$L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p<\infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.

The damped wave equation ∂ 2 t u − ∆u + ∂ t u = 0 is known as a model describing the wave propagation with friction, and studied for long years. In particular, for the Cauchy problem where (u 0 , u 1 ) is a given function, the asymptotic behavior of the solution has been investigated by many mathematicians after the pioneering work by Matsumura [24]. Matsumura [24] applied the Fourier transform to (1.1) and obtained the formula where D(t) is defined by sin(t |ξ| 2 − 1/4) (1.2) Using the above formula, he proved the so-called Matsumura estimates (L p -L q estimates) , where 1 ≤ q ≤ 2 ≤ p ≤ ∞, t := (1 + |t| 2 ) 1/2 , [n/2] denotes the integer part of n/2, and the notation f g stands for f ≤ Cg with some constant C > 0. The first term and the second term in the right-hand side are corresponding to the low and high frequency part of the solution, respectively. The estimates (1.3) indicate that the low frequency part of the solution behaves like that of the heat equation Here, we recall the well-known L p -L q estimates for the heat equation where 1 ≤ q ≤ p ≤ ∞, g ∈ L q (R n ), and G(t) := F −1 e −t|ξ| 2 F . Namely, G(t)g is the solution of (1.4) with v(0) = g (see [7]). Also, we see from (1.3) that the high frequency part causes derivative losses like the wave equation We also recall the estimates for the wave equation: with some constant δ ′ p > 0, where 1 < p < ∞, g ∈ H β−1 p (R n ), β = (n − 1)| 1 2 − 1 p |, and W(t) := F −1 sin(t|ξ|) |ξ| F , namely, W(t)g is the solution of (1.6) with (w, ∂ t w)(0) = (0, g) (see [33,25]). Here, we set H s p (R n ) := {f ∈ S ′ (R n ); f H s p = ∇ s f L p < ∞}. However, in contrast to the estimates (1.5) and (1.7), the Matsumura-type estimate (1.3) requires the restriction q ≤ 2 ≤ p, and the derivative losses of the high frequency part seems not sharp. Therefore, we expect that the estimate (1.3) can be improved.
Indeed, in the following we give an improvement of the Matsumura-type estimate (1.3). Let χ ≤1 (∇) and χ >1 (∇) be the cut-off Fourier multipliers defined by (1.18) for low and high frequency, respectively. Our first result reads as follows.
We will prove this theorem in the next section. The main ideas of the proof are the following. To remove the restriction of the exponent 1 ≤ q ≤ 2 ≤ p ≤ ∞, we derive a pointwise estimate for the convolution kernel for low frequency part. Also, to make the derivative losses sharp, we apply the estimate (1.7) to the high frequency part. Remark 1.1. Chen, Fan and Zhang [2,3] stated similar L p -L q estimates for the damped fractional wave equation. However, unfortunately, the proof seems incomplete. For the damped wave equation, we give a complete proof. Moreover, our argument remains valid for the damped fractional wave equation, through minor modifications.
In the proof of Theorem 1.1, we explicitly write a leading part of the convolution kernel of D(t)g in the low frequency part, which has the same coefficient as that of the heat kernel (see Lemmas 2.3 and 2.8 below). Accordingly, the difference in the low frequency part satisfies a better time-decay estimate. Theorem 1.2. Let 1 ≤ q ≤ p < ∞ with p = 1, s 1 ≥ s 2 , and β = (n − 1)| 1 2 − 1 p |. Then, there exists δ p > 0 such that for t ≥ 1, provided that the right-hand side is finite.
Remark 1.2. Nishihara [30] gives an improvement of the estimate of (1.3) in the 3-dimensional case of the form (1.11) where 1 ≤ q ≤ p ≤ ∞ (for other space dimensions, see [23,13,27,32]). In other words, D(t)f is asymptotically expressed as and it implies the high frequency part causing the derivative loss is explicitly given by W(t)f when n = 3. Therefore, combining (1.11) and (1.7), we can obtain similar estimates as (1.8), (1.10). However, our approach is direct and has broad utility.
Our next purpose is the application of Theorems 1.1 and 1.2 to the Cauchy problem of the nonlinear damped wave equation x) ∈ (0, ∞) × R n , u(0, x) = εu 0 (x), ∂ t u(0, x) = εu 1 (x), x ∈ R n , (1.12) where N (u) denotes the nonlinearity, (u 0 , u 1 ) is a given function, which denotes the shape of the initial data, and ε is a positive parameter, which denotes the size of the initial data.
Our concern is to prove the local and global existence of the solution, asymptotic behavior, and blow-up of solutions when the initial data do not belong to L 1 (R n ) in general. More precisely, we show the existence of the global solution with small data even for the critical nonlinearity.
