A NOTE ON A WEAKLY COUPLED SYSTEM OF STRUCTURALLY DAMPED WAVES

. In this note, we ﬁnd the critical exponent for a system of weakly coupled structurally damped waves.

1. Introduction. Recently, K. Nishihara and Y. Wakasugi [17] studied the Cauchy problem of the weakly coupled system of damped wave equations where p, q > 1. They proved that global existence of small data solutions to (1) holds, in any space dimension n ≥ 1, if Condition (2) is equivalent to max{p, q} (min{p, q} + 1 − p F (n)) > p F (n), where p F (n) = 1 + 2/n is the Fujita critical exponent for the damped wave equation [18]. They also estimated the lifespan of the solution in the case of blow-up in finite time, which occurs for α > n/2. The result in [17] extended the previous one obtained in space dimension n ≤ 3 by K. Nishihara [16]. The bound on the space dimension does not appear in the weakly coupled system of heat equations, and it is related to the regularity problems which appear in the estimates available for the linear damped wave equations where a > 0. Using weighted energy estimates, as in [12,18], this limitation has been overcame in [17]. We address the interested reader to the reference therein, for a more detailed overview about systems of damped waves.

A SYSTEM OF STRUCTURALLY DAMPED WAVES 321
The main purpose of this note is to derive an analogous result for the system of structurally damped waves where a, b > 0. In order to do that, we take advantage of the better linear decay estimates available for linear wave equations with structural damping (−∆) 1 2 u t (see, for instance, [1,15]): In particular, for any n ≥ 2, we show that global existence of small data solutions to (5) holds if max{p, q} (min{p, q} where p K (n) = 1 + 2/(n − 1) is the critical exponent for the wave equation with structural damping (−∆) 1 2 u t and power nonlinearity (see [7,9,10]). We remark that p K (n) is also the critical exponent obtained, by comparison method, by Kato for wave equations with variable coefficients and compactly supported data [13]. Condition (7) is equivalent to Theorem 1. Let n ≥ 2 and p, q > p K (n)−1, be such that (7) holds. Then, there exists ε > 0 such that, for any (u 0 , v 0 ) ∈ L 1 ∩ H 1 ∩ L ∞ and (u 1 , v 1 ) ∈ L 1 ∩ L n , with there exists a solution for any r ∈ [1, ∞], and where provided that p, q = p K (n). Here γ(p) and γ(q) represent the possible loss of decay with respect to the corresponding linear estimates for u and v (see (22)-(24)). If p = p K (n) or, respectively, q = p K (n), then the loss of decay (1 + t) γ(p) or, respectively, (1 + t) γ(q) , is replaced by log(e + t).
Remark 1. We notice that (7) implies that at least one among γ(p) and γ(q) is zero.

Remark 2.
We may weaken the assumptions on the space of the data in different ways, if we are not interested for the solution to be in L 1 , or in L ∞ , or if we are not interested in an estimate for the energy (see the proof of Theorem 1). However, a reasonable assumption for the regularity of the solution to guarantee the same critical exponent, is at least In fact, in this case, linear L 1 − L q and L 1 − L p estimates may be successfully applied to deal with the nonlinear term, via Duhamel's principle.
The critical exponent in (7) is sharp. Indeed, we have the following.
We remark that min{p, q} may be smaller than 1 in Theorem 2. We leave open the problem to study whether a nonexistence result holds in the limit case α = (n − 1)/2.
It is clear that one may also study the global existence of small data solutions for a system where a wave equation with classical damping is coupled with a structurally damped wave, where a, b > 0. In this note, we limit the study of (17) to space dimension n = 2 and we assume q > p F (2) = 2. The restriction q > 2 allow us to avoid the use of both weighted energy estimates and L 1 − L ∞ estimates for the wave equation with classical damping, since standard energy estimates on the L 2 basis are sufficient. We recall that p K (2) = 3. Proposition 1. Let n = 2 and p, q > 2 be such that Then, there exists ε > 0 such that, for any there exists a solution to (17). Moreover, where γ(p) = (3 − p) + , as in (13), represents the possible loss of decay with respect to the corresponding linear estimates for u.
Remark 3. It is clearly possible to extend Proposition 1 to space dimension n ≥ 3, but some technical difficulties appear in the application of the linear estimates, due to the coupling of waves with different damping terms. In particular, the loss of decay should be carefully managed; as far as the employed estimates allow to keep this latter equal to (1 + t) γ(p) , one may expect the critical exponent to be given by for n ≥ 3 and q > p F (n).
2. Proof of Theorem 1. In order to prove Theorem 1, by Duhamel's principle, we may reduce to consider the linear equation in (6). We recall the linear estimates (see, for instance, [6,15]): for any r ∈ [1, ∞], and In the following, we denote by E 0 (t, x) and E 1 (t, x) the fundamental solutions to the linear equation, corresponding to the two initial data, namely, is the solution to (6).
Due to we reduce to estimate and analogously for Gu(t, ·) L ∞ . We remark that the second term may be controlled by the first one. As we did for (33), we may distinguish two cases, and derive the desired estimate We proceed similarly for (∂ t , ∇)F v L 2 and (∂ t , ∇)Gu L 2 , using (24) in [0, t/2] and (25) in [t/2, t]. Now (31), (32) come into play. This concludes the proof of (27).

