a wave-like lifespan. This is in contrast with the estimates presented in Subsection 1.1, where, under the assumption we have heat-like lifespans in low dimensions, more precisely if n = 1 or if n = 2 and 1 < p ≤ 2, with a logarithmic gain if n = p = 2. Let us consider now the Cauchy problem for the scale-invariant damped wave equation Since µ 1 = 2 and µ 2 = 0, the condition h = 0 is equivalent to f +g = 0. In low dimensions n = 1 and n = 2, Kato, Takamura and Wakasa [17] and Imai, Kato, Takamura and Wakasa [15] proved that, if the initial data satisfy then the lifespan estimates in 1-dimensional case are These estimates are greatly different from the ones presented in Subsection 1.3, which hold under the assumption In dimension n = 1, we have no more a heat-like behaviour, but a wave-like one appears for p < 2, whereas for p > 2 we have a mixed-like behaviour, accordingly with the notation introduced in Subsection 1.1. Indeed, in the latter case, even if the lifespan is related to the heat-like one, an additional p appears. In dimension n = 2, we have no more a heatlike behaviour, but a wave-like one. The strange exponent in the critical lifespan can be explained by the same phenomenon underlined in Remark 1.4. We are ready to exhibit our results, which give upper lifespan estimate in the subcritical case when h = 0. It is easy to see that our estimates coincide with the ones just showed above in the respective cases. Going on with the exposition followed until now, we will present firstly the particular massless case, then the more general one where also the mass is considered. For the simplicity, we will consider only non-negative damping coefficients.
Let us introduce the exponent 15) and observe that, for p > 1 and n + µ 1 = 1, . (1.16) The following results hold. See Figure 2 for a graphic representation of the claim in Theorem 3.
About the relation between p * and r * , we have that, for 0 ≤ δ < 1, if √ δ ⋚ θ then p * ⋚ r * . Remark 1. 10. We conjecture that the estimates in the previous two theorems are indeed optimal, except in dimension n = 1 for Theorem 3 on the transition curve defined by and for Theorem 4 on the transition surface Moreover, in the critical case we expect, due to the wave-like and mixed-like behaviours, except for √ δ = n − d * (n + µ 1 ) and p = p µ 1 ,δ (n), where the lifespan should be different.
Remark 1.11. The conditions (1.12) on the initial data in Theorem 1 & 2 can be replaced by the less strong conditions where the positive function φ 1 (x) is defined later in (3.9).

Similarity can be done for the initial conditions of Theorem 3 & 4, requiring
It will be clear from the proof of our theorems that these weaker hypothesis are sufficient.