Based on the linear estimates (1.3), many mathematicians studied the global existence and blow-up of solutions (see [9,10,11,12,13,15,16,18,19,20,21,22,23,24,27,30,34,36,37] and the references therein). In particular, from these studies, the critical exponent was determined as p c (n) = 1 + 2 n , provided that the initial data decay sufficiently fast at the spatial infinity. Here, the critical exponent means the threshold of the global existence and the blow-up of solutions for small initial data. More precisely, if p is larger than the critical exponent p c , then for any shape (u 0 , u 1 ), there exists ε 0 > 0 such that the solution exists globally in time for any ε ∈ (0, ε 0 ), and if p is smaller than the critical exponent p c , then there exists a shape (u 0 , u 1 ) and ε 0 > 0 such that the solution blows up in finite time for any ε ∈ (0, ε 0 ).
However, there are only few results when the initial data slowly decay, namely, do not belong to L 1 (R n ), at the spatial infinity. Nakao and Ono [26] studied the case (u 0 , u 1 ) ∈ H 1 (R n ) × L 2 (R n ) and they proved the global well-posedness with small data when p ≥ 1 + 4 n . We also refer the reader to [28] for the global existence of solutions with slowly decaying initial data in modulation spaces, but the nonlinearity should be a polynomial of u. Ikehata and Ohta [17] proved the global existence of solutions for small data (u 0 , u 1 ) ∈ (H 1 (R n )∩L r (R n ))×(L 2 (R n )∩ L r (R n )) when p > 1 + 2r n . Here r ∈ [1,2] (n = 1, 2), They also proved for any n ≥ 1 and for the nonlinearity N (u) = |u| p−1 u with 1 < p < 1 + 2r n , there exists (u 0 , u 1 ) ∈ (H 1 (R n ) ∩ L r (R n )) × (L 2 (R n ) ∩ L r (R n )) such that there is no global solution even if the size of the initial data ε is arbitrary small. Here we shall give a remark on their results. In the supercritical case p > 1 + 2r n , their solution belongs only to C([0, ∞); H 1 (R n )) ∩ C 1 ([0, ∞); L 2 (R n )) and we do not know whether u(t) ∈ L r (R n ). It is a natural question whether the solution u has the same integrability near the spatial infinity as the initial data. Narazaki and Nishihara [29] further considered the asymptotic profile of the solution under the assumption (u 0 , u 1 ) ∼ x −kn with k ∈ (0, 1]. They proved that when n ≤ 3 and p > 1 + 2 kn (which corresponds to the condition p > 1 + 2r n in terms of the Lebesgue space L r (R n )), the small data global existence holds and the solution is approximated by εG(t)(u 0 + u 1 ). Moreover, in [14], we extended the above results to higher dimensional cases in terms of the weighted Sobolev spaces where the symbol ∇ s stands for the Fourier multiplier F −1 ξ sf (ξ) (x). We showed that if p > 1+ 2r n and if the initial data satisfy (u 0 , ) and sufficiently small, then the global solution uniquely exists. However, in this setting, we cannot treat the critical case p = 1 + 2r n . In the present paper, based on the improved L p -L q estimates given in Theorem 1.1, we further generalize the results of [14] when the initial data belong to L r (R n ) with r ∈ (1, 2]. In particular, we prove the small data global existence in the critical case p = 1 + 2r n . This result is completely new when r ∈ (1, 2). We recall that when r = 1, p = 1 + 2 n , and N (u) = |u| p , the local solution blows up in a finite time even if the size of the initial data ε is arbitrary small, provided that the shape of the initial data (u 0 , u 1 ) ∈ (H 1 (R n ) ∩ L 1 (R n )) × (L 2 (R n ) ∩ L 1 (R n )) has positive integral average (see [37]). Namely, when r = 1, the critical power p = 1 + 2 n belongs to the blow-up case. On the other hand, when r = 2, as we explained before, Nakao and Ono [26] showed that the critical power p = 1 + 4 n belongs to the global-existence case. Our main result for the nonlinear problem (Theorem 1.4) shows that for r ∈ (1, 2), the critical exponent p = 1 + 2r n belongs to the global-existence case, although some restriction on the range of r is imposed. Also, we refer the reader to [35] in which the global existence of solutions to the critical semilinear heat n was proved, when the initial datum belongs to L r (R n ) and is sufficiently small.
To state our results, we first define a solution. We say that a function u ∈ L ∞ (0, T ; L 2 (R n )) is a mild solution of (1.12) if u satisfies the integral equation We assume that there exists p > 1 such that N ∈ C p0 (R) with some integer p 0 ∈ [0, p] and Before going to the global existence results, we first prepare the local existence of a unique H s ∩L r -mild solution based on the linear estimates in Theorem 1.1. We note that an H s ∩ L r -mild solution is also an H s -mild solution for any r ∈ (1, 2]. After introducing the existence of H s ∩L r -mild solution, we also discuss the blow-up alternative for H s -mild solution Theorem 1.3 (Local existence). Let n ≥ 1 and p > 1, and assume (1.13). Let s ≥ 0 and r ∈ (1, 2] satisfy [s] ≤ p 0 , r ≥ 2(n−1) n+1 and 1 < p < ∞, if 2s ≥ n, Let β = (n − 1) 1 r − 1 2 and let the initial data satisfy . Then, for any ε > 0, there exists T > 0 such that the problem (1.12) admits a unique H s ∩ L r -mild solution Moreover, for the lifespan of the H s -mild solution defined by we have the blow-up alternative: if T 2 (ε) < ∞, then the solution satisfies namely, the derivative loss in the linear estimate does not exceed 1. (ii) In the previous result [14], the local existence requires p ≥ max{1 + r n , 1 + r 2 }, which comes from estimates involving weighted Sobolev norms. Theorem 1.3 removes this condition and we do not need any restriction from below on p.
