THE LIFESPAN OF SOLUTIONS TO SEMILINEAR DAMPED WAVE EQUATIONS IN ONE SPACE DIMENSION

. In the present paper, we consider the initial value problem for semilinear damped wave equations in one space dimension. Wakasugi [7] have obtained an upper bound of the lifespan for the problem only in the subcritical case. On the other hand, D’Abbicco & Lucente & Reissig [3] showed a blow-up result in the critical case. The aim of this paper is to give an estimate of the upper bound of the lifespan in the critical case, and show the optimality of the upper bound. Also, we derive an estimate of the lower bound of the lifespan in the subcritical case which shows the optimality of the upper bound in [7]. Moreover, we show that the critical exponent changes when the initial data are odd functions.

1. Introduction. In this paper we consider the initial value problem for semilinear damped wave equations: where F (v) = |v| p or F (v) = |v| p−1 v with p > 1, f, g have appropriate regularity, and ε > 0 is a "small" parameter. When F (v) = |v| p and 1 < p < 3, Wakasugi [7] showed that an upper bound of the lifespan which is the maximal existence time of solutions of (1) is Cε −(p−1)/ (3−p) for some positive initial data, where C is a positive constant independent of ε. The proof of [7] is based on the "test function method". (See 165p. in [7].) To be more precise, the upper bound of the lifespan in general spatial dimensions (n ≥ 1) is given by a constant times ε −(p−1)/{2−n(p−1)} for 1 < p < p F (n) := 1 + 2/n, where p F (n) (the Fujita exponent) is the critical exponent for the semilinear heat equation.
In the case of F (v) = |v| p with p = 3, D'Abbicco & Lucente & Reissig [3] showed that the solution of (1) blows up in finite time if the initial data satisfy some positivity conditions, and have a compact support. However, they did not discuss about the estimates of the lifespan. We note that D'Abbicco [2] has obtained a global existence result for (1) in the case of p > 3 if ε is "small" enough.

KYOUHEI WAKASA
In the proof of [3], they reduced the problem to the following semilinear wave equations: by setting u(x, t) = (1 + t)v(x, t), where F (u) = |u| p . Actually in [3], (2) is studied in general spatial dimensions: where n ≥ 1. Let p c (n) = max {p F (n), p 0 (n + 2)} , where p 0 (n) is the Strauss exponent, that is the positive root of the quadratic equation (n − 1)p 2 − (n + 1)p − 2 = 0. When n = 2, 3, they showed that the problem (3) has a global in-time solution for p > p c (n) if ε is "small" enough and (f, g) have a compact support. On the other hand, the solution of (3) blows up in finite time if 1 < p ≤ p c (n) in the case of n ≥ 1. Since p 0 (3) = 1 + √ 2 < 3 = p F (1), we see that p c (1) = 3.
To state our result, we define the lifespan T ε of the C 2 -solution of (2) by T ε ≡ T ε (f, g) := sup{T ∈ [0, ∞) : There exists a unique solution u ∈ C 2 (R × [0, T )) of (2).} for arbitrarily fixed (f, g) ∈ C 2 (R) × C 1 (R). Our purpose in the present paper is to show the followings for the problem (2). The first one is to derive an estimate of the upper bound of the lifespan in the case of p = 3, and show the optimality of the upper bound. Namely we give an estimate of the lifespan from below which has the same order with respect to ε as the upper bound. The second one is to show the optimality of the result of [7] in the case of 1 < p < 3. The third one is to give an alternative proof of the estimates of [7]. Finally, the forth one is to show the critical exponent changes to 1 + √ 2 from 3 when the initial data are odd functions. Our proof is based on the iteration argument which was introduced by John [4]. For the critical case in Theorem 1.2 and Theorem 1.4, we apply the "slicing method" which was introduced by Agemi & Kurokawa & Takamura [1].
The following theorem shows that the optimality of the upper bound of [7] in the subcritical case 1 < p < 3. (2). Assume that both f ∈ C 2 (R) and g ∈ C 1 (R) have compact support contained in {x ∈ R : |x| ≤ 1}. Then, there exists a positive constant c = c(f, g, p) such that holds for ε > 0.
To derive a blow-up result, we require the following assumptions on the data: Let f ≡ 0 and g ∈ C 1 (R) does not vanish identically.
Then, we have the following.
Remark 1.1. Due to (7), the lower bound in the case of p = 3 in (5) is optimal. However, if the initial data are odd functions, we obtain different estimates of the lifespan and different critical exponent.
Remark 1.2. One can also obtain a similar estimate to (9) in the case of 1 < p ≤ 2. See Remark 5.1.

