Existence and nonexistence of global solutions for a structurally damped wave system with power nonlinearities

Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped wave equations. Main goal is to find the threshold, which classifies the global (in time) existence of small data solutions or the nonexistence of global solutions under the growth condition of the nonlinearities.


Introduction
In this paper, let us consider the following Cauchy problem for weakly coupled system of semilinear structurally damped wave equations: x P R n , t ě 0, up0, xq " u 0 pxq, u t p0, xq " u 1 pxq, x P R n , vp0, xq " v 0 pxq, v t p0, xq " v 1 pxq, for any δ 1 , δ 2 P r0, 1s and for nonlinearities with powers p, q ą 1. The special case of (1) with δ 1 " δ 2 " 1 2 in the form # u tt´∆ u`p´∆q 1 2 u t " |v| p , v tt´∆ v`p´∆q 1 2 v t " |u| q , up0, xq " u 0 pxq, u t p0, xq " u 1 pxq, vp0, xq " v 0 pxq, v t p0, xq " v 1 pxq, (2) was well-studied by D'Abbicco in [1]. In the cited paper, he succeeded to determine the critical exponent for (2). For details, the author proved the global (in time) existence of small data solutions to (2) in any space dimensions n ě 2 if the condition 1`maxtp, qu pq´1 ă n´1 2 holds by using sharp decay estimates for solutions to the linear corresponding Cauchy problem. Moreover, the above condition is sharp because a nonexistence result of global (in time) weak solutions to (2) was also discussed if this condition is no longer true. The proof of blow-up result is based on a contradiction argument by using the test function method (see, for example, [1,10]). The fact is that for this purpose some difficulties arise. In general, standard test function method, i.e. test functions with compact support, is not directly applicable since this method relies on pointwise control of derivatives of test functions. In addition, the fractional Laplacian operators p´∆q δ for any δ P p0, 1q are well-known non-local operators, it follows that suppp´∆q δ φ is bigger than suppφ for any φ P C 8 0 pR n q in general. However, this application linked to the estimate p´∆q δ φ ℓ ď ℓφ ℓ´1 p´∆q δ φ for δ P p0, 1q, ℓ ě 1 and for all φ ě 0, φ P C 8 0 pR n q is possible to (2) due to the following key observation: Any local or global solution to (2) is nonnegative with the assumption of nonnegative initial data u 1 , v 1 and u 0 " v 0 " 0, which was investigated by D'Abbicco-Reissig in [4]. Thanks to this essential property, the above inequality works well to extend the test function method to (2). Unfortunately, we cannot expect nonnegative solutions to (1), which contains the nonlocal terms p´∆q δ for any δ P p0, 1q. For this reason, the first main motivation of this paper is to prove the global (in time) existence of small data solutions to (1), where the parameters δ 1 and δ 2 are not necessary to be equal. More in details, we would like to explain the impact of the flexible choice of the parameters δ 1 , δ 2 on our global (in time) existence results and the range of admissible exponents p, q as well. To establish this, we have in mind to take advantage of the better decay estimates available for the corresponding linear wave equations with structural damping p´∆q δ u t of (1) in the following form: w tt´∆ w`p´∆q δ w t " 0, wp0, xq " w 0 pxq, w t p0, xq " w 1 pxq, where δ " δ 1 or δ " δ 2 . From