Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions

In this paper, we treat the coupled system of wave equations whose nonlinearities 
are $|u|^{p_j}|v|^{q_j}$ and propagation speeds may be different from each other. 
We study the lower bounds of $p_j$ and $q_j$ to assure the global existence 
of a class of small amplitude solutions which includes self-similar solutions. 
The exponent of self-similar solutions plays crucial role to find the lower bounds. 
Moreover, we prove that the discrepancy of propagation speeds allow us to bring them down. 
Conversely, if such conditions for the global existence do not hold, then no self-similar 
solution exists even for small initial data.

Before we proceed further, we briefly recall the known results concerning the Cauchy problem for the single equation: For simplicity, we take n = 3. (For the other space dimensional case, see [12], [13], [30], [23], [32], [11], and references cited therein). Let u 0 (x, t) be the solution of the homogeneous wave equation with compactly supported initial data. Then, as is well known, u 0 (x, t) is supported in {(x, t)| |x| ≤ t + R} with R the diameter of a ball containing the support of the data, and Therefore, the first iterate u 1 (x, t), i.e., the solution of (1.7) Employing those estimates and applying the successive iteration, F. John [16] was able to show that p 0 (3) = 1 + √ 2 is the critical exponent of the problem in the above sense. (See also [29] for the critical case). This exponent had already been conjectured by W. Strauss in [31]. Namely, the exponent p 0 (n) is given by the positive root of (n − 1)p 2 − (n + 1)p − 2 = 0. (1.8) Moreover, F. Asakura [3] proved the small data global existence for a wider class of the initial data. Roughly speaking, we have only to assume that the Cauchy data (f, g) decay faster than (|x| −m0 , |x| −m0−1 ) as |x| → ∞. Here m 0 = 2/(p − 1).
To understand the number m 0 , it is convenient to consider the self-similar solution u(x, t), which is a solution of (1.6) verifying u(x, t) = λ m0 u(λx, λt) for any λ > 0 and (x, t). The existence of the self-similar solution of (1.6) is proved in [28] for p > 2 + √ 5, in [25] for p > ( √ 13 + 4)/3 and in [26] and [14] for p > 1 + √ 2. Pecher [26] also showed that even for small data no self-similar solutions in general exist if p ≤ 1 + √ 2. If u(x, t) is a self-similar solution, we easily see that Now, heuristically, we may say that the condition p > p 0 (3) for the small data global existence comes from in view of both (1.9), which is connected with the most slowly decaying data, and (1.7), which is the estimate for the compactly supported data. For general n ≥ 2, replacing the right hand side as we can recover the condition p > p 0 (n). We remark that this type of observation holds true for the Schrödinger and heat equations. (For the existence of self-similar solutions for those equations and related topics, see Cazenave-Weissler [4], [5] and Ribaud-Youssfi [27]). Turning to our problem, we need to study the self-similar solution (u(x, t), v(x, t)) of (1.1)-(1.4), which is a solution of (1.1)-(1.4) satisfying for any λ > 0 and (x, t), where Here we assumed (p 1 − 1)(q 2 − 1) − q 1 p 2 = 0. Then the analogue to (1.10) is Actually, if we put p 1 = q 2 = 0 in (1.11), then we get Λ < 0 as we desired.
Next we turn our attention to the role of propagation speeds. The small data global existence for systems of nonlinear wave equations with different propagation speeds has been well developed when the nonlinear terms depend only on the derivatives of unknown functions but not on unknown functions themselves in the works of [18], [2], [15], and [33]. Those result suggest that the discrepancy of propagation speeds may relax the condition (1.11) to prove the small data global existence also for our problem (1.1)- (1.4).
In this spirit, the works of [20], [21] and [6] deal with (1.1)-(1.4) with s = 1 and p 2 = q 1 = 0, and found that Λ still works as the Fujita-type critical exponent. However, when s = 1 and either p 1 > 0 and p 2 > 0 or q 1 > 0 and q 2 > 0, it was shown in [22] that something different happens for some peculiar values of p j , q j (j = 1, 2). Here we try to formulate the condition in general. Since m 1 and m 2 satisfy m 1 + 2 = m 1 p 1 + m 2 q 1 , m 2 + 2 = m 1 p 2 + m 2 q 2 , (1.12) we see that (1.11) is equivalent to We suppose that the following weaker version of (1.13) is sufficient to prove the small data global existence: 14) and will realize the conjecture for the case n = 3 in this paper. Notice that if p 2 = q 1 = 0, then (1.14) coincides with (1.13) , hence (1.11).
Remark: Since we have strict inequality in (1.11) and (1.14), we are able to show a kind of the stability of self-similar solutions in Theorem 3.3 below when n = 3, as in [28], [25], [26] and [14]. Besides, at least in the formal level, we could derive such conditions as in (1.11) and (1.14), even though the number of unknown functions increase. This paper is organized as follows. In section 2, following Pecher's argument in [26], we prepare several estimates which is connected with pointwise estimates for the solution. For that reason, we restrict ourselves to the case n = 3 in what follows. The proof of estimates for the solution of inhomogeneous wave equation is simplified (see Proposition 2.2 and its proof). In section 3, we prove the small data global existence for a class of data which includes homogeneous data f j and g j of degree −m j and −m j − 1, respectively. Hence the existence of self-similar solutions is shown, and the solutions are not necessarily radially symmetric. Moreover, we show the existence of asymptotically self-similar solutions. In section 4, we show that even for small data no self-similar solutions in general exist if (p 1 , q 1 ) ∈ Ω s or (p 2 , q 2 ) ∈ Ω s , where we have set for s = 1: and for s = 1: This result suggests that the conditions (1.11) for s = 1 and (1.14) for s = 1 with n = 3 are sharp for the existence of self-similar solutions. We conclude this introduction by giving some notations. For s > 0, we define the operators as follows: We define the space V 1 , V s and V 1,s as follows: where we have set R 3 * = R 3 \ {0} and R + = (0, ∞).