Proof of Theorem 2.
To prove Theorem 2, we may use the test function method (first used for the damped wave equation with power nonlinearity in [19]), following the approach in [7,10]. An essential role in the proof of Theorem 2 is played by the fact that any local or global solution to (15) with non-negative initial data u 1 , v 1 , is non-negative. This property is used to extend the test function method to the system in (15), which contains the nonlocal term (−∆) 1 2 u t . To get this property, we set a = b = 1 and u 0 = v 0 ≡ 0 in (5) (see [10]). The requirement of non-negativity of the solution does not appear for (1).
Proof of Theorem 2. We assume by contradiction that (u, v) ∈ L q loc ×L p loc is a global solution to (15). Therefore, for any test function where Let φ ∈ C ∞ c ([0, ∞)) be a nontrivial, nonincreasing function, compactly supported in [0, 1], and let > max{p , q }. For any R > 1, we set ψ(t, x) = φ(t/R) φ(|x|/R) in (35). Recalling that φ, −φ , u 1 , and v 1 are nonnegative and that (see [3]) for any θ ∈ (0, 1] and > 1, we may derive We remark that, by virtue of the nonnegativity of I R and J R , it follows that the functions u ψ −1 h(t/R, |x|/R) and v ψ −1 h(t/R, |x|/R) have nonnegative integrals over [0, ∞) × R n , so that we can apply the triangle inequality for integrals. We notice that ψ (t, x)h(t/R, |x|/R) is a bounded function with compact support in [0, R] × B R , for any > 0, since h is bounded.
Setting p = ( − 1)/ − /p and q = ( − 1)/ − /q, by Hölder's inequality, we obtain: Let q ≥ p. Combining (36a) and (36b), we obtain R . The power of R is negative if, and only if, condition (16) holds. Therefore, in this case, being pq > 1, by 5. Some open problems. It is possible to study the general case of two wave equations with structural damping (−∆) δ u t , with δ ∈ [0, 1], namely, where δ, σ ∈ [0, 1] and a, b > 0. However, some difficulties arise. On the one hand, linear L 1 − L p estimates for u tt − ∆u + 2a(−∆) δ u t = f (t, x), (47) appear to be not good enough for any exponent p > 1 in any space dimension n ≥ 1, when δ = 1/2. On the other hand, the use of weighted energy estimates introduced in [12,18] for the wave equation with classical damping cannot be directly extended if δ = 0.
Another generalization consists into replace the power nonlinearites |v| p and |u| q in (46) with other nonlinearities, for instance, with powers of derivatives of u and v, as in [4]. Also, one may replace one or both power nonlinearities with nonlinear memory terms obtained by their Riemann-Liouville fractional integral, i.e. t 0 (t − s) −α |u(s, x)| p ds, with α ∈ (0, 1). For damped wave equations with a nonlinear memory term it has been shown [5,6] that this latter brings, in general, a loss of decay in the estimates with respect to the linear problem. This effect would then be taken into account together with the loss of decay described by (13), (14) in Theorem 1.