Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.


Introduction
Let N ≥ 2. We consider the wave equation with space-dependent damping term in an exterior domain Ω ⊂ R N with a smooth boundary:    u tt − ∆u + a(x)u t = 0, x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω, t > 0, (u, u t )(x, 0) = (u 0 , u 1 )(x), x ∈ Ω, (1.1) where we denote by ∆ the usual Laplacian in R N and by u t and u tt the first and second derivative of u with respect to the variable t, and u = u(x, t) is a real-valued unknown function. The coefficient of the damping term a(x) satisfies a ∈ C 2 (Ω), a(x) > 0 on Ω and lim |x|→∞ x α a(x) = a 0 (1.2) with some constants α ∈ [0, 1) and a 0 ∈ (0, ∞), where y = (1 + |y| 2 ) 1 2 for y ∈ R N . In this moment, the initial data (u 0 , u 1 ) are assumed to have compact supports in Ω and to satisfy the compatibility condition of order k ≥ 1: where u ℓ is successively defined by u ℓ = ∆u ℓ−2 − a(x)u ℓ−1 (ℓ = 2, . . . , k). We note that existence and uniqueness of solution to the problem (1.1) have been discussed (see e.g., Ikawa [2,Theorem 2]). It is proved in Matsumura [4] that if Ω = R N and a(x) ≡ 1, then the solution u of (1.1) satisfies the energy decay estimate where the constant C depends on the size of the supprot of initial data. Moreover, it is shown in Nishihara [7] that u has the same asymptotic behavior as the one of the problem v t − ∆v = 0, x ∈ R N , t > 0, v(x, 0) = u 0 (x) + u 1 (x), x ∈ R N .
(The solution has an asymptotic behavior similar to the solution of the usual wave equation without damping). Therefore one can expect that diffusion phenomena occur only when a(x) ≥ C x −α for α ≤ 1.
In this paper, we discuss precise decay rates of the weighted energy and β > 0) which is introduced by Todorova and Yordanov [12] based on the ideas in [11] and in [3]. They proved weighted energy estimates when a(x) is radially symmetric and satisfies (1.2). After that, Radu, Todorova and Yordanov [8] extended it to higher-order derivatives. In [13], the second author proved diffusion phenomena for (1.1) with Ω = R N and a(x) = x −α (α ∈ [0, 1)) by comparing the solution of the following problem In [10], diffusion phenomena for (1.1) with an exterior domain and for general radially symmetric damping term are obtained. However, the weighted energy estimates and diffusion phenomena for (1.1) with non-radially symmetric damping are still remaining open. The difficulty seems to come from the choice of auxiliary function A in the weighted energy, which strongly depends on the existence of positive solution to the Poisson equation ∆A(x) = a(x). In fact, an example of non-existence of positive solution to ∆A = a for non-radial a(x) is shown in [10]. Radu, Todorova and Yordanov [9] considered the case Ω = R N and used a solu- is a subsolution of the equation ∆A = a. In general one cannot obtain the optimal decay estimate via this choice because of the luck of the precise behavior of a(x) at the spatial infinity which can be expected to determine the precise decay late of weighted energy estimates. Our main idea to overcome this difficulty is to weaken the equality ∆A = a and consider the inequality (1 − ε)a ≤ ∆A ≤ (1 + ε)a, and to construct a solution having appropriate behavior, we employ a cut-off argument. The aim of this paper is to give a proof of Ikehata-Todorova-Yordanov type weighted energy estimates for (1.1) with non-radially symmetric damping and to obtain diffusion phenomena for (1.1) under the compatibility condition of order 1 and the condition (1.2) (without any restriction).
This paper is originated as follows. In Section 2, we discuss related elliptic and parabolic problems. The weighted energy estimates for (1.1) are established in Section 3 (Proposition 3.5). Section 4 is devoted to show diffusion phenomena (Proposition 4.1).

Related elliptic and parabolic problems
2.1. An elliptic problem for weighted energy estimates. As we mentioned above, in general, existence of positive solutions to the Poisson equation ∆A(x) = a(x) is false for non-radial a(x). Thus, we weaken this equation and consider the following inequality where ε ∈ (0, 1) is a parameter. Here we construct a positive solution A of (2.1) satisfying for some constants A 1ε , A 2ε > 0.
Proof. Firstly, we extend a(x) as a positive function in C 2 (R N ); note that this is possible by virtue of the smoothness of ∂Ω. To simplify the notation, we use the same symbol a(x) as a function defined on R N . We construct a solution of approximated equation Then we have Let ε ∈ (0, 1) be fixed. Then by (2.5) there exists a constant R ε > 0 such that where N is the Newton potential given by Then we easily see that ∆B 1ε (x) = b 1 (x) and ∆B 2ε = η ε (x)b 2 (x). Moreover, noting that supp (η ε b 2 ) is compact, we see from a direct calculation that there exist a constant M ε > 0 such that This yields that B ε := B 1ε + B 2ε is bounded from below and positive for x ∈ R N with sufficiently large |x|. Moreover, we have Using the same argument as in the proof of [10, Lemma 3.1], we can see that there exists a constant λ ε ≥ 0 such that A ε (x) := λ ε + B ε (x) satisfies (2.1)-(2.3).

A parabolic problem for diffusion phenomena.
Here we consider L p -L q type estimates for solutions to the initial-boundary value problem of the following parabolic equation x ∈ Ω. (2.6) Here we introduce a weighted L p -spaces which is quite reasonable because the corresponding elliptic operator a(x) −1 ∆ can be regarded as a symmetric operator in L 2 dµ . The L p -L q type estimates for the semigroup associated with the Friedrichs' [10]. The proof is based on Beurling-Deny's criterion and Gagliardo-Nirenberg inequality. and

Weighted energy estimates
In this section we establish weighted energy estimates for solutions of (1.1) by introducing Ikehata-Todorova-Yordanov type weight function with an auxiliary function A ε constructed in Subsection 2.1.
To begin with, let us recall the finite speed propagation property of the wave equation (see [2]).
Before introducing a weight function, we also recall two identities for partial energy functionals proved in [10].
and let u be a solution to (1.1). Then, we have Here we introduce a weight function for weighted energy estimates, which is a modification of the one in Todorova-Yordanov [12].
where A ε is given in Lemma 2.1. And define for t ≥ 0,
Now we are in a position to state our main result for weighted energy estimates for solutions of (1.1).
Proof. As in the proof of [10, Lemma 3.6], by integration by parts we have Noting that we have (3.4).
In order to clarify the effect of the finite propagation property, we now put Then Lemma 3.7. For t ≥ 0, we have Proof. By a(x) −1 ≤ a −1 1 x α ≤ a −1 1 (1 + |x|) α and the finite propagation property we have Using the Cauchy-Schwarz inequality and the above inequality yields (3.6): We can prove (3.7) in a similar way.
Proof of Proposition 3.5. Firstly, by (3.7) we observe that By using the above estimate, we prove the assertion via mathematical induction.