(iii) In Theorem 1.3, we show the blow-up criterion only for the H s -mild solution. It is difficult to obtain the blow-up criterion for the H s ∩L r -mild solution for r ∈ [1, 2) because the derivative loss prevents us from extending the local solution. Namely, the solution does not have the persistence property, which means that the solution u(t) belongs to the same space as the initial data with continuous dependence on the time variable.
We prove Theorem 1.3 in Section 3. Our proof is based on the L p -L q estimates given in Theorem 1.1 and the contraction mapping principle. To control the nonlinear term, we introduce an appropriate norm for the nonlinearity (see (3.3)) , which is inspired by Hayashi, Kaikina and Naumkin [9]. Then, to estimate the derivative of the nonlinearity, we apply the fractional chain rule.
Moreover, in the critical or supercritical case p ≥ 1 + 2r n , we have the global existence of the H s ∩ L r -mild solution for the small initial data. Theorem 1.4 (Global existence of H s ∩ L r -mild solution for small data). In addition to the assumption of Theorem 1.3, we suppose Then, there exists ε 0 = ε 0 (n, p, r, s, u 0 H s ∩H β r , u 1 H s−1 ∩L r ) > 0 such that for any ε ∈ (0, ε 0 ], the problem (1.12) admits a unique global H s ∩ L r -mild solution The reason why the global solution exists even in the critical case p = 1 + 2r n is that the nonlinearity N (u) decays faster than the linear part at the spatial infinity. More precisely, we see that N (u) ∈ L σ1 (R n ) with σ 1 = max{1, r p } < r (see Section 3), while the linear part of the solution satisfies εD(t)(u 0 +u 1 )+ε∂ t D(t)u 0 ∈ L r (R n ). This enables us to control the nonlinearity even in the critical case p = 1 + 2r n . On the other hand, if we consider the global existence of the H s -mild solution and do not require that u ∈ C([0, ∞); L r (R n )), then we do not need to impose r ≥ 2(n−1) n+1 . , 2 , and Let the initial data satisfy Then, there exists ε 0 = ε 0 (n, p, r, s, u 0 H s ∩L r , u 1 H s−1 ∩L r ) > 0 such that for any ε ∈ (0, ε 0 ], the problem (1.12) admits a unique global H s -mild solution satisfying Remark 1.4. Theorem 1.5 states that we can relax the condition r ∈ [ 2(n−1) n+1 , 2] in the case n ≥ 5 if we do not require that u ∈ C([0, ∞); L r (R n )). Indeed, concerning the assumption on the range of r, we see that if and only if n ≥ 5.
Therefore, for n ≥ 5, the assumption of r in Theorem 1.5 is weaker than that of Theorem 1.4.
Furthermore, in the supercritical case p > 1 + 2r n , we prove that the solution is approximated by that of the linear heat equation (1.4) with the initial data ε(u 0 + u 1 ). This extends the results by [29] to all space dimensions. Theorem 1.6 (Asymptotic behavior of global solutions). Let u be the global H s ∩ L r -mild solution constructed in Theorem 1.4. We further assume p > 1+ 2r n . Then, for any δ > 0 and t ≥ 1, , where q = r if 2s ≥ n and q = min{r, 2n p(n−2s) } if 2s < n, and the implicit constant depends on δ.
We next handle the subcritical case p < 1 + 2r n . We have the sharp lower bound and an almost sharp upper bound of the lifespan. To state the result, we define the lifespan of H s ∩ L r -mild solution by  Then, there exists ε 1 = ε 1 (n, p, r, s, u 0 H s ∩H β r , u 1 H s−1 ∩L r ) > 0 such that for any ε ∈ (0, ε 1 ], the lifespan of H s ∩ L r -mild solution T r (ε) is estimated as where ω = 1 p−1 − n 2r and the implicit constant is independent of ε. We prove Theorem 1.7 in Section 6. The proof is a slight modification of Theorem 1.3.
The rate −1/ω of ε in the estimate (1.17) is optimal in the sense that we cannot obtain the estimate T r (ε) ε −1/ω−δ for any δ > 0 in general. More precisely, we give the following upper estimate of T 2 (ε) (see also Remark 1.5). . In addition to the assumption of Theorem 1.3, we assume that N (u) = ±|u| p with p < 1 + 2r n . Then, for any δ > 0, there exist initial data (u 0 , ) and a constant ε 2 = ε 2 (n, p, r, s, δ) > 0 such that for any ε ∈ (0, ε 2 ], the lifespan of the H s -mild solution defined by (1.14) is estimated as where the implicit constant is dependent on δ but independent of ε.
