The growth or decay estimates for nonlinear wave equations with damping and source terms

: The spatial decay or growth behavior of a coupled nonlinear wave equation with damping and source terms is considered. By defining the wave equations in a cylinder or an exterior region, the spatial growth and decay estimates for the solutions are obtained by assuming that the boundary conditions satisfy certain conditions. We also show that the growth or decay rates are faster than those obtained by relevant literature. This kind of spatial behavior can be extended to a nonlinear system of viscoelastic type. In the case of decay, we also prove that the total energy can be bounded by known data.

The wave equations have attracted many attentions of scholars due to their wide application, and a large number of achievements have been made in the existence of solutions (see [1,[5][6][7][8][9][10][11][12]). The fuzzy inference method is used to solve this problem. The algebraic formulation of fuzzy relation is studied in [13,14]. In this paper, we study the Phragmén-Lindelöf type alternative property of solutions of wave equations (1.1)- (1.2). It is proved that the solution of the equations either grows exponentially (polynomially) or decays exponentially (polynomially) when the space variable tends to infinite. In the case of decay, people usually expect a fast decay rate. The Phragmén-Lindelöf type alternative research on partial differential equations has lasted for a long time (see [2,[15][16][17][18][19][20][21][22][23]).
It is worth emphasizing that Quintanilla [2] considered an exterior or cone-like region. Under some appropriate conditions, the growth/decay estimates of some parabolic problems are obtained. Inspired by [2], we extends the research of to the nonlinear wave model in this paper. However, different from [2], in addition to condition A1 and condition A2, we also consider a special condition of ρ. The appropriate energy function is established, and the nonlinear differential inequality about the energy function is derived. By solving this differential inequality, the Phragmén-Lindelöf type alternative results of the solution are obtained. Our model is much more complex than [2], so the methods used in [2] can not be applied to our model directly. Finally, a nonlinear system of viscoelastic type is also studied when the system is defined in an exterior or cone-like regions and the growth or decay rates are also obtained.

The Phragmén-Lindelöf type alternative result under A1
Letting that Ω(r) denotes a cone-like region, i.e., and that B(r) denotes the exterior surface to the sphere, i.e., Equations (1.1) and (1.2) also have the following initial-boundary conditions where g 1 and g 2 are positive known functions, x = (x 1 , x 2 , x 3 ) and τ > 0. Now, we establish an energy function Let r 0 be a positive constant which satisfies r > r 0 ≥ R 0 . Integrating E(z, t) from r 0 to r, using the divergence theorem and Eqs (1.1) and (1.2), (2.1) and (2.2), we have where ω is positive constant. Now, we show how to bound E(r, t) by ∂ ∂r E(r, t). We use the Hölder inequality, the Young inequality and the condition A1 to have We consider inequality (2.8) for two cases. I. If ∃r 0 > R 0 such that E(r 0 , t) ≥ 0. From (2.5), we obtain E(r, t) ≥ E(r 0 , t) ≥ 0, r ≥ r 0 . Therefore, from (2.8) we have Integrating (2.9) from r 0 to r, we have Combing (2.4) and (2.10), we have II. If ∀r > R 0 such that E(r, t) < 0. The inequality (2.8) can be rewritten as Integrating (2.12) from r 0 to r, we have Inequality (2.13) shows that lim r→∞ −E(r, t) = 0. Integrating (2.5) from r to ∞ and combining (2.13), we obtain We summarize the above result as the following theorem. Theorem 2.1. Let (u, v) be solution of the (1.1), (1.2), (2.1), (2.2) in Ω(R 0 ), and ρ satisfies condition A1. Then for fixed t, (u, v) either grows exponentially or decays exponentially. Specifically, either (2.11) holds or (2.14) holds.

The Phragmén-Lindelöf type alternative result under A2
If the function ρ satisfies the condition A2, we recompute the inequality (2.6) and (2.7). Therefore and Inserting (3.1) and (3.2) into (2.3) and combining (2.5), we have By following a similar method to that used in Section 2, we can obtain the Phragmén-Lindelöf type alternative result.
Remark 3.1. Clearly, the rates of growth or decay obtained in Theorems 2.1 and 3.1 depend on ω. Because ω can be chosen large enough, the rates of growth or decay of the solutions can become large as we want.
Remark 3.2. The analysis in Sections 2 and 3 can be adapted to the single-wave equation and the heat conduction at low temperature In this section, we suppose that ρ satisfies ρ(q 2 ) = b 1 + b 2 q 2β , where b 1 , b 2 and β are positive constants. Clearly, ρ(q 2 ) = b 1 + b 2 q 2β can not satisfy A1 or A2. In this case, we define an "energy" function Computing as that in (2.4) and (2.5), we can get and Using the Hölder inequality and Young's inequality, we have and where we have chosen 1 2(β+1) > 1 n+1 . Inserting (4.4) and (4.5) into (4.1), we obtain where 0 . Next, we will analyze Eq (4.6) in two cases I. If ∃r 0 ≥ R 0 such that F (r 0 , t) ≥ 0, then F (r, t) ≥ F (r 0 , t) ≥ 0, r ≥ r 0 . Therefore, (4.6) can be rewritten as Using Young's inequality, we have Inserting (4.8) and (4.9) into (4.7), we have Integrating (4.11) from r 0 to r, we get Dropping the third term on the left of (4.12), we have we have from (4.13) (4.14) Combining (4.2) and (4.14), we have II. If ∀r ≥ R 0 such that F (r, t) < 0, then (4.6) can be rewritten as Without losing generality, we suppose that m > n > 1.
Inserting (4.17) and (4.18) into (4.16), we get where Integrating (4.20) from R 0 to r, we obtain Dropping the first and fourth terms on the left of (4.21), we obtain (4.23) From (4.15) and (4.23) we can obtain the following theorem. Obviously, in this case of ρ(q 2 ) = b 1 + b 2 q 2β , the decay rate obtained by Theorem 4.1 is slower than that obtained by Theorem 2.1 and Theorem 3.1.

A nonlinear system of viscoelastic type
In this section, we concern with a system of two coupled viscoelastic equations which describes the interaction between two different fields arising in viscoelasticity. In (5.1) and (5.2), 0 < t < T and h 1 , h 2 are differentiable functions satisfying h 1 (0), h 2 (0) > 0 and Messaoudi and Tatar [24] considered the system (5.1) and (5.2) in a bounded domain and proved the uniform decay for the solution when t → ∞. For more special cases, one can refer to [25][26][27]. They mainly concerned the well-posedness of the solutions and proved that the solutions decayed uniformly under some suitable conditions. However, the present paper extends the previous results to Eqs (5.1) and (5.2) in an exterior region. We consider Eqs (5.1) and (5.2) with the initial-boundary conditions (2.1) and (2.2) in Ω.

Conclusions
In this paper, we have considered several situations where the solutions of Eqs (1.1) and (1.2) either grow or decay exponentially or polynomially. We emphasize that the Poincaré inequality on the cross sections is not used in this paper. Thus, our results also hold for the two-dimensional case. On the other hand, there are some deeper problems to be studied in this paper. We can continue to study the continuous dependence of coefficients in the equation as that in [31]. These are the issues we will continue to study in the future.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.