these appearing difficulties as mentioned above, the second main motivation of this paper is to find the precise critical exponents to (1) with general cases of δ 1 , δ 2 P r0, 1s, especially we are interested in facing up to the proof of blow-up result, where the requirement of nonnegativity of solutions does not appear for (1). In order to overcome this difficulty, the crux of our ideas is to apply a modified test function method effectively in dealing with the fractional Laplacian p´∆q δ 1 and p´∆q δ 2 . Moreover, concerning the linear equation (3) and some of its semi-linear equations with the power nonlinearity |u| p we want to point out the paper [4] of D'Abbicco-Reissig. The authors have proposed to distinguish between "parabolic like models" in the case δ P r0, 1 2 s, the so-called effective damping, and "hyperbolic like models" or "wave like models" in the case δ P p 1 2 , 1s, the so-called noneffective damping. To the best of author's knowledge, it seems that nobody has ever succeeded to determine really critical exponent to semi-linear structurally damped wave equations with noneffective damping. Hence, it is still an open problem as far as to explore. From this observation, in order to give a partial positive answer to the open problem in [1], it is quite natural that we may restrict ourselves to consider only (1) with effective damping, i.e. the assumption of δ 1 , δ 2 P r0, 1 2 s is of our interest in this paper.
1.1. Notations. We use the following notations throughout this paper.
‚ We write f À g when there exists a constant C ą 0 such that f ď Cg, and f « g when g À f À g. ‚ As usual, the spaces H a and 9 H a with a ě 0 stand for Bessel and Riesz potential spaces based on L 2 spaces. Here D a and |D| a denote the pseudo-differential operators with symbols ξ a and |ξ| a , respectively. We denote p f pt, ξq :" F xÑξ`f pt, xq˘as the Fourier transform with respect to the space variable of a function f pt, xq. ‚ For a given number s P R, we denote rss :" max k P Z : k ď s ( and rss`:" maxts, 0u as its integer part and its positive part, respectively. ‚ We put x :" a 1`|x| 2 , the so-called Japanese bracket of x P R n . ‚ We fix the constant m 0 :" 2m 2´m , that is, 1 m 0 " 1 m´1 2 with m P r1, 2q. ‚ Finally, we introduce the spaces A :"`L m X H 1˘ˆ`Lm X L 2˘w ith the norm }pu 0 , u 1 q} A :" }u 0 } L m`}u 0 } H 1`}u 1 } L m`}u 1 } L 2 , where m P r1, 2q.
Finally, in order to show the optimality of our exponents to (1), we have the following blow-up results.
. We assume that we choose the initial data u 0 " v 0 " 0 and u 1 , v 1 P L 1 satisfying the following relations: ż R n u 1 pxqdx ą ǫ 1 and where ǫ 1 and ǫ 2 are suitable nonnegative constants. Moreover, we suppose the following conditions: Then, there is no global (in time) Sobolev solution pu, vq P C`r0, 8q, L 2˘ˆC`r 0, 8q, L 2˘t o (1).
The outline of this article is presented as follows: Section 2 is to provide pL m X L 2 qĹ 2 estimates and L 2´L2 estimates for solutions to (2), with m P r1, 2q, and some of essential properties of a modified test function method from the recent papers of Dao [2] and Dao-Reissig [5], respectively. In Section 3, we prove the global (in time) existence of small data solutions to (1). Finally, we devote to the proof of nonexistence result of global solutions to (1) in Section 4.

