On asymptotic behavior of solution to a nonlinear wave equation with Space-time speed of propagation and damping terms

In this paper, we consider the asymptotic behavior of solution to the nonlinear damped wave equation with space-time speed of propagation and damping potential. We obtained L 2 decay estimates via the weighted energy method and under certain suitable assumptions on the functions a ( t, x ) and b ( t, x ) . The technique follows that of Lin et al.[8] with modi ﬁ cation to the region of consideration in R n . These decay result extends the results in the literature.

In the case of scalar coefficients and bounded smooth domains Ω, there is an extensive literature on energy dacay results. For the semi-linear wave equation u tt − ∆u + u t = |u| p , (1.4) Todorova and Yordanov [18] showed that C n = 1 + 2 n is the critical exponent(Fujita exponent) for p < ∞ (n < 3) and p < 1 + 2 n (n ≥ 3). Nishihara in his paper [11] showed that the decay rate of solution to the damped linear wave equation follows that of self similar solution of its corresponding heat equation for n = 3 and showed this by obtaining L p − L q estimates on their difference. For similar results on 1-dimension and 2-dimensions, see Marcati and Nishihara [9] and Hosono and Ogawa [5] respectively, and in any dimension, see Narazaki [10]. Hence, it is expected that the behavior of the solution to equation (1.4) is similar to that of the corresponding heat equation

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On asymptotic behavior of solution to a nonlinear wave equation ... 1617 whose similarity solution u a (t, x) has the form t −1 p−1 F (xt − 1 2 ) with a = lim |x|→∞ |x| 2 p−1 f (x) ≥ 0 provided that p < 1 + 2 n . In the case of time dependent potential type of damping, with equations of the form u tt − ∆u + b(t)u t + |u| p−1 u = 0, (1.6) there are also several results on the decay rate of the solution. Nishihara and Zhai [13], used a weighted energy method similar to those in [18] and obtained decay estimates of the form under the assumption that b(t) ≈ (1 + t) −β . For Cauchy problem of the form u tt − a 2 (t)∆u + b(t)u t + c 0 |u| p−1 u = 0, (1.8) it is well known that the interplay between the coefficient a 2 (t) and the term b(t)u t induces different effect on the asymptotic behavior of the energy E(t) given by For more details see [2,3,4,20] and the references therein. In [1] Bui considered the asymptotic behavior of the nonlinear problem (1.8) with a(t) = (1 + t) and b(t) = µ(1 + )(1 + t) −1 , > 0, c 0 = 0 and obtained the following estimate . In the case of damped wave equation with space dependent potential type of damping; Todorova and Yordanov [19] investigated the decay rate of the energy when 0 ≤ α < 1. They obtained several decay rate types for solutions of (1.11) depending on p and α. These decay rates take the form , for t > 1, where δ is a constant. Nishihara [12] also considered the asymptotic behavior of solution to the semi-linear wave equation (1.11) (1.14) and obtained decay rates of the following type where α ∈ [0, 1). Ikehata and Inoue [6] studied nonlinear wave equations with b(x) = b 0 (1 + |x|) −1 and showed that solutions to (1.11) depend on the coefficient b 0 and their decay estimate takes the form ifb 0 ≥ 1. Moreover, for damped wave equations with space-time dependent potential type of damping Lin et al. [8] considered decay rates of solution to (1.17) and showed using the weighted energy method that the L 2 norm of the solution decays as On asymptotic behavior of solution to a nonlinear wave equation ... 1619 For nonlinear wave equations with variable coefficients which exhibit a dissipative term with a space dependent potential under the assumption that where b 0 > 0, b 1 > 0 and β ∈ [0, 2). R. Ikehata et al. [7] obtained long time asymptotics for solutions to (1.19)-(1.20) as a combination of solutions of wave and diffusion equations under certain assumptions on b in an exterior domain, see also [15].
Said-Houari [17] considered a viscoelastic wave equation with spacetime dependent damping potential and an absorbing term and by using a weighted energy method, they showed that the L 2 decay rates are the same as those in [8].
More recently, Roberts [16] under the assumption that obtained energy decay estimates of solution to the dissipative non-linear wave equation using a modification of the weighted multiplier technique introduced by Todorova and Yordanov [14].
In this paper, by using the weighted L 2 -energy method similar to that of [8], we obtain decay estimates of the energy of the solution to (1.1), where a(t, x) and b(t, x) have the form in (1.2)-(1.3) above. In [8], the space R n was divided into two zones and Z c (t; L, t 0 ) = R n \Z(t; L, t 0 ). To obtain boundedness on certain estimates on Z, a further division of Z was required. Here, we split the domain into two zones and Ω c (t, L, t 0 ) = R n \Ω(t, L, t 0 ) which depend on the weighted function for A = 2(1+β+γ) 2−(α+δ) and positive constants L, t 0 . With this choice, we overcome the challenge of splitting the first zone in order to obtain boundedness for every estimate on Ω(t; L, t 0 ) in the proof.