Remark 1.3.
We note that the estimates of (9) in the case of 2 < p < 1 + √ 2 holds when f, g are odd functions, and satisfy the same assumptions in Theorem 1.1. Hence, the estimates of (9) is an improvement of (5) for small ε, because is equivalent to p > 1.
To derive a blow-up result when the initial data are odd functions, we require the following assumptions on the data: Then, we have the following.
. Assume (11). Then, there exist positive constants ε 0 = ε 0 (f, g, p) and C = C(f, g, p) such that holds for any ε with 0 < ε ≤ ε 0 . Remark 1.4. We note that Theorem 1.3 and Theorem 1.4 do not hold for F (u) = |u| p , because |u(x, t)| p is not an odd function with respect to x.
Remark 1.5. There exists some initial data which satisfies the assumptions in Theorem 1.1 and Theorem 1.4. However, the estimates (5) does not contradict to (12) for small ε, because of (10).
Remark 1.6. Making use of the iteration argument in [4] and [1], it would be able to get the optimal estimates of the lifespan in the case of two and three space dimensions.
This paper is organized as follows. In the next section, we prepare some definitions and lemmas. The proofs of Theorem 1.1 and Theorem 1.2 shall be discussed in Section 3 and Section 4, respectively. The proofs of Theorem 1.3 and Theorem 1.4 are obtained in Section 5 and Section 6, respectively.

2.
Preliminaries. In this section, we give some definitions and useful lemmas.
We define When F (u) = |u| p−1 u and (f, g) are odd functions, if u is the C 2 -solution of (2) then −u(−x, t) is the solution to the problem (2). Making use of the uniqueness of the C 2 -solution to (2), we see that u(x, t) is odd function with respect to x. Therefore, in that case, we see that it is sufficient to consider the following integral equation: where we set Next, we prepare some useful lemmas for proving Theorem 1.3.

THE LIFESPAN OF SOLUTIONS TO SEMILINEAR DAMPED WAVE EQUATIONS 1269
For the proof, see e.g. Lemma 2.1 in Kubo & Osaka & Yazici [5].
Lemma 2.2. Let p > 2 and σ > 0, and let E σ (τ ) be a function defined by for τ ≥ 0. Then there exists a positive constant C p,σ such that Proof. We set which is the left-hand side of (20). We shall estimate I(x, t) on the following two domains: (ii) Estimation in D 2 (x, t). Since 1 + s ≤ 1 + s + y for y ≥ 0, we have Replacing the domain of integration by and changing the variables in the above integral by we get Making use of Lemma 2.1 with setting ν = p − 2 > 0, we have Then, the β-integral is dominated by . This completes the proof.
3. Proof of Theorem 1.1. In this section, we prove Theorem 1.1. First of all, we introduce a Banach space which is equipped with a norm We shall construct a solution of the integral equation (15) in X under suitable assumption on T such as (27) below. Define a sequence of functions {u n } n∈N ⊂ X by where F (u) = |u| p or F (u) = |u| p−1 u, L and u 0 are given by (14) and (13), respectively. It follows that The following a priori estimate plays a key role in the proof of Theorem 1.1.
for τ ≥ 0. Then, there exists a positive constant C p such that Proof. We divide the proof into two cases, 1 < p < 2 and 2 ≤ p ≤ 3.
(i) Estimation in the case of 1 < p < 2. The left-hand side in (26) is dominated by . Therefore, we get (26) in the case of 1 < p < 2.
(ii) Estimation in the case of 2 ≤ p ≤ 3. Making use of the support condition for V ∈ X, we have for (x, t) ∈ R × [0, T ]. Therefore, we get (26) in the case of 2 ≤ p ≤ 3. This completes the proof of Lemma 3.1. Now, we move on to the proof of Theorem 1.1. First of all, we take T > 0 such that where C p is the one in Lemma 3.1.
Analogously to the proof of Theorem 1.2 in [6] (see p.16 in [6]), we see from Lemma 3.1 that {u n } n∈N is a Cauchy sequence in X, provided (27) holds. Since X is complete, there exists u ∈ X such that u n converges to u in X. Therefore u satisfies the integral equation (15), so that u is the C 2 -solution of (2). Hence, the proof of Theorem 1.1 is completed.
4. Proof of Theorem 1.2. In this section, we prove Theorem 1.2. We show that the solution to the following integral equation blows up in finite time: is a solution of (28), then u satisfies u(x, t) ≥ 0 for (x, t) ∈ R × [0, ∞) by g(x) ≥ 0 for all x ∈ R. Therefore, this u must solve the equation (15) with F (u) = |u| p−1 u by the uniqueness of solutions to (2). Before proving Theorem 1.2, we prepare some definitions and lemmas. For T > 0, we define the following domains: where Lemma 4.1. Let p > 1, c 0 > 0, and let us define a sequence {C p,j } ∞ j=1 by where and Then, we have the following relation: Since this lemma follows from Lemma 3.1 in [6], if C a,j , F p,a , E p,a and k a are replaced by C p,j , F p , E p , and k p , respectively, we omit the proof. Next, we derive a lower bound of the solution to (28) which is a first step of our iteration argument (for the proof, see e.g. Lemma 3.2 in [6]).
where c 0 = 1 2 1 −1 g(y)dy > 0. Our iteration argument will be done by using the following estimates.
for (x, t) ∈ Σ 0 , and for (x, t) ∈ Σ j , where Σ 0 and Σ j are defined in (29). Here C p,j is the one in (31) with c 0 = 1 2 1 −1 g(y)dy > 0 and a j is defined by