Preliminary estimates.
In this section, we mention preliminary estimates.
where k > 1, then the following estimate holds for (x, t) ∈ R 3 × R + with |x| = st: where C 0 is a constant depending only on k and s.
This proposition is the estimate for the solution of the homogeneous wave equation. For the proof, see, e.g., Proposition 2.1 in [26] and [24].
where C is a constant depending only on s, κ 1 , κ 2 , µ, µ 1 and µ 2 . If This proposition is the estimate for the nonlinear terms. Before carrying out a proof of Proposition 2.2, we prepare several lemmas. First one is a fundamental identity concerning the spherical mean. For the proof of Lemma 2.3, see, e.g., [24].

6)
where C is a constant depending only on µ.
Proof: We prove only (2.5), since the other can be handled analogously. If t ≥ 2r, then t − r ≥ (r + t)/3, hence t+r t−r If 0 < t ≤ 2r, then we have r/(r + t) ≥ 1/3. Therefore we obtain (2.5) by a direct calculation. This completes the proof.
3. Main results. In this section, we first prove the existence of global solutions to the Cauchy problem (1.1)-(1.4) including self-similar solutions. More precisely, we consider the corresponding integral equation: Throughout this section, we assume that so that |u| p1 |v| q1 and |u| p2 |v| q2 satisfy the Lipschitz condition. Besides, we require that with (p 1 − 1)(q 2 − 1) − q 1 p 2 = 0. If m 1 ≤ 1 or m 2 ≤ 1, we have to modify the function space V 1 or V s defined by (1 .19) and (1.20) as in [3]. But we do not go further in this direction, since we are interested in values of (p j , q j ) which are close to Ω s in the p-q plane. Actually, it is clear that (p j , q j ) / ∈ Ω s if m 1 ≤ 1 or m 2 ≤ 1.
In addition, we introduce a set from where we take the Cauchy data: (3.5) Then we have the following.
We next consider the existence of a class of solutions which asymptotically behave like a self-similar solution.
Finally we consider the problem (1.1)-(1.4) for smooth initial data at the origin. The proof of the following result is analogous to that of Theorem 3.1, so is omitted.
Theorem 3.4. Assume that p j , q j (j = 1, 2) satisfy the assumptions of Theorem 3.1 and that f j (

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where C 0 is the number in Proposition 2.1, and 4. A counterexample. The following example shows that even for small data no self-similar solution in general exists if (p 1 , q 1 ) ∈ Ω s or (p 2 , q 2 ) ∈ Ω s , where Ω s is defined by (1.15) and (1.16). We consider the Cauchy problem: via its corresponding integral equation: We assume that (u, v) is a local solution of integral equation (4.5)-(4.6) which is measurable in (x, t) ∈ R 3 × (0, T ). By symmetry, we have only to treat the case (p 1 , q 1 ) ∈ Ω s . Besides, we assume s = 1 in the following, since the case of s = 1 can be handled analogously.