The diffusion phenomenon for damped wave equations with space-time dependent coefficients

We introduce a method to study the long-time behavior of solutions to damped wave equations, where the coefficients of the equations are space-time dependent. We show that solutions exhibit the diffusion phenomenon, connecting their asymptotic behaviors with the asymptotic behaviors of solutions to corresponding parabolic equations. Sharp decay estimates for solutions to damped wave equations are given, and decay estimates for derivatives of solutions are also discussed.

1. Introduction. We introduce a method to study solutions to damped wave equations, where the coefficients of the equations depend on space and time. To demonstrate this method, we consider the problem u tt + u t − ∇ · a(x, t)∇u = 0, x ∈ R N , t > 0, (u, u t )(x, 0) = (u 0 , u 1 )(x), The global well-posedness and regularity of (1) are found in Ikawa [3]. We prove that the solution to (1) exhibits the diffusion phenomenon, meaning that u asymptotically behaves like a solution to v t − ∇ · a(x, t)∇v = 0, x ∈ R N , t > 0, v(x, 0) = u 0 (x) + u 1 (x), x ∈ R N .
As a corollary to the diffusion phenomenon, a sharp decay estimate for u(x, t) L 2 x is obtained. Decay estimates for L 2 x norms of derivatives of u(x, t) are also discussed. Matsumura [14] considered solutions to the equation u tt + u t − ∆u = 0 and used Fourier methods to establish L p x − L q x sharp decay estimates for u(x, t) and its space and time derivatives. Following Matsumura's results, many authors have considered many variants of the equation u tt +u t −∆u = 0, including variants where ∆ is replaced with a more general, time-independent operator. Many authors have also considered variants of the form u tt + a(x)b(t)u t − ∆u = 0. ( Note that until recently, (3) was considered with either space-or time-dependent damping coefficients, and the methods used for these two types of problems are incompatible.
Ikehata and Nishihara [9], followed by Chill and Haraux [1], considered the problem u tt + u t + Bu = 0 in a Hilbert space H, where B is a nonnegative self-adjoint operator in H. For B = ∆, their results are comparable to Matsumura's in low spatial dimensions.
Reissig and Wirth [25], Wirth [29,30] considered (3) with a slowly changing, time-dependent damping coefficient, and using Fourier methods, Wirth obtained L p x − L q x sharp decay estimates for solutions. Yamazaki [31] studied abstract wave equations with time-dependent damping.
For damped wave equations with slowly changing, space-dependent coefficients, Radu, Todorova and Yordanov [23] proved the exact gain in the decay rate for all higher order energies in terms of the first order energy.
Todorova and Yordanov [27] considered (3) with a constant damping coefficient and the nonlinear term |u| p . They developed a weighted energy method that uses a special weight related to the fundamental solution of the equation u tt +u t −∆u = 0.
Radu, Todorova and Yordanov [22] proved the diffusion phenomenon for the abstract problem u tt + u t + Bu = 0 in a Hilbert space H. Ikehata, Todorova and Yordanov [10] showed a more complex diffusion phenomenon for abstract wave equations with strong damping. Then Radu, Todorova and Yordanov [24] proved the diffusion phenomenon for the problem Cu tt + u t + Bu = 0 in a Hilbert space H, where B and C are two noncommuting self-adjoint operator on H, which excludes the use of the spectral theorem. Instead, they used consecutive approximations with conveniently defined diffusion solutions. They also expanded their decay gains that originated in [23], giving the exact gain in the decay rate for ∂ n t u in terms of u .
By resolvent arguments, Nishiyama [20] showed the diffusion phenomenon for the problem u tt + Au t + Bu = 0 in a Hilbert space H. Here A and B are two noncommuting self-adjoint operator on H, satisfying some additional conditions. Sobajima and Wakasugi [26] considered (3) with radially symmetric, slowly decaying space-dependent damping. They gave a sharp decay estimate for a modified L 2 x norm of the solution. We present the L 2 x sharp decay rate for the solution to (1) and decay rates for its derivatives. The coefficient a(x, t) is neither assumed to be separable in space and time, nor radially symmetric.
There are three key tools used in this present work. The first tool is the improved decay, which specifically refers to the gains in the decay rates for space and time derivatives of u in terms of u. This gain in decay is expressed in a weighted average sense. Note that the improved decay was discussed in [23] and [24].
