Lifespan Estimates of Solutions for a Coupled System of Wave Equations with Damping terms and Negative Mass terms ∗

The main purpose of this paper is to study the formation of singularity for a coupled system of wave equations with damping terms, negative mass terms, and divergence form nonlinearities. Upper bound lifespan estimates of solutions to the system are obtained by using the iteration method. The results are the same as the corresponding coupled system of the wave equation with power nonlinearities | v | p and | u | q . To the best of our knowledge, the results in Theorems 1–5 are new. In addition, the variation trend of the wave is analyzed by using numerical simulation.


Introduction
In this paper, we mainly consider the coupled system of wave equations with damping terms and negative mass terms where Here, Δ � n i�1 z 2 /zx 2 i is the Laplace operator. b 1 (t), b 2 (t), b 3 (t), b 4 (t) ∈ C([0, T)) ∩ L 1 ([0, T)) are nonnegative functions. ε is a small positive parameter. Initial values u 0 , u 1 , v 0 , v 1 are non-negative functions. It holds that supp(u 0 , u 1 , v 0 , v 1 ) ⊂ B R (0), where B R (0) � x||x| ≤ R { } and R > 2. The exponents in the nonlinear terms satisfy p, q > 1. It is well known that the classical wave equation is related to the Strauss conjecture. The Strauss exponent p S (n) is the positive solution of the following quadratic equation: − (n − 1)p 2 +(n + 1)p + 2 � 0.
More precisely, if 1 < p ≤ p S (n), there is no global solution to the Cauchy problem for (3) with small initial data. The solution exists globally when p > p S (n), which is equivalent to Let us briefly review the small data Cauchy problem for the wave equation with scattering damping term which has attracted more attention (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Wirth et al. [13] obtained the energy solution to the Cauchy problem. Wirth et al. [14] proved a structural property of solution to the Cauchy problem.
Lai and Takamura [4] obtained an upper bound lifespan estimate of the solution to the problem with a nonlinear term f(u, u t ) � |u| p in the subcritical case by introducing the multiplier and using the iteration method. Wakasa and Yordanov [15] proved the lifespan estimate of the solution for the variable coefficient wave equation with scattering damping in the critical case by using the rescaled test function method and iterative technique. Utilizing some multiplier to absorb the damping term, Lai and Takamura [16] successfully established an upper bound lifespan estimate of the solution, which is the same as the case without a damping term. Lai and Takamura [17] studied the problem with combined nonlinearities f(u, u t ) � |u t | p + |u| q . It is shown that the scattering damping has no effect on the behavior of the solution compared with the solution to wave equation.
Let us briefly review the blow-up dynamics of the coupled system for semilinear classical wave equations.
Which has been studied in ( [18][19][20][21][22][23][24][25][26][27][28][29]). Santo et al. [28] proved that there exists a critical curve in the (p, q) plane, which divides the plane into two pieces when n ≥ 2. One piece is the range which shows the global existence of small amplitude solutions. Another piece is the range of nonexistence of global solutions. We observe that the critical curve is determined by a cubic relation between p and q, which possesses a cusp at p � q. Agemi et al. [18] obtained lifespan estimates for solutions when n � 3 and p, q > 2. Moreover, lower and upper bounds lifespan estimates for solutions are derived in the subcritical case. Santo and Mitidieri [29] investigated the lifespan estimate of the solution to the problem when p � q. Georgiev et al. [21] studied the formation of singularities solutions to the problem in the case n ≥ 4. Kurokawa and Takamura [25] acquired the upper bound lifespan estimates of solutions to coupled semilinear wave equations in the high dimensional critical case by making use of the iteration method.
Recently, the small data Cauchy problem for coupled wave equations with damping terms have been investigated (see [30][31][32][33][34][35][36][37][38][39][40][41]). Takamura and Palmieri [37] considered lifespan estimates of solutions to the problem with scattering damps term and power nonlinear terms f(v, v t ) � |v| p , f(u, u t ) � |u| q in the subcritical and critical cases. The critical curve of exponential pair (p, q) in nonlinear terms is the same as that of coupled semilinear wave equations without damping terms. Palmieri and Takamura [36] studied the blow-up of solutions to the semilinear damped wave equations with mixed nonlinear terms f(v, v t ) � |v| p , f(u, u t ) � |u t | q . Lifespan estimates of solutions in the subcritical case are obtained by employing the iterative method. The formation of singularities of solutions in the critical case is established by making use of the auxiliary functions, which are different from the test functions in [32]. Palmieri and Takamura [38] investigated upper bound lifespan estimates of solutions for semilinear damped wave equations with derivative type nonlinearities in the subcritical and critical cases.
Inspired by the works in [10,37,[42][43][44][45], we mainly considered blow-up dynamics of solutions to the coupled wave equations with divergence form nonlinearities in the subcritical and critical cases. It is worth noting that Han [44] studied the problem with divergence form nonlinearities when n � 4. We extended the problems studied in [44] to the coupled system of the semilinear wave equation in n space dimensions. Palmieri and Takamura [37] studied upper bound lifespan estimates of solutions to the coupled system of wave equations. Blow-up results of solutions in the subcritical and critical cases are obtained. Ming et al. [10] established upper bound lifespan estimates of the solution to the variable coefficient wave equation with scattering damping and divergence nonlinear terms in the case n � 4. We extended the problem studied in [10,37,44] to problem (1) with damping terms, negative mass terms, and divergence form nonlinear terms. We used the iterative method and slicing method to derive upper bound lifespan estimates of solutions in the subcritical and critical cases, respectively. To the best of our knowledge, the results in Theorems 1-5 are new conclusions. In addition, we describe the variation of waves through numerical simulation.
We present the definition of weak solutions as follows: It holds that and Employing integration by parts in (11)-(12) and letting t ⟶ T, we get and Throughout the paper, we denote Discrete Dynamics in Nature and Society The main results of this work are stated as follows: Then, the upper bound estimate for the lifespan satisfies where, C is a positive constant independent of ε.
Then, the lifespan estimates in (17) can be improved as Then, the lifespan estimates in (17) can be improved as Then, the lifespan estimates in (17) can be improved as Suppose that problem (1) Then, the upper bound estimates for the lifespan satisfies

