The Pointwise Estimates of Solutions for Semilinear Dissipative Wave Equation

In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the initial value problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution.


Introduction
In this paper we consider the initial value problem for the semilinear dissipative wave equation in n(n ≥ 1) dimensions, ( + ∂ t )u(x, t) = f (u), x ∈ R n , t > 0, (1.1) 1a with initial condition
There have been many results on the equation ( 1a 1a 1.1) and its variants corresponding to the different forms of f (u). By employing the weighted L 2 energy method and the explicit formula of solutions, Ikehata, Nishihara and Zhao INZ [7] obtained that the behavior of solutions to ( 1a 1a 1.1) as t → ∞ is expected to be same as that for the corresponding heat equation, Nishihara Ni0 [15] studied the global asymptotic behaviors in three and four dimensions, and Nishihara [9] studied the decay property of solutions to ( 1a 1a 1.1) by using the energy method combined with L p − L q estimates, and Ono Ono1 [20] derived sharp decay rates in the subcritical case of solutions to ( 1a 1a 1.1) in unbounded domains in R N without the smallness condition on initial data. Also, recently Nishihara,etc. in Ni1,Ni2 [16, 17] studied the following semilinear damped wave equations with time or space-time dependent damping term, 1), and obtained the global existence and the L 2 decay rate of the solution by using the weighted energy method. (  [3,6,8,13,14,18], for studies on the case f (u) = |u| θ+1 , see Ik,LZ,Ono2,Ono3, TY, Z [5,12,21,22,23,25], and for studies on the global attractors, see BP, KS [1,10] and the references cited there. The main purpose of this paper is to study the pointwise estimates of solutions for ( 1.2), and obtained the pointwise estimates of solutions. In this paper, we first obtain the global-in-time solutions by energy method combined with the fixed point theorem of Banach, and then obtain the optimal pointwise decay estimates of the solutions by using the properties of the Green function proved in

LW
[11] combined with Fourier analysis. One point worthy to be mentioned is that, different from that for solutions to the corresponding linear problem, the order of derivatives with respect to time variable t of solutions does not contribute to the decay rate of solutions due to the presence of the semilinear term, which could be seen from ( and we denote its inverse transform by F −1 . For 1 ≤ p ≤ ∞, L p = L p (R n ) is the usual Lebesgue space with the norm · L p . Let s be a nonnegative integer. Then H s = H s (R n ) denotes the Sobolev space of L 2 functions, equipped with the norm In particular, we use · = · L 2 , · s = · H s . Here, for a multi-index α, D α x denotes the totality of all the |α|-th order derivatives with respect to x ∈ R n . Also, C k (I; H s (R n )) denotes the space of k-times continuously differentiable functions on the interval I with values in the Sobolev space Finally, in this paper, we denote every positive constant by the same symbol C or c without confusion.

Main theorems
The first main result is about the global existence of solutions to the initial value problem ( 1a 1a

1.2).
ge Theorem 2.1 (Global existence). Let θ > 0 be an integer. Assume that Then if E 0 is suitably small, ( 1a 1a , then there exists T > 0 and a unique solution to ( 1a 1a The proof of the local existence result is based on the fixed point theorem of Banach and standard argument, so the detail is omitted.
Based on the a priori assumption where s > n is an integer andδ < 1 is a small constant, the following a priori estimate is obtained.

The global existence of solutions
First we give a lemma which will be used in our next energy estimates.
L Lemma 3.1. Let n ≥ 1, 1 ≤ p, q, r ≤ ∞ and 1 p = 1 q + 1 r . Then the following estimate holds: Proof. The estimate ( d1 d1 3.1) can be found in a literature but we give here a proof. To prove ( d1 d1 3.1), it is enough to show that, for k 1 ≥ 1, k 2 ≥ 1 and k 1 + k 2 = k, the following estimate holds: Since θ 1 + θ 2 = 1, we have 1 p = 1 p 1 + 1 p 2 . By using the Hölder inequality and the Gagliardo-Nirenberg inequality, we have In the last inequality, we have used the Young inequality. Thus ( By multiplying ( 1a 1a 1.1) with u t and integrating on R n × (0, t) with respect to (x, t), we get By taking sum for α with 1 ≤ |α| ≤ s, it yields that By multiplying ( 1a 1a 1.1) with u and integrating on R n × (0, t) with respect to (x, t), by virtue of ( 3a0 3a0