We will prove more general blow-up results in Section 7. The proof is based on the argument of [5], in which the blow-up of solutions to the semilinear wave equation with time-dependent damping was studied via an analysis of ordinary differential inequality. Remark 1.5. By the definitions of lifespans of H s -mild solution (1.14) and H s ∩ L r -mild solution (1.16), we immediately have T r (ε) ≤ T 2 (ε). From this and Theorems 1.7 and 1.8, we see that which gives an almost optimal estimate for both T r (ε) and T 2 (ε).
The rest of the paper is organized as follows. In Section 2, we prove our L p -L q estimates. Theorems 1.3 and 1.4, namely, the local and global existence of H s ∩ L rmild solution are proved in Section 3. Then, Section 4 is devoted to the proof of Theorem 1.5, that is, the global existence of H s -mild solution. In Section 5, we give a proof of Theorem 1.6. Finally, in Sections 6 and 7, we give proofs of Theorems 1.7 and 1.8, respectively.
We introduce notations used throughout this paper. For the variable x = (x 1 , . . . , x n ) ∈ R n , we use the notation of derivatives ∂ j = ∂ ∂xj (j = 1, . . . , n). Let 1 I be the characteristic function of I ⊂ R. The notation X ∼ Y stands for X Y and Y X. Let χ ∈ C ∞ 0 (R) be a cut-off function satisfying χ(r) = 1 for |r| ≤ 1 and χ(r) = 0 for |r| ≥ 2. We write For a function f : R n → C, we define the Fourier transform and the inverse Fourier transform by Moreover, for a measurable function m = m(ξ), we denote the Fourier multiplier m(∇) by For s ∈ R and p ∈ (1, ∞), we denote the usual Sobolev space by H s p (R n ) and its homogeneous version byḢ s p (R n ).

Proof of the L p -L q estimates
We divide D(t) into low and high frequency parts Let d be the multiplier of D 1 , namely 2.1. L p -L q estimates for linear damped wave equation. First, we focus on the low frequency part. The L p -L q estimates of the low frequency part is similar to that of the heat propagator. The first step is to get the pointwise estimate for the kernel d, which gives the value of the L r -norm of the kernel d. The second step is to get the L p -L q estimates whose proof is based on Young's inequality and the value of the L r -norm of the kernel d.
We have the following pointwise estimate of the kernel d.
Moreover, for any j ∈ N, For the proof of Proposition 2.1, we observe the following lemmas.
Proof. By changing variable η = t 1 2 ξ, we have When t 1 2 σ ≥ 1 2 , the integral on the right hand side is bounded by which concludes the proof.
Proof. For k = 0, we have C 0,0 = 1. We assume that (2.3) holds for some k ∈ Z ≥0 . For simplicity, we define C Then, a direct calculation yields Hence, the constants C (k+1) l,m are defined by Proof of Proposition 2.1. We prove the inequality with respect to the right side in the minimum in (2.1) and (2.2), i.e.
for any s ≥ 0. Since It remains to prove the inequality with respect to the left side in the minimum in (2.1) and (2.2). To obtain the decay with respect to |x|, we divide d into two Without loss of generality, we may assume that |x| ∼ |x 1 |. First, we prove the estimate for d 1 with respect to the left side in the minimum in (2.2). Namely, we show By (2.5) and Lemma 2.3, for k ∈ Z ≥0 and |ξ| ≤ 1 4 , we have (2.7) Since ∂ k 1 χ < 1 8 (|ξ|) 1 for k ∈ Z ≥0 , integration by parts, (2.7), and Lemma 2.2 yield Secondly, we show the inequality for d 1 with respect to the left side in the minimum in (2.1) i.e. ||∇| s d 1 (t, x)| |x| −(s+n) . For s ≥ 0, we assume |x| ≥ t 1 2 , otherwise the desired bound follows from (2.6). Then, we further divide the multiplier d 1 into two parts By (2.7), we have We set j := [s] + n + 1 for simplicity. Integration by parts j-times yields Finally, Lemma 2.2 concludes At last, we go on to the estimate for d 2 with respect to the left side in the minimum in (2.1) and (2.2). More precisely, we prove the better estimate ||∇| s d 2 (t, x)| |x| −j for any j ∈ N. Since L(t, ξ) is smooth with respect to ξ, 2] (|ξ|) for |ξ| ≤ 2 and k ∈ Z ≥0 . Since Hence, integration by parts yields t for any j ∈ N. Combining (2.6) and (2.8)-(2.11), we obtain the desired bound.
In (2.7), we neglect the decay factor e − t 2 of d 1 . By keeping this factor, the proof of Proposition 2.1 yields that for any s ≥ 0 and j ∈ Z ≥0 , we have Proposition 2.1 and Young's inequality lead to the following linear estimate for the low frequency.