Proof of Theorem 1
Let We are in the position to establish the lower bound for the spatial integral of nonlinear terms |v| p , |u| q . It holds that 4 Discrete Dynamics in Nature and Society Multiplying both sides of (29) with m(t) yields Then, we have It follows that Setting we have Direct calculation gives rise to where, Multiplying (35) with e 2t , we obtain Applying a comparison argument, we have U 1 (t) > 0 for all t > 0. Similar to the calculation in [46], we arrive at Finally, we derive the lower bound of U 1 (t) in the form Similarly, we have By using (27) and (28) and Holder inequality, we have Direct calculation shows From (38), (40), and (42), we acquire Discrete Dynamics in Nature and Society 5 Applying (39), (41), and (42) yields Choosing ϕ � ϕ(x, s) � 1 and ψ � ψ(x, s) � 1 in (11) and (12), we find Equivalently, we obtain Differentiating (46) and (47) with respect to t, we arrive at Multiplying (48) by m 1 (t) gives rise to Using the assumption u 1 ≥ 0, we obtain Similar to Section 3 in [46], we acquire where Employing integration by parts yields Utilizing the Holder inequality yields where, where, where, a j j∈N , b j j∈N , D j j∈N , α j j∈N , β j j∈N , Δ j j∈N are suitable sequences of non-negative real numbers to be determined afterwards. 6 Discrete Dynamics in Nature and Society Plugging (44) into (55), we obtain where Substituting (61) into (63), we arrive at where Taking into account (61) and (62), for an odd integer j, we have If j is even, from (58), we obtain Thus, from (65) and (66), we recognize that for all j ≥ 1, it holds that where, B 0 � B 0 (p, q, n) and B 0 � B 0 (p, q, n) are positive constants. The next step is to derive lower bounds for D j and Δ j . From the definition of D j and Δ j , we obtain Due to (67)-(69), we acquire Employing an inductive argument yields Discrete Dynamics in Nature and Society

Proofs of Theorems 2-4
Bear in mind We note that U and V are convex functions. Therefore, we deduce U(t) ≥ U ′ (0)t and V(t) ≥ V ′ (0)t. Consequently, using the Holder inequality If n � 1, 2and 1 < p < 2, we define If n � 1, 2and 1 < q < 2, we set 8 Discrete Dynamics in Nature and Society We replace (78) by Analogously, we acquire The proofs of Theorems 2-4 are finished.

Proof of Theorem 5
We are in the position to recall a pair of auxiliary functions from [47]. Let λ 0 ∈ (0, α/2] and r > − 1. We introduce the functions where, λ 0 is a fixed positive parameter and φ λ (x)is defined in (26).

Lemma 3.
Let F(n, p, q) � 0. Then, for all t ≥ 3/2, the following estimates hold: Discrete Dynamics in Nature and Society Lemma4. If p � q, utilzing (57) and (101), we derive (111) Substituting (111) into (112) yields Similarly, taking advantage of (42) and (104), we get If p < q, applying (114), we find Applying (106), we arrive at We finish the proof of Lemma 3. Assume that where, l j � 2 − 2 − (j+1) , a j j∈N , and b j j∈N , C j j∈N are sequences of non-negative real numbers. We have If p > q, combining (109) and (117), for all s ≥ l 2j+1 , we obtain (120) We obtain the trend of waves in Figure 1. Figure 1 shows the trend of the two waves in the time 0 ∼ 4s. We define the wave with the initial position of x � 0, y � 0.5 as a large wave, and the wave with x � 0, y � − 1.3 as a small wave. When t � 0s, it represents the initial state of the wave, in which the peak of the large wave is 0.9989, and the peak of the small wave is 0.9388. When t � 0.25s, the peaks of these two waves both begin to decline. At this time, the peak of the large wave is 0.8792, and the peak of the small wave is − 0.1137. Comparing the big wave with the small wave, it is obvious that the small wave has a fasterdescending speed. The peak of the big wave is still above the horizontal plane, while the small wave has fallen below the horizontal plane within 0.25s. When t � 0.5s, the peaks of the two waves continue to move downward. The peak of the large wave is 0.5753, and the peak of the small wave is − 0.08573. When t � 1s, the crests of these two waves both drop below the horizontal plane, the crests of the small wave begin to rise upwards as well. At the same time, these two waves begin to meet. The small wave begins to spread outward, and the spread of the big wave is affected. When t � 2s, both waves are concave down at the same time. The wave continues to move downward when t � 2 ∼ 4s. By observing the wave changes from 0 ∼ 4s, we can see that when the two waves move at the same time, the changing trend of the wavelet is obvious.

Data Availability
The data used in this study are available upon request from the author via email (jieyang199520@163.com).

Conflicts of Interest
The authors declare that they have no conflicts of interest.