Preliminaries
In this section, we state some basic assumptions used in this paper. First, we introduce the following notations. L p (R n ), 1 ≤ p ≤ ∞, the Lebesgue space with norm k · k p and H 1 ρ (R n ) the Sobolev space defined by There exist a constant C > 0 such that the inequality holds for all u ∈ C ∞ 0 (R n ) if and only if the following relations hold: the Lemma is referred to as the Gagliardo-Nirenberg inequality.
We define the weighted function ψ(t, x) as follows: and consequently, we have In the sequel, we will denote the function ψ(t, x) by ψ for simplicity. To begin, we state the following lemmas which will be needed in the proof of the main result. First, we define the functions E(t) and H(t) associated to problem (1.1) by and respectively. Then for the function E(t) in (2.6), we have the following result.
Lemma 2.2. Let u be a solution of (1.1), then the function Proof. Multiplying (1.1) by e 2ψ u t and using (2.5), we obtain where we have used By employing Schwartz inequality, we observe that Hence, using (2.5) in (2.11) and substituting the resulting estimate in (2.9), we obtain and multiplying (2.12) by η(t), we get 2 Now, for the function H(t), we have the following lemma.
On asymptotic behavior of solution to a nonlinear wave equation ... 1623 Lemma 2.3. Let u be a solution of (1.1), then the function Proof. Multiplying (1.1) by e 2ψ u and using the estimate (2.5), we get where we have used (2.16) Using Schwartz inequality for the second to the last term on the right hand side of (2.15), we have the following estimate In a similar way, using (2.5) in (2.17), and substituting the resulting estimate in (2.15), we get and multiplying (2.18) by η(t), we obtain

Main result
In this section, we consider the long time behavior of the solution to (1.1). The result here is obtained via a weighted energy method and the technique follows that of Lin et al. [8]. For local existence result, the compactness condition on the support of the initial data is replaced by the following condition: With respect to the size of (1 + |x| 2 ) and (1 + t) and considering the weighted function ψ, we partition the space R n into the following zones: which is a modification of the zones as inspired by Lin et. al. [8], where A = 2(1+β+γ) 2−(α+δ) . Since α + β + δ + γ ∈ [0, 1), it follows that A < 2.
[Proof of Theorem 3.1] We split the proof into three parts, the first part considers the case x ∈ Ω(t, L, t 0 ), the second part covers the case x ∈ Ω c (t, L, t 0 ) and the third part combines the two results . We state the result in each of the zones in the form of a lemma.
Lemma 3.2. Let u be a solution of (1.1) and the functions E(t) and H(t) be defined as in (2.6) and (2.7) above, then for x ∈ Ω(t, L, t 0 ), the function E ψ (Ω(t, L, t 0 )) satisfies where k 0 is a positive constant to be determined later. Furthermore, we have (3.9) Observe that β + αA 2 ≤ β + α < 1 since A < 2 and α + β + δ + γ < 1. Now, multiplying (2.14) by ν (where ν < b 0 ) and adding the resulting estimate to (3.9), we get d dt where we have used Schwartz inequality to obtain the following estimates for the third and last term on the right hand side of (2.14) respectively: (3.12) By a suitable choice of ν sufficiently small as mentioned earlier, we can now choose a positive constant k 0 such that the estimates below are satisfied (3.13) this gives the desired estimate (3.7). We now integrate the estimate (3.7) over Ω(t; L, t 0 ) to obtain where H 3 (t; Ω(t; L, t 0 )) Define the function H E by (3.16) It can be proved easily that for positive constants k 1 , k 2 , the following inequality is satisfied: Now, multiplying (3.14) by (t 0 + t) m for m a constant which will be determined later, we obtain d dt  The term on the right hand side is estimated as On asymptotic behavior of solution to a nonlinear wave equation ... 1629 where we have used ψ t ≤ 0. From (3.13), it can be easily seen that we can choose t 0 large enough, such that Therefore, the first term on the right hand side of (3.19) yields To estimate the second term on the right hand of (3.19), we apply Young's inequality to obtain where C = C(m, b 0 , k 2 , p) and k p = k p (k 0 , p). Define J by Thus, if α(p+1) (p−1) > n, it follows that and if α(p+1) (p−1) < n, we obtain (3.25) Hence, we have that where ν is a small positive constant to be determined later. In addition, define where − → n is the unit outward normal vector of ∂Ω c (t; L, t 0 ). We can now state the next lemma.
Lemma 3.3. Let u be a solution of (1.1) and the functions E(t) and H(t) be defined as in (2.6) and (2.7) above, then for x ∈ Ω c (t; L, t 0 ), the function where k 0 is a positive constant to be determined later. Moreover, we have that (3.34) For the second term on the right hand of (3.34), by using Schwartz inequality, we obtain and observe here that 1 < 1. Also, by using the Schwartz inequality, we obtain the following estimates for the second to the last term and the last term on the right hand side of (3.34) respectively: and (3.37) Therefore, substituting the estimates (3.35) -(3.37) in (3.34), we get (3.38) Now, just as in the Case 1, we choose a suitable value for ν which is sufficiently small and a positive constant k 0 such that the estimates we have below are satisfied.
39) which gives the desired estimate. Therefore by integrating the estimate (3.31) over Ω c (t, L, t 0 ), we obtain (3.42) It can be proved in a similar way as in Case 1 that for positive constants k * 1 , k * 2 , the following inequality holds.
Multiplying (3.40) by (t 0 + t) m for the same constant m as in Case 1, we have The term on the right hand side is estimated as (3.45) It can be seen from (3.39) that we can suitably choose k 0 such that mk * 2 ≤ λk 0 (1 + β + γ). Therefore the first term on the right hand side of (3.45) yields 50) for 0 < < 1 and integrating (3.49) over [0, t], we obtain h H E (t; Ω(t; L, t 0 )) + H E (t; Ω c (t; L, t 0 ))   Remark 3. The decay result in Theorem 3.1 coincides with that of [8] for the case δ = γ = 0 and with that of [13] for the case δ = γ = α = 0.