Proof. It reduces to show the bound for s ≥ 0 and 1 ≤ p ≤ ∞. Indeed, (2.12) and Young's inequality show It remains to show (2.12). The case p = ∞ is a direct consequence of Proposition 2.1. For s > 0 and 1 ≤ p < ∞, Proposition 2.1 implies On the other hand, for s = 0, Proposition 2.1 with j = n + 1 implies This finishes the proof.
Next, we consider the high frequency part of D(t). To estimate the high frequency part, we reduce the high frequency part to the wave propagator by using Mikhlin's multiplier theorem and apply the L p -estimate of the wave propagator.
This proposition follows from the L p bound for linear wave solutions, which was proved by Sjöstrand [33] and improved by Miyachi [25] and Peral [31].
Theorem 2.6 (L p estimates for the wave equation). Let 1 < p < ∞. Then, for Proof of Proposition 2.5. Since we focus on the high frequency and sin(t Because a simple calculation shows for ξ = 0 and α ∈ Z n ≥0 . Hence, Mikhlin's multiplier theorem (see [8,Theorem 6.2.7]) shows that Owing to Theorem 2.6, this estimate yields (2.13).
The estimate (1.8) in Theorem 1.1 follows from Propositions 2.4 and 2.5.
2.2. L p -L q estimates for the derivative of the solution. Next, we prove the second statement (1.9) of Theorem 1.1. We use the same notation as in Remark 2.1. We define the Fourier multipliers D Therefore, Remark 2.1 and Proposition 2.4 imply the desired estimate for the low frequency part ∂ t D 1 (t)g. Moreover, the same argument as in the proof of Proposition 2.5 shows the estimate for the high frequency part ∂ t D 2 (t)g.

2.
3. L p -L q estimates for the difference. Now, we prove Theorem 1.2. Set We recall that d is the multiplier of the low frequency part of D.
We show the pointwise decay estimates for m. .
For the proof of Proposition 2.7, we observe the following two lemmas.
k,l are the constants in Lemma 2.3.
Proof. For k = 1, we have D 1,1 = −2, because ∂ 1 e −t|ξ| 2 = −2tξ 1 e −t|ξ| 2 . We assume that (2.14) holds for some k ∈ N. For simplicity, we define D Then, a direct calculation yields Hence, the constants D  Furthermore, for k ∈ N and |ξ| ≤ 1 4 , we have Proof. We note that for |ξ| ≤ 1 4 , where in the second inequality we used The triangle inequality with (2.5) implies Similarly, Lemmas 2.3 and 2.8 yield that, for k ∈ N, Proof of Proposition 2.7. Lemmas 2.2 and 2.9 give To obtain the decay with respect to |x|, we assume |x| ≥ t 1 2 , otherwise the desired bound follows from (2.15). We divide m into four parts m = m 1 + m 2 + m 3 + m 4 : Without loss of generality, we may assume that |x| ∼ |x 1 |.
We set j := [s] + n + 3 for simplicity. Integration by parts j-times yields , we proceed the calculation with Lemma 2.9 to obtain Furthermore, Lemma 2.2 gives provided that |x| ≥ t Proof. We divide G(t) into low and high frequency parts Since Proposition 2.7 gives for s ≥ 0 and 1 ≤ p ≤ ∞, the same argument as in the proof of Proposition 2.4 yields Set r := −(s 1 − s 2 ) + 2([s 1 − s 2 ] + n + 1). Since r > n, integration by parts gives Hence, Young's inequality and the well-known L p -L q estimates for the heat equation imply for t ≥ 1 and for any large N ∈ N.

Local and global existence
Based on the linear estimates, we define the following function spaces. For T ∈ (0, ∞], s ≥ 0, and r ∈ (1, 2], we define We also consider a wider function space with the norm u Z(T ) := u L ∞ (0,T ;L 2 (R n )) .
Then, we can see that X(T, M ) is a closed subset of Z(T ) for T ∈ (0, ∞) (see Lemma A.1). We shall find a local solution by constructing an approximate sequence in the ball X(T, M ) and prove its convergence with respect to the metric To this end, for the estimate of the nonlinear term satisfying (1.13) and |N (u)| |u| p , we define an auxiliary space. For T ∈ (0, ∞], s ≥ 0, r ∈ (1, 2], and 1 < p ≤ 2n/(n − 2s) if n > 2s and 1 < p < ∞ if n ≤ 2s we define the space Y (T ) as follows. When s > 1, we define and when 0 < s < 1 we define We remark that the condition p ≤ 1 + min{n,2} n−2s if 2s < n implies σ 1 ≤ σ 2 . We also note that the choices of the parameters η, σ 1 , σ 2 are quite natural. Indeed, in the proof of Theorems 1.3 and 1.4, we use the norm of Y (T ) with ψ = N (u), that is, the nonlinear term of the equation (1.12). Roughly speaking, if u belongs to X(∞), then |u(t)| p L 2 = u(t) p L 2p decays like t − n 2 ( p r − 1 2 ) , and hence, we expect |∇| s−1 |u(t)| p L 2 t −η . Also, roughly speaking, if u ∈ X(∞), then the Sobolev embedding implies u(t) ∈ L r (R n ) ∩ L 2n/(n−2s) (R n ), and hence, we expect |u(t)| p ∈ L r/p (R n ) ∩ L 2n/(p(n−2s)) (R n ) and |u(t)| p L γ decays like t − n 2 ( p 3.1. Local and global existence. Hereafter, we assume the condition in Theorem 1.3. n for any T > 0 and ψ ∈ Y (T ).
