Sharp blow-up for semilinear wave equations with non-compactly supported data

. This paper corrects Asakura’s observation on semilinear wave equations with non-compactly supported data by showing a sharp blow-up theorem for classical solutions. We know that there is no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak, in spite of (cid:12)nite propagation speed of the linear wave. Our theorem clari(cid:12)es the (cid:12)nal criterion on such a phenomenon.

1. Introduction.We consider the initial-value problem for semilinear wave equation where n ≥ 2 and u = u(x, t) is a scalar unknown function of space-time variables.The assumptions on the nonlinear term F will be given precisely later, but at this moment we may assume that F (u) = |u| p or, F (u) = |u| p−1 u with p > 1.
In the case where the initial data (f, g) has compact support, we have the following Strauss' conjecture.There exists a critical number p 0 (n) such that (1) has a global in time solution for "small" data if p > p 0 (n) and has no global solution for "positive" data if 1 < p ≤ p 0 (n).As in Section 4 in Strauss [16], p 0 (n) is a positive root of the quadratic equation (n − 1)p 2 − (n + 1)p − 2 = 0.
This conjecture was first verified by John [6] for n = 3 except for p = p 0 (3).Later, Glassey [4,5] verified this for n = 2 except for p = p 0 (2).Both critical cases were studied by Schaeffer [14].In high dimensions, n ≥ 4, the subcritical case was proved by Sideris [15] and the supercritical case was proved by Georgiev, Lindblad and Sogge [3].Finally, the critical case in high dimensions was obtained 2 H.TAKAMURA, H.UESAKA AND K.WAKASA by Yordanov and Zhang [23] or, Zhou [24] independently.We note that the blowup results in high dimensions are available only for the positive nonlinear term, F (u) = |u| p .All the cited works on this conjecture are summarized in the following table. 1 [23], [24] indep.
[3] On the contrary, if the support of the initial data (f, g) is non-compact, we may have no global solution even for the supercritical case.Actually we have the following Asakura's observation.There exists a critical decay κ 0 of the initial data such that (1) has no global solution provided (f, g) satisfies that for some constant C > 0, and has a global solution provided (f, g) satisfies that is sufficiently small with κ ≥ κ 0 and p > p 0 (n).This was first proved by Asakura [2] in n = 3 except for the critical case clarifying The critical case in n = 3 was studied by Kubota [12] or, Tsutaya [21] independently.
For n = 2, the nonexistence part was verified by Agemi and Takamura [1], and the existence part was verified by Kubota [12] or, both parts by Tsutaya [19,20] independently.In high dimensions, only the radially symmetric solution has been studied.The nonexistence part was proved by Takamura [17], and the existence part was proved by Kubo and Kubota [10,11], and Kubo [9].All the cited works on this observation are summarized in the following table.[20] indep.[12], [19] indep.[12], [20] [10] and [11] It is remarkable that the critical decay κ 0 does not depend on space dimensions n.We also note that the nonlinear equation is invariant under a scaling u (5) In view of this fact, we can say that the blow-up of the subcritical decay is equivalent to the blow-up for large data at spatial infinity.As suggested by this fact, (1+|x|) 1+κ in (3) cannot be replaced by, for example, (1 + |x|) 1+κ0 log −l (2 + |x|) with any l > 0. See Kurokawa and Takamura [13].Moreover, we point out that (3) on α = 0 is strong.
In order to correct Asakura's observation, we need a blow-up result for the case where f ≡ 0 and g ≡ 0. The breakthrough was obtained by Uesaka [22] in low dimensions under some decaying assumption on f .Later, in our previous paper [18], the result was extended to all space dimensions, and the relation between such an assumption and Asakura's observation was clarified.Finally we propose the following corrected criterion for the global existence of the solution.
(1) has no global solution provided there exists a constant R > 0 such that (f, g) satisfies or for |x| ≥ R with 0 < κ < κ 0 and φ(x) ≡ positive const., or κ = κ 0 , φ is positive, monotonously increasing and lim On the other hand, (1) has a global solution provided (f, g) satisfies that with κ ≥ κ 0 and p > p 0 (n) is sufficiently small.
In high dimensions we have to treat radially symmetric solutions, so that there is no loss of the decay of the initial data by the representation formula of the solution.Hence the existence part has been already obtained in [9,10,11].In low dimensions, a non-symmetric assumption may make the loss of the decay, but easy modifications of [2,12,19,20,21] overcome it.See [18].Therefore we summarize the results only on the blow-up part in the following table according to the combinations of the assumptions.
Then we have the following theorem.
Theorem 2.1.Let u be a C 3 -solution of (12).Suppose that ( 13) is fulfilled.Then u cannot exist globally in time.

Remark 1.
As in [18], the simple example can be constructed by F (u) = |u| p or |u| p−1 u with p > 1 and For this example, R in the assumption ( 7) is directly computed.
Remark 2. The same result for C 2 -solution is available under the stronger assumption on f .See Sections 6 and 7 in [18] for details.
3. Positive C 3 solution.In order to prove Theorem 2.1, we need positivity of a solution of (12).
Lemma 3.1.Assume that there exists a positive constant R such that Then there is a positive constant δ = δ(n) such that a C 3 -solution u of ( 12) satisfies as far as u exists.Moreover, u in Σ satisfies where m is an integer part of n/2, namely m = [n/2].This is Lemma 4.1 in [18].4. Blow-up of C 3 solution.Let u be a global in time C 3 -solution of (12).We will see that this is false under the assumption of Theorem 2.1 by following up the basic iteration argument by John [6].
The assumption (13) on F and f enable us to make use of Lemma 3.1.Hence, cutting the domain of the integral, we have in Σ by monotonicity of φ.This is the first step of our iteration.
First we assume that u t has an estimate where all a, b, c are positive constants.This is true with a = 1, b = l, c = 1/8, d = 1 as we see.Integrating this inequality with respect to t, we obtain, by u(r, 0) = f (r) > 0, that Then we can put ( 17) and ( 18) into the second term in the right hand side of ( 16) because its domain of the integral is included in Σ.Hence, neglecting the first term by positivity, we have by ( 13) that In order to repeat this procedure infinitely many times, one should compare ( 17) with (19) and define sequences {a j }, {b j }, {c j }, {d j } by Therefore we have ( It is easy to find a point (r 0 , t 0 ) ∈ Σ such that K(r 0 , t 0 ) > 0 because of the monotonicity of φ and κ 0 + 2 = 2p/(p − 1).Therefore, letting j → ∞, we get a contradiction u t (r 0 , t 0 ) → ∞.The proof is now completed.