Preliminaries
In this section, we collect some preliminary knowledge needed in our proofs.

Linear estimates.
Main purpose is to recall pL m X L 2 q´L 2 and L 2´L2 estimates for solutions and some of their derivatives to (3) from the recent paper of Dao [2]. Using partial Fourier transformation to (3) we have the following Cauchy problem: The characteristic roots are The solutions to (27) are written by the following form (here we assume λ 1 ‰ λ 2 ): For this reason, we may read the solutions to (3) as follows: Here we want to underline that although all the decay estimates from Proposition 2.1 are available for any space dimensions n ě 1, under a constraint condition to space dimensions n ą 2m 0 δ we may conclude the better decay estimates. Namely, we obtain the following result.
We recognize that the decay rates from Propositions 2.1 and 2.2 coincide with those in [4]. Moreover, the optimality of those from Proposition 2.1 is also guaranteed by the study of asymptotic profile of solutions to (3) in [3]. From this observation, these estimates play really a fundamental role in the proofs of global (in time) existence results for (1) in Section 3.

2.2.
A modified test function. Main aim of this section is to provide some auxiliary properties of the modified test function φ " φpxq :" x ´r for some r ą 0 from the recent paper of Dao-Reissig [5] which are key tools in the proof of our blow-up result in Section 4. 7,9]). Let s P p0, 1q. Let X be a suitable set of functions defined on R n . Then, the fractional Laplacian p´∆q s in R n is a non-local operator given by as long as the right-hand side exists, where p.v. stands for Cauchy's principal value, C n,s :" is a normalization constant and Γ denotes the Gamma function.
Lemma 2.1 (Lemma 2.3 in [5] with m " 0). Let s P p0, 1q and r ą 0. Then, the following estimates hold for all x P R n :ˇˇp´∆ Lemma 2.2. Let s P p0, 1q. Let ψ be a smooth function satisfying B 2 x φ P L 8 . For any R ą 0, let φ R be a function defined by where κ ą 0. Then, p´∆q s pφ R q satisfies the following scaling properties for all x P R n : p´∆q s pφ R qpxq " R´2 κs`p´∆ q s φ˘`R´κx˘.
Proof. We follow the proof of Lemma 2.4 in [5] with minor modifications to conclude the desired statement. Lemma 2.3 (Lemma 2.7 in [5]). Let s P R. Let φ 1 " φ 1 pxq P H s and φ 2 " φ 2 pxq P H´s. Then, the following relation holds: ż 3. Global (in time) existence of small data solutions 3.1. Proof of Theorem 1.1. At first, let recall the fundamental solutions K 0,δ pt, xq " F´1 ξÑx`p K 0,δ pt, ξq˘and K 1,δ pt, xq " F´1 ξÑx`p K 1,δ pt, ξqd efined in Section 2 to represent the solutions of the corresponding linear Cauchy problems with vanishing right-hand sides to (1) By applying Duhamel's principle, the formal implicit representation of the solutions to (1) can be read as follows: vpt, xq " v ln pt, xq`ż t 0 K 1,δ 2 pt´τ, xq˚x |upτ, xq| q dτ ": v ln pt, xq`v nl pt, xq.
Due to the condition p ď 1`2 m n´2mδ 2 in (7), it implies immediately that the term p1`τ q´n 2mp1´δ 2 q pp´1q`p δ 2 1´δ 2 is not integrable. For this reason, may estimate p1`tq´n 2p1´δ 1 q p 1 where ε is a sufficiently small positive number. Thanks to the condition n ď 4m 2´m in (5), we may verify that´n 2p1´δ 1 q p 1 m´1 2 q`δ 1 1´δ 1 ě´1. Hence, we derive p1`tq´n 2mp1´δ 2 q pp´1q`p δ 2 1´δ 2 ż t t{2 p1`t´τ q´n 2p1´δ 1 q p 1 where ε is a sufficiently small positive number. Therefore, combining the above estimates we may conclude the following estimate: pp,δ 2 q }pu, vq} p Xptq . In order to control ∇u nl , we use the pL m X L 2 q´L 2 estimates if τ P r0, t{2s and the L 2´L2 estimates if τ P rt{2, ts from Corollary 2.1 to arrive at where we used again the estimates (35) and (36) linked to the relation (37). In the same treatment of u nl , we obtain the following estimate for first integral: Moreover, the remaining integral can be dealt with the following way: 2p1´δ 1 q`ε pp,δ 2 q due to δ 1 ě δ 2 . Consequently, we have shown that › › ∇u nl pt,¨q › › L 2 À p1`tq´n 2p1´δ 1 q p 1 m´1 2 q´1´2 δ 1 2p1´δ 1 q`ε pp,δ 2 q }pu, vq} p Xptq . By analogous arguments as we estimated ∇u nl we also derive › › u nl t pt,¨q › › L 2 À p1`tq´n 2p1´δ 1 q p 1 m´1 2 q´1´2 δ 1 1´δ 1`ε pp,δ 2 q }pu, vq} p Xptq . Similarly, we may conclude the following estimates for j, k " 0, 1 with pj, kq ‰ p1, 1q: provided that the conditions from (4) to (7) are satisfied for q. Therefore, from the definition of the norm in Xptq we have proved that the inequality (34) holds. Let us now indicate the inequality (33). For two elements pu, vq and pū,vq from Xptq, we get N pu, vqpt, xq´N pū,vqpt, xq "`u nl pt, xq´ū nl pt, xq, v nl pt, xq´v nl pt, xq˘.