Proof. The first assertion is easily derived by modifying the following argument, and we omit its proof.
Finally, we estimate III. Theorem 1.1 with p = r, s 1 = s 2 = 0, and q ∈ [σ 1 , σ 2 ] determined later implies First, we have (3.8). Next, for the first term, we choose q as with sufficiently small ǫ > 0. Consequently, we have n , which completes the proof.
for any T > 0 and u ∈ X(T ).
Proof. Let γ ∈ [σ 1 , σ 2 ]. Then, by the assumption (1.13), we have |N (u)| |u| p and we calculate By the definition of σ 1 and σ 2 , we see that This and the interpolation between L r and L 2 in the case that pγ ∈ [r, 2] and betweenḢ s and L 2 in other case imply t [9,Lemma 2.3]
Then, we obtain Next, we consider the case of s > 1. Lets be the fractional part of s, namely, s := s − [s]. Using the Faá di Bruno formula (see the proof of Lemma 2.5 in [14]), the fractional Leibniz rule, and the fractional chain rule (see for example [4, Proposition 3.1, 3.3]), we see that where q 0 and q j (k) (j = 1, 2, . . . , [s]) are given so that if 2s < n, (3.11) We postpone the proof of the existence of such exponents to Appendix B. We also note that when [s] = 1, the above inequality is interpreted as Finally, by the interpolation, we obtain . The proof is complete. Now we prove the local existence of the solution.
We show that u ∈ C([0, T ); H s (R n ) ∩ L r (R n )). The solution u satisfies the integral equation Since the linear part of the solution obviously satisfies this property, it suffices to show that t 0 D(t − τ )N (u(τ )) dτ ∈ C([0, T ); H s (R n ) ∩ L r (R n )).

By Theorem 1.1, we have
for t ∈ [0, T ) and τ ∈ [0, t] and the right-hand sides are bounded independently of t. Therefore, we can apply the dominated convergence theorem in the Bochner integral and thus the continuity holds. We also prove that ∂ t u ∈ C([0, T ); H s−1 (R n )). Since the linear part of the solution obviously satisfies this property, it suffices to show that We note that N (u) Y (T ) is bounded. This and Theorem 1.1 implies for t ∈ [0, T ) and τ ∈ [0, t], and the right-hand sides are bounded. Therefore, for any fixed t ∈ [0, T ), D(t − ·)N (u(·)) ∈ L ∞ (0, t; H s (R n )) holds. Moreover, this also leads to ∂ t D(t − ·)N (u(·)) ∈ L ∞ (0, t; H s−1 (R n )). Indeed, by (1.9) in Theorem 1.1, we have for t ∈ [0, T ) and τ ∈ [0, t], and the right-hand sides are bounded independently of t. Therefore, by the Lebesgue convergence theorem in the Bochner integral and D(0) = 0, we see that which implies (3.20). Next, we show that under the assumptions of Theorem 1.3, for any fixed T 0 > 0, the H s -mild solution on the interval [0, T 0 ) is unique. This also implies the uniqueness of H s ∩ L r -mild solution, because an H s ∩ L r -mild solution is also an H s -mild solution. Let T 0 > 0 and fix it, and let u, v be H s -mild solutions of (1.12) with same initial data ε(u 0 , u 1 ) ∈ H s (R n ) × H s−1 (R n ). Let T 1 ∈ (0, T 0 ) be an arbitrary number. We define Then, there exists a constant M > 0 such that u X2(T1) + v X2(T1) ≤ M . From this and the same argument as deriving (3.16) with r = 2, we can see that for any T ∈ [0, T 1 ] with some constant C 3 = C 3 (n, s, p) > 0. Indeed, for t ∈ [0, T ], we can obtain in the same manner as (3.19), where q and γ are defined in (3.17) and (3.18). Thus, the Gronwall inequality implies u ≡ v on [0, We next prove the locally Lipschitz continuity of the solution map Let M > 0 and we consider the ball for the initial data. Then, by the proof of the existence part above, we find T > 0 depending only on M such that for each ε(u 0 , u 1 ), there exists a unique solution u ∈ X(T, 2M ). Let ε(u 0 , u 1 ), ε(v 0 , v 1 ) ∈ B(M ) and let u, v be the associated solutions, respectively. From this and the same argument before, we see that Therefore, the Gronwall inequality implies which shows the locally Lipschitz continuity of the solution map.