3.2) we get
By taking sum for α with 1 ≤ |α| ≤ s and in view of ( 3a 3a

3.7) holds with
1.1), by using induction argument we could prove that the following two equalities hold for k ≥ 1, Let t = 0 in ( even even

Estimates on Green function
In this section, we list some formulas and properties of the Green function obtained in

LW
[11] to make preparation for the next section about the pointwise estimates of solutions.
The Green function or the fundamental solution to the corresponding linear dissipative wave equation (i.e. f (u) = 0 in ( 1a 1a By Fourier transform we get that, The symbol of the operator for equation ( 1a 1a 1.1) is τ and ξ correspond to ∂ ∂t , and 1 √ −1 ∂ ∂x j , j = 1, 2, · · · , n. It is easy to see that the eigenvalues of ( 2a 2a Let be the smooth cut-off functions, where ε and R are any fixed positive numbers satisfying 2ε < R − 1. Set andĜ ± i (ξ, t) = χ i (ξ)Ĝ ± (ξ, t), i = 1, 2, 3. We are going to study G ± i (x, t), which is the inverse Fourier transform corresponding toĜ ± i (ξ, t). Denote B N (|x|, t) = (1+ |x| 2 1+t ) −N . First we give two propositions regarding to G 1 (x, t) and G 2 (x, t), the proof can be seen in LW, WY [11,24]. 21 Proposition 4.1. For sufficiently small ε, there exists constant C > 0, and N > n such that 22 Proposition 4.2. For fixed ε and R, there exist positive numbers m , C and N > n such that Next we will come to consider G 3 (x, t). Now we list some lemmas which are useful in dealing with the higher frequency part.

can be seen in WY
[24].
The following Kirchhoff formulas can be seen in E,HZ [2,4].
25 Lemma 4.5. Assume that w(x, t) is the fundamental solution of the following wave equation with c = 1, There are constants a α , b α depending only on the spatial dimension n ≥ 1 such that, if h ∈ C ∞ (R n ), then for odd n, and for even n. Here dS z denotes surface measure on the unit sphere in R n .
we have the following proposition.
29 Proposition 4.7. For R sufficiently large, there exists distribution The proof of Proposition ], we have that where N > n can be big enough.

Pointwise estimates
In this section, we aim at verifying that the solution obtained in Theorem We denote the solution to the corresponding linear dissipative wave equation asū, then then the solution u to ( 1a 1a 1.1) can be expressed as: u =ū −ũ. In

LW
[11], the following pointwise estimate of the solutionū to the linear problem is obtained.

LW
[11], we obtain the following estimate for D α x J 1 , As for the estimate to D α x J 2 , we divide it as following , Next we estimate J 2i (i = 1, 2, 3, 4, 5) respectively by using Proposition 210 210 4.8 (with N ≥ r) and the fact that B(|x|, t) ≤ 1 and is an increasing function of t and decreasing function of |x| .
By the definition of M(T ), we have that Now we estimate J 21 in two cases. Case 1. |x| 2 ≥ t. We have here in the second inequality we used Lemma 41 41 here in the last inequality we used Lemma 41 41 5.2 (2). Combining the two cases, we have that For J 22 , we have noticing that if |x| ≥ 2|y|, then |x − y| ≥ |x| 2 , it yields that For J 23 , by using the monotonic properties of B(|x|, t) with respect to |x| and t, we have that For J 24 , similar to J 22 we have that ≤ CM(T ) θ+1 (ϕ α (x, t)) −1 .
Thus Theorem pe pe 2.4 is proved.