The weighted energy method developed in [27] in the second key tool. Note that a special weight is used. One important consequence of this weighted energy method is that solutions to damped wave equations decay exponentially for x outside of the ball B 0 (t + 1) (1+δ)/2 , where δ > 0 can be arbitrarily small.
The third key tool is the fundamental solution of (2). The fundamental solution of (2) encodes desirable parabolic decay properties. We prove that u−v, the difference between the solutions of (1) and (2), can be expressed in terms of the fundamental solution of (2) acting on derivatives of u. This representation of u − v permits the three key tools to work together.
1.1. Assumptions for the hyperbolic problem (1). We assume the data (u 0 , We also assume that a(x, t) ∈ C 2 R N +1 and its first and second order derivatives are bounded and continuous in R N +1 ; this includes the mixed space-time derivatives. In addition, we assume where the constants a 1 , a 2 , a 3 , and a 4 > 0.
1.2. Existence, uniqueness and regularity for the hyperbolic problem (1). These are given by where v(x, t) is a prescribed solution to (2), and the constant C depends on a 1 , . . . , a 4 , N, and R 0 .
The prescribed solution to (2) will be shown to have the property v(x, t) . Combining this with Theorem 1.2 gives the following corollary. Corollary 1. Let u(x, t) be the solution to (1), where the assumptions in subsection 1.1 hold. Then for t ≥ 0, the following hold: where the constant C depends on a 1 , . . . , a 4 , N, and R 0 .
Remark 1. Using the same method, it is possible to consider a more general problem where a, b, c ∈ C 2 R N +1 have bounded first and second order derivatives and each satisfies assumptions similar to A1, A2, and A3. Conclusions analogous to Theorem 1.2 and Corollary 1 can be achieved via analogous proofs.
This paper is structured as follows: section two is devoted to the proofs of the improved decay. Estimates coming from the weighted energy method are proved in section three. A representation formula for the difference between solutions of (1) and (2), in terms of the fundamental solution of (2), is derived in section four. In section five, the main result and its corollary are proved via the three key tools.
2. Improved decay for dissipative wave equations.   2.2. Improved decay. The purpose of this subsection is to obtain the gains in the decay rates for derivatives of u in terms of u. These gains in decay are expressed in a weighted average sense. Definition 2.3. For i = 1, 2, respectively, we use the first and second energies: The improved decay for the first energy E 1 (t; u) is proved in the following proposition. Proposition 1. Let u(x, t) be the solution to (1), and let the assumptions in subsection 1.1 be satisfied. For r ≥ 0 and θ ≥ 0, where C depends on a 2 , a 3 , and θ.
Proof. We begin by taking the L 2 x (R N ) inner product of equation (1) and 2u t . Then apply assumption (A2) and get Similarly, we take the L 2 x (R N ) inner product of equation (1) and u to obtain Next, define the continuously differentiable function x ; for the regularity, see Lemma 1.1. Then combine (6) with (7) and add θ t+1 Y (t) to both sides. This gives Notice Similarly, . Apply (10) and a∇u , ∇u L 2 x ≤ 2E 1 (t; u) to the RHS of (8) and obtain Multiply both sides of (11) by the integrating factor (t + 1) θ to see that Next integrate both sides of (12) with respect to t, from 0 to r. To complete the proof, we estimate the integrals of the first and last terms of (12) by the initial data. Note that (9) and (10), followed by assumption (A1) give Apply assumption (A1) and then the energy inequality Lemma 2.1 to the RHS of (13), obtaining for all r ≥ 0. Therefore, the proof of (5) is complete.
The improved decay for u t 2 L 2 x is shown in the following proposition. Proposition 2. Let u(x, t) be the solution to (1), and let the assumptions in subsection 1.1 be satisfied. For r ≥ 0 and θ ≥ 0, where C depends on a 2 , a 3 , and θ.
Proof. Add θ+1 t+1 E 1 (t; u) to both sides of (6) to obtain θ x from above by 2E 1 (t; u), and then multiply both sides of the resulting inequality by the integrating factor (t + 1) θ+1 . This gives Next, integrate both sides of this inequality with respect to t, from 0 to r, and note that To complete the proof of (14), apply the improved decay Proposition 1 to the term r 0 (θ + 1 + 2a 3 )(t + 1) θ E 1 (t; u)dt, obtaining the last term on the RHS of (14).
The next two propositions show the improved decay for E 2 (t; u) and u tt respectively. These propositions are analogous to Propositions 1 and 2, except with larger weights on their left-hand sides. Proposition 3. Let u(x, t) be the solution to (1), and let