Finally, we prove the blow-up alternative for H s -mild solution, namely, T 2 (ε) < ∞ implies (1.15). Let us suppose T 2 (ε) < ∞ and Then, there exist a constant M > 0 and a sequence {t m } ∞ m=1 ⊂ [0, T 2 (ε)) such that t m → T 2 (ε) (m → ∞) and We note that, from the above proof of the local existence of the H s -mild solution in the case r = 2, we deduce that there exists T 1 > 0 independent of {t m } ∞ m=1 such that we can construct the solution u ∈ C([t m , t m + T 1 ); H s (R n )), ∂ t u ∈ C([t m , t m + T 1 ); H s−1 (R n )).
However, letting m → ∞, this contradicts the definition of the lifespan T 2 (ε). This completes the proof.
Proof of Theorem 1.4. Let p ≥ 1 + 2r n . Let T > 0 be an arbitrary finite number. We define M (ε) by (3.13) and consider the mapping (3.14) on X(T, M (ε)). Then, applying Lemmas 3.1 and 3.2, we have for u, v ∈ X(T, M (ε)), with some constants C 1 , C 2 > 0 independent of T , instead of (3.15) and (3.16), respectively. Indeed, the first estimate is a direct consequence of Lemmas 3.1 and 3.2. The second estimate is obtained by a similar way to the proof of (3.13). More precisely, we have where γ and q are defined by (3.18) and (3.17) and in the last inequality we have used the Sobolev inequality with s ′ = n 1 2 − 1 q(p−1) and the interpolation inequality to obtain Here we also note that the definition of q implies 0 ≤ s ′ ≤ s. Therefore, it suffices to show holds under the condition p ≥ 1 + 2r n . To prove (3.22), we divide the integral into The term A is estimated as By noting that − n 1, since 1 γ = 1 q + 1 2 and p ≥ 1 + 2r n . Next, the term B is estimated as When 2s ≥ n, 1 γ = 1 2 + δ with sufficiently small δ, and hence, − n 1, since 1 γ = 1 q + 1 2 and p ≥ 1 + 2r n . When 2s < n, the definition of γ is 1 γ = 2n (p−1)(n−2s) 1, since 1 γ = 1 q + 1 2 and p ≥ 1 + 2r n . Thus, we have (3.22). Therefore, by taking ε 0 > 0 so that holds, the mapping Ψ becomes a contraction mapping on X(T, M (ε)) with respect to the metric of Z(T ), and thus we have the solution u ∈ X(T, M (ε)). Moreover, the uniqueness has been already proved in the proof of Theorem 1.3. Since T is arbitrary, the solution is global. Moreover, u ∈ X(∞, M (ε)). Indeed, since we have u X(T ) ≤ M (ε) for arbitrary T ∈ (0, ∞) from (3.21) and (3.23), we get u X(∞) ≤ M (ε). We also have since Lemma 3.2 holds for any T ∈ (0, ∞).
From this and the assumption p ≥ 1 + 2r n , it follows that r > 2 p , which also implies σ 1 < r. Therefore, we can apply the same argument as Lemmas 3.1 and 3.2. We note that the norm ofX(T ) does not involve L r -norm, and hence, we do not need to estimate the L r -norm of the solution. From this and repeating the same procedure as the proof of Theorem 1.4, we can prove the existence of a global solution.

Asymptotic behavior of the global solution
In this section, we give a proof of Theorem 1.6.
Proof of Theorem 1.6. Let u be the global solution constructed in Theorem 1.4 and let p > 1 + 2r/n. We see that From Theorems 1.1 and 1.2, we obtain . Therefore, it suffices to estimate w N L (t). We note that Lemma 3.2 gives Theorem 1.4 and (3.24)). In the same manner as in the proof of Lemma 3.1, we see that Hereafter, δ denotes an arbitrary small positive number, and the implicit constants are dependent on δ. We also have where q = min{r, σ 2 } = min{r, 2n p(n−2s)+ } ∈ [σ 1 , σ 2 ] and σ 1 , σ 2 are defined in (3.5), (3.6). Here we note that q > r/p holds under the assumption of Theorem 1.6.

Consequently, we have
. This completes the proof.

Lower bound of the lifespan
In this section, we prove Theorem 1.7.