Nonexistence result via modified test function method
In order to prove the blow-up results, we shall apply a modified test function method from Section 2 which plays a significant role in the following proofs.
Let R be a large parameter in r0, 8q. We define the following test function: where ϕ R ptq :" ϕpR´αtq and ψ R pxq :" ψpR´βxq for some α, β which we will fix later. We define the functionals and I R,t :" Let us assume that pu, vq "`upt, xq, vpt, xq˘is a global (in time) Sobolev solution from C`r0, 8q, L 2˘Ĉ`r 0, 8q, L 2˘t o (1). We multiply the first equation to (1) by η R " η R pt, xq and carry out partial integration to get Employing Hölder's inequality with 1 q`1 q 1 " 1 we can proceed as follows: After performing the change of variablest :" R´αt andx :" R´βx, we calculate straightforwardly to obtain where we used ϕ 2 R ptq " R´2 α ϕ 2 ptq and the assumption (38). Now let us focus our considerations to deal with I 2R and I 3R . First, since ψ R P H 2 and u P C`r0, 8q, L 2˘, we apply Lemma 2.3 to arrive at the following relations: ż As a consequence, it implies immediately that Applying Hölder's inequality again as we estimated J 1 leads to and To estimate the above two integrals, the key tools rely on results from Lemmas 2.1 and 2.2. More in detail, in the first step we use the change of variablest :" R´αt andx :" R´βx to derive where we notice that ∆ψ R pxq " R´2 β ∆ϕpxq. Hence, we deduce the following estimate: Now let us come back to estimate I 3R in the second step. After carrying out again the change of variablest :" R´αt andx :" R´βx and applying Lemma 2.2, we may estimate I 3R by where we used ϕ 1 R ptq " R´αϕ 1 ptq and the assumption (38). In order to control the last integral, we employ Lemma 2.1 with q " n`2δ 0 and s " δ 1 to have Thanks to the assumption (22), there exists a sufficiently large constant R 1 ą 0 such that it holds ż R n u 1 pxqψ R pxq dx ą 0 (43) for all R ą R 1 . As a result, combining the estimates from (39) to (43) gives for all R ą R 1 . In the same arguments we may conclude the following estimate for all R ą R 1 : Without loss of generality we can assume δ 1 ě δ 2 . Now let us fix α :" 2´2δ 1`δ 1´δ2 2p1´δ 2 q pnq´n´2qqpn´2q 1`q and β :" 1´δ 1´δ2 2p1´δ 2 q nq`2´n 1`q . For this choice, we may verify that 2α ď´2β,´α´2δ 1 β ď´2β and´α´2δ 2 β ď´2β.

From (44) and (45) it follows immediately that
Therefore, we arrive at β`α`n β q 1`p´2 β`α`n β p 1 q 1 β`α`n β p 1`p´2 β`α`n β q 1 q 1 It is obvious that the assumption (23) is equivalent to γ 2 ď 0. For this reason, we shall divide our attention into two subcases.
Case 1: Let us consider the subcritical case of γ 2 ă 0. Then, we let R Ñ 8 in (47) to obtain which follows u " 0, a contradiction to the assumption (22). This means that there is no global (in time) Sobolev solution to (1) in the subcritical case.
Case 2: Let us now come back to the critical case of γ 2 " 0. At first, we introduce the following constants: After repeating some arguments as we have proved in the subcritical case, we may conclude the following estimates: Thus, it follows that For this reason, we obtain immediately where D 0 :"´D p 1 D 1 p q 1¯p q pq´1 is a positive constant, and By replacing (50) into the left-hand side of (48), a direct calculation leads to q`ppqq 2 .
Then, we use iteration arguments to arrive at the following estimate for any integer j ě 1: q`ppqq 2`¨¨¨`p pqq j " (51) Let us now choose the constant in the assumption (22). This means that there exists a sufficiently large constant R 2 ą 0 such that ż R n v 1 pxqψ R pxq dx ą ǫ 2 for all R ą R 2 . We can see that the above assumption is equivalent to Hence, passing j Ñ 8 in (51) gives J R Ñ 8. This is a contradiction to the boundedness of J R in (49). As a consequence, we may conclude the nonexistence of global (in time) Sobolev solution to (1) in the critical case. Summarizing, the proof of Theorem 1.3 is completed.

4.2.
Proof of Theorem 1.4. We follow the ideas from the proof of Theorem 1.3. We introduce the test functions ϕ " ϕptq as in Theorem 1.3 and ψ " ψp|x|q :" x ´n´2δ . Then, we may repeat exactly, on the one hand, the proof of Theorem 1.3 to conclude the following estimates: ż R n u 1 pxqψ R pxq dx`I R ď C q 1 J 1 q R´R´2 α`α`n β q 1`R´α´2 δβ`α`n β q 1`R´2 β`α`n β q 1¯, ż R n v 1 pxqψ R pxq dx`J R ď C p 1 I 1 p R´R´2 α`α`n β p 1`R´α´2 δβ`α`n β p 1`R´2 β`α`n β p 1¯, where C p 1 :"´ż R n x ´n´2δ dx¯1 p 1 and C q 1 :"´ż R n x ´n´2δ dx¯1 q 1 .
Let us now fix α :" 2´2δ and β :" 1. As a result, from the both above estimates we obtain ż R n On the other hand, because of the assumption (25), the following estimate holds: ż for all R ą R 0 , where R 0 ą 0 is a sufficiently large number and C 1 is a suitable positive constant. In the same way we also derive ż R n v 1 pxqϕ R pxq dx ě C 2 ǫ 0 R n´n`ǫ 2 m (55) 1 p 0´1 p`s n 1 p 0´1 p 1`σ n and s σ ď θ ď 1.
For the proof one can see [6].