It follows from
Since 1 ≤ (t − x)/l 1 for Σ 1 , the β-integral is estimated as follows: Therefore, (39) holds for j = 1. Assume that (39) holds for some j ∈ N. Let (x, t) ∈ Σ j+1 . Define Replacing the domain of integration in (41) by D j (x, t), and making use of (21), we have Noticing that D j (x, t) ⊂ Σ j for (x, t) ∈ Σ j+1 and putting (39) into the integral above, we have Analogously to the case of j = 1, we get in Σ j+1 . Making use of integration by parts in the integral above, we obtain

THE LIFESPAN OF SOLUTIONS TO SEMILINEAR DAMPED WAVE EQUATIONS 1275
in Σ j+1 . By (44) and recalling the definition of l j , given by and (30), we have Making use of (36), we get in Σ j+1 . Therefore, (39) holds for all j ∈ N.
The proof of Proposition 4.1 is now completed.
(ii) Upper bound of the lifespan in the case of p = 3.

5.
Proof of Theorem 1.3. In this section, we prove Theorem 1.3. First of all, we define the following weighted L ∞ space. For γ > 0 and 0 < T ≤ ∞, we define for (x, t) ∈ [0, ∞) 2 . Next we prepare the following lemmas which play a key role in the proof of Theorem 1.3.
Assume that (f, g) ∈ Y κ and f, g are odd functions. Then there exists a positive constant C κ such that where u 0 is defined in (13).
Proof. Noticing that f and g are odd functions, and making use of Lemma 2.1, we get

THE LIFESPAN OF SOLUTIONS TO SEMILINEAR DAMPED WAVE EQUATIONS 1277
Let 0 ≤ x ≤ t. Since f is an odd functions, we have Analogously to the above, we get Next we consider the case of 0 ≤ t ≤ x. Then we have Hence, we get Thus we find (50) is valid via (13).
are odd functions and U ∈ X p−2 . Then there exist positive constants C p,κ and C p such that where E σ (T ) is the one in (19), L is the one in (17), and u 0 is the one in (13).
is the solution of (54), then u := U + U 0 satisfies (16). Since U 0 exists globally in time, it suffices to examine the lifespan of U . We shall construct a solution of the integral equation (54) in X p−2 with p > 2 under suitable assumption on T such as (57) below. Define a sequence of functions We take ε 0 = ε 0 (f, g, p, κ) > 0 so small that For a fixed ε ∈ (0, ε 0 ], we take T > 0 such that Analogously to the proof of Theorem 1.2 in [6] (see p.16 in [6]), we see from Lemma 5.1 and Lemma 5.2 that {U n } n≥2 is a Cauchy sequence in X p−2 , provided (56) and (57) hold. Since X p−2 is complete, there exists U ∈ X such that {U n } n∈N converges to U in X p−2 . Therefore U satisfies the integral equation (54). Hence, the proof of Theorem 1.3 is completed.
Remark 5.1. Analogously to the proof of Theorem 1.3, one can get a lower bound of the lifespan in the case of 1 < p ≤ 2. In fact, we have only to change the weight function by for (x, t) ∈ [0, ∞) 2 .
6. Proof of Theorem 1.4. In this section, we prove Theorem 1.4. We show that the solution of (16) blows up in finite time. First of all, we prepare some definitions and lemmas. For T > 0 and δ ∈ (0, 1), we define the following domains:  )) − log E p 1/(p−1) } (j ≥ 2), C p,1 = c p 1 k p ε p ,
However this is a contradiction to the definition of t 1 . Therefore we have u > 0 in D(x 0 , t 0 ), so that we get u > 0 in (0, ∞) 2 .
Next, we derive a lower bound of the solution to (16) which is a first step of our iteration argument.
Our iteration argument will be done by using the following estimates.