Proof of Theorem 1.7. By Lemma 3.1, when 1 < p < 1 + 2r/n, we see that there exists a constant C 1 > 0 such that for any T > 0 and ψ ∈ Y (T ). Also, it follows from Lemma 3.2 that N (u) Y (T ) ≤ C 1 u p X(T ) for any T > 0 and u ∈ X(T ). Based on these estimates, we repeat the argument in the proof of Theorem 1.3. First, we have (3.12) for any T > 0. We define a constant M (ε) by (3.13). Let Ψ be a mapping on X(T, M (ε)) given by (3.14). Then, instead of (3.15) and (3.16), in this case we obtain with some constant C 2 > 0 for any u, v ∈ X(T, M (ε)) and T > 0. Therefore, as long as holds, the mapping Ψ is contractive on X(T, M (ε)) with respect to the metric of Z(T ), and we can construct a unique local solution. We take ε 1 > 0 sufficiently small so that namely, (6.1) formally holds for T = 0 and ε = ε 1 . Let ε ∈ (0, ε 1 ] and letT (ε) be the first time which gives the identity in the condition (6.1), that is, where ω = 1 p−1 − n 2r and C 3 = C 3 (n, s, r, p, u 0 H s ∩H β r , u 1 H s−1 ∩L r ) > 0 is a constant independent of ε. Since we can construct a solution until the timeT (ε), we haveT (ε) ≤ T r (ε). Finally, retaking ε 1 > 0 smaller if needed so that C 3 ε −1/ω 1 ≥ 4/3, we have for any ε ∈ (0, ε 1 ], which gives the desired estimate (1.17).

Upper bound of the lifespan
In this section, we prove Theorem 1.8. More precisely, we will give more general blow-up result (see Proposition 7.2), and Theorem 1.8 will follow as a corollary of it. A similar result in L 1 (R n )-data setting and in the Fujita-subcritical case, i.e. p < 1 + 2 n was obtained in [5]. We define a smooth compactly supported function ψ ∈ C ∞ ([0, ∞); [0, 1]) as Set ψ(x) := ψ(|x|) for x ∈ R n , and for R > 0 let ψ R (x) = ψ(x/R). For p > 1, and A > 0, we define µ = µ(p, A) by For n ∈ N, p > 1, l ∈ N satisfying l > 2p ′ , where p ′ := p/(p − 1), and for φ ∈ C ∞ 0 (R n ) with φ ≥ 0, we also define A(n, p, l, φ) as We derive an ordinary differential inequality for the weighted average of the solution, i.e.
We recall a blow-up result obtained in [5], which gives an upper estimate of the lifespan of solutions to (1.12) in a general setting. We note that in the following theorem, we do not need any condition on p such as p < 1 + 2r n , but we impose certain condition on the test function φ. Proposition 7.1 (Proposition 1.3 in [5]). Let s ≥ 0, p ∈ (1, ∞) and (u 0 , u 1 ) ∈ H s (R n ) × H s−1 (R n ), and let u be the associated H s -mild solution to (1.12) constructed in Theorem 1.3 (the solution obtained by applying Theorem 1.3 with r = 2). Assume that there exists φ ∈ S(R n ; [0, ∞)) such that the inequalities A(n, p, l, φ) .
Then, the estimate is valid for t ∈ [0, T 2 (ε)). Moreover, the lifespan T 2 (ε) of the H s -mild solution u is estimated as Remark 7.1. From the estimate (7.3), we can expect that the blow-up rate of the solution u is similar to that of the second order ordinary differential equation y ′′ (t) = y(t) p , which indicates the wave-like behavior of the solution near the blowup time. However, we remark that the estimate (7.3) does not directly imply the blow-up rate of the solution, because there is a possibility that the blow-up time of (u, ∂ t u)(t) H 1 ×L 2 is earlier than that of J φ (t).
Proposition 7.1 means that the condition (7.2) is a sufficient condition for the blow-up of a solution. Indeed, we prove that if p ∈ (1, 1 + 2r n ), we take the test function φ = ψ R(ε) with an appropriate scaling parameter R(ε), which will be defined later, we ensure the condition (7.2) for any ε > 0 and show an upper estimate of the lifespan of solutions to (1.12) for sufficiently small ε > 0.
Appendix A.
In this appendix, we prove that X(T, M ) is a closed subset of Z(T ) if T > 0 is finite. Let s ≥ 0, r ∈ (1, 2], T ∈ (0, ∞). Since T is finite, we note that the topology of X(T ) with respect to the norm (3.1) is the same as the usual topology of L ∞ (0, T ; H s (R n ) ∩ L r (R n )). Proof. First, it is obvious that X(T ; M ) ⊂ Z(T ). Therefore, it suffices to show that for any sequence in X(T ; M ) converging in Z(T ), its limit belongs to X(T ; M ). Let {u j } ∞ j=1 ⊂ X(T ; M ) converges in Z(T ) and let u be its limit. We note that L ∞ (0, T ; H s (R n ) ∩ L r (R n )) = L 1 (0, T ; H −s (R n ) + L r ′ (R n )) * , where r ′ = r/(r − 1). This and the separability of L 1 (0, T ; H −s (R n ) + L r ′ (R n )) (in general, the sum the two separable normed space is separable) enable us to apply the sequential Banach-Alaoglu theorem [ converge in the space of the distribution D ′ ((0, T ) × R n ) and hence, we obtain Thus, by the uniqueness of the limit of distribution implies u = v, which shows u ∈ X(T ; M ).
Appendix B.
First, we consider the case of n > 2s. Therefore, a j < 1/2 for all j. We define q j (k) such that 1 q j (k) = 1 2 − a j .