Wave equation with scattering damping and negative mass
Finally, in this subsection we want to continue the study of a problem examined by the authors in [20,21]. In these two works, we considered the Cauchy problem for the wave equation with scattering damping and negative mass term, thus where ν 1 ≥ 0, ν 2 < 0, α ∈ R and β > 1.
In Subsection 1.3 we already observed that, if the damping is of scattering type, the solution of the homogeneous damped wave equation "scatters" to the one of the wave equation. For the equation with power non-linearity, according to the results by Lai and Takamura [22] and Wakasa and Yordanov [47], the solution again seems to be wave-like both in the critical exponent and in the lifespan estimate.
In [20], the authors took in consideration (1.19) with α > 1 and observed a double scattering phenomenon, in the sense that both the damping and the mass terms seem to be not effective. Hence, the solution behaves like that of the wave equation with power non-linearity u tt − ∆u = |u| p . More precisely, supposing f, g > 0 for the simplicity, we established the blow-up for 1 < p < p S (n) and the upper bound for the lifespan estimates: where a ≡ a(ε) satisfies ε 2 a 2 log(1 + a) = 1, although in the case n = p = 2 more technical conditions were required. In [21], the authors studied the case α < 1, discovering a new behaviour in the lifespan estimate. Indeed, we proved blow-up for every p > 1 and the upper lifespan estimate As observed in Remark 2.1 of [21], a less sharp but more clear estimate for the lifespan in the case α < 1 is Hence, if the negative mass term with α > 1 seems to have no influence on the behaviour of the solution, on the contrary if α < 1 the negative mass term becomes extremely relevant, implying the blow-up for all p > 1 and a lifespan estimate which is much shorter, compared to the ones introduced previously.
We come now to the case α = 1. This is particular and was not deepened in our previous works. Indeed in Subsection 3.5, after introducing a multiplier to absorb the damping term, we will show that we can get blow-up results and lifespan estimates for this problem by reducing ourself to calculations similar to the ones we will perform to prove the results in the previous subsections. Roughly speaking, we will find out that (1.19) with α = 1 has the same behaviour of (1.1) with µ 1 = 0 and µ 2 = ν 2 e ν 1 /(1−β) . Therefore, in the following we will consider the Cauchy problem where ν 1 ≥ 0, ν 2 < 0 and β > 1.
Definition 2. We say that u is an energy solution of (1.20) with any test function φ ∈ C ∞ 0 (R n × [0, T )) for t ∈ [0, T ). We have the following result. See Figure 3 for a graphic representation of them.
and let 1 < p < p δ (n), with Assume that f ∈ H 1 (R n ), g ∈ L 2 (R n ) are non-negative and not both vanishing. Suppose that w is an energy solution of (1.20) on [0, T ) that, for some R ≥ 1, satisfies Then, there exists a constant ε 5 = ε 5 (f, g, β, ν 1 , ν 2 , n, p, R) > 0 such that the blow-up time T ε of problem (1.20), for 0 < ε ≤ ε 5 , has to satisfy: √ δ) is in the blue area, T ε ε −2p(p−1)/γ S (p,n) , hence the lifespan estimate is wave-like. Otherwise, if (p, √ δ) is in the red area, T ε ε −(p−1)/γ F (p,n−(1+ √ δ)/2) and the lifespan is heat-like. Note that this figure represents also the results of Theorem 2 for the case µ 1 = 0, µ 2 ≤ 1/4. Remark 1.12. As a direct consequence of Remark 1.7 & 1.8, we expect that p δ (n) is the critical exponent and that the lifespan estimates presented in Theorem 5 are optimal, except on the transition curve (in the (p, δ)-plane) defined by on which we presume a logarithmic gain can appear. Moreover, we expect that, if p = p δ (n), then for some constant C > 0. If √ δ = n − d * (n) and p = p δ (n), we presume a lifespan estimate of different kind.

Kato's type lemma
The principal ingredient we will employ in the demonstration of our theorems is the following Kato's type lemma. Although this tool is well known and used in the literature, here we will reformulate it in such a way, in the following sections, we can directly apply it to obtain not only the condition to find the possible critical exponent, but also the upper lifespan estimate. We will prove it using the so called iteration argument.
Assume that F ∈ C([0, T )) satisfies, for t ≥ T 0 , where c, T 0 ≥ 0 and E, A, B > 0. Then, for T ≥ T 0 we have that where C is a constant independent of E.
Proof. Let T be as in the statement of the lemma and start with the ansatz where D j , a j , b j , c j are positive constants to be determined later. Due to hypothesis (2.1), observe that (2.4) is true for j = 1 with where [x] ± := (|x| ± x)/2. Plugging (2.4) into (2.2), we get and then we can define the sequences to establish (2.4) with j replaced by j + 1. It follows from the previous relations and (2.5) that for j ≥ 1 In particular, we obtain that Moreover S j is a sequence definitively increasing. Hence we obtain that for j sufficiently large. Let us turning back to (2.4) and let C > 1 a constant to be determined later. Supposing t ≥ C T , so that in particular t − T ≥ (1 − 1/C)t, and considering (2.3), we have . Since γ > 0, we can choose C > 1 big enough, such that J > 1. Letting j → +∞ in (2.7), we get F (t) → +∞. Then, T < C T as claimed.
Remark 2.1. We can observe that the previous lemma is still true if in (2.2) an arbitrary number of integrals appear, more precisely if we replace (2.2) with

Proof for Theorems
We come now to the demonstration for Theorems 2 & 4. In the next two subsections, we will prove some key inequalities which will be employed in the machinery of the Kato's type lemma. Applying the latter, we will find a couple of results, which will be compared in Subsection 3.4 to find the claimed ones. The proof of Theorems 1 & 3 are clearly omitted, since they are corollaries of Theorems 2 & 4 respectively, just setting the mass equal to zero. In the end, we will sketch the proof for Theorem 5 in Subsection 3.5.

Key estimates
Let us define the functional Choosing the test function φ = φ(x, s) in (1.2) to satisfy which yields, by taking derivative with respect to t, we obtain that (3.2) is equivalent to Integrating twice the above equality over [0, t], we get where Define the functional F (t) := (1 + t) κ+λ F 0 (t) and observe that F 0 and F implies the same blow-up results, so we will study the latter functional. Since and they are not both equal to zero, we want to prove that exists a time T 0 > 0, independent of ε, such that, for t ≥ T 0 , the following estimates hold: Thanks to the Hölder inequality and using the compact support of the solution (1.13), we have for t 1. Considering L and recalling the definition (1.12) of H 0 we obtain So, from the condition on the initial data we get, for t 1 sufficiently large, that Finally, we will prove (3.6) in the next section.

Weighted functional
Let us introduce where ψ 1 is the test function presented by Yordanov and Zhang in [54], which satisfies the following inequality (equation (2.5) in [54]): We want to establish the lower bound for F 1 . From the definition of energy solution (1.2), we have that Integrating the above inequality over [0, t], and in particular using integration by parts on the second term in the first line and on the first term in the second line, we get Hence we obtain from (3.11) from which, after a derivation, Let us define the multiplier Then, multiplying equation (3.12) by m(t), using for convenience the change of variables z := 1 + t and denoting B(z) := m(t)F 1 (t), (3.13) we obtain that B satisfies the nonlinear modified Bessel's equation with initial data (3.15) and where Now we want to estimate B. Homogeneous problem. Let us firstly consider the homogeneous Cauchy problem The fundamental solutions are the modified Bessel's functions B + √ δ/2 (z) := I √ δ/2 (z) and B − √ δ/2 (z) := K √ δ/2 (z). Then we have where, thanks to equations (9.6.15) and (9.6.26) from Chapter 9 in [1], it holds Due to the assumptions on the initial data and recalling that B + ν (z), B − ν (z) > 0 when ν > −1 and z > 0, we can observe that c + > 0 (see also Remark 1.11). Exploiting the asymptotic expansions for the modified Bessel's functions (equations (9.7.1) and (9.7.2) from Chapter 9 in [1]), we have that Then, there exist two constants C > 0 and z 0 ≥ 1, both not depending on ε, such that Inhomogeneous problem. Let us consider now the Cauchy problem Exploiting the method of variation of parameters, we have that Recalling that N(z) ≥ 0 and using the fact that B + √ δ/2 (z) is increasing and B − √ δ/2 (z) is decreasing respect to the argument for z > 0 (due to relations (9.6.26) from Chapter 9 in [1]), we get that B N (z) ≥ 0 for z ≥ 1. (3.17) Since the solution B to the Cauchy problem (3.14)-(3.15) is the sum of B 0 and B N and from estimates (3.16) and (3.17), we get So, recalling the definition (3.13) of B and changing again the variables, we reach (3.18) By Hölder inequality and using estimates (3.10) and (3.18), we obtain which inserted in (3.3) and recalling that L(t) is positive for t great enough, give us for a suitable T 1 > 0, where we define q ≡ q(p) := κ − (n + µ 1 − 1) p 2 + n + 1. (3.19) We obtain, for large time t 1, that: Summing all up, we deduce the relations in (3.6).

Application of Kato's type lemma
Now we will proceed applying the Kato's type lemma, as presented in Section 2, two times to two different couples of inequalities, and subsequently we will infer which result is optimal. The calculations of this subsection are all elementary (and quite tedious), so we will only sketch them. Apply Lemma 1 to the inequalities (3.4) and (3.5), with We chose p ∈ (1, p c ) since this is equivalent to γ > 0 for p > 1. Then, for every p ∈ (1, p c ), we have T ε T ≡ T (ε), with Apply Lemma 1 to the inequalities (3.4) and (3.6), with where q is the one in (3.19) and p c ∈ (1, +∞] is defined as the exponent such that γ > 0 for 1 < p < p c (we will explicitly define this exponent later). Then, for every p ∈ (1, p c ), we have T ε S ≡ S(ε), with (3.21) In both cases, since (3.4), (3.6) and (3.5) are true for t ≥ T 0 with some time T 0 , and since we need to require T , S ≥ T 0 to apply the Kato's type lemma, we need to impose also that ε is sufficiently small. From these computations, we deduce the blow-up for 1 < p < p k := max{p c , p c } and the upper lifespan estimate T ε min{ T , S}. We will go further in the analysis to clarify these values. Before to move forward, in order to understand the definition of S we need to write down more explicitly the definitions of c, p c and γ, since they depend on q and therefore on the exponent p. Firstly, recall the definition (1.15) of p * = p * (n + µ 1 , n − √ δ) and that, by (1.16), for p > 1 and µ 1 + n = 1, it holds p = p * ⇐⇒ q(p) = 0 ⇐⇒ γ S (p, n + µ 1 ) = 2γ F (p, n + κ).
We will consider several cases, due to the generality of the constants involved, but what lies beneath is the elementary comparison between the parabola γ S (line in the case µ 1 + n = 1) and the line 2γ F . Also, since we want to be in the hypothesis of Kato's type lemma, our interest is directed to γ > 0, and so we explicit its definition only for the range 1 < p < p c .
At the end, recalling the definitions of γ, γ, c and c in the various cases and summing all up, we conclude the proof for Theorem 2 and Theorem 4.

Proof for Theorem 5
We will only sketch the demonstration, since it is a variation of the previous one. Let us introduce the functional G 0 (t) = R n w(x, t)dx and, as in [20,21], the bounded multiplier m(t) := exp ν 1 (1 + t) 1−β 1 − β .
Choosing the test function φ = φ(x, s) in (1.21) to satisfy (3.1), deriving respect to the time and multiplying by m, we get that  It is simple to see, by comparison argument, that G 0 is positive. Indeed, by the hypothesis on initial data, we know that G 0 (0) = R n f (x)dx and G ′ 0 (0) = R n g(x)dx are non-negative and not both zero. If G 0 (0) > 0, by continuity G 0 is positive for small time. If G 0 (0) = 0 and G ′ (0) > 0, then G 0 is increasing and again positive for small time t > 0. If we suppose that there exist a time t 0 > 0 such that G 0 (t 0 ) = 0, calculating (3.22) in t = t 0 we get a contradiction, since the left-hand term would be zero and the right-hand term would be strictly positive. Then, G 0 is positive for any time t > 0. Define now the functional G 0 as the solution of the integral equation  Since m(0) < m(t) < 1 for any t > 0 and ν 2 < 0, we have that R n ν 2 (1 + t) 2 w(x, t)φ(x, t)dx + m(t) R n |w(x, t)| p φ(x, t)dx. Setting φ(x, t) = ψ 1 (x, t) = e −t φ 1 (x) on supp w and recalling the bounds on the multiplier m(t), we obtain G ′ 1 (t) + 2G 1 (t) ≥ m(0)G ′ 1 (0) + 2m(0)G 1 (0)

By integration on [0, t] we get
Integrating the above inequality over [0, t] after a multiplication by e 2t , we get |w(x, r)| p φ(x, r)dx, from which, thanks again to a comparison argument, we infer that G 1 is non-negative, and so, neglecting the last two term in the above inequality, it is easy to reach G 1 (t) ε for t 1.
Hence, we have also Finally, we obtain the inequality analogous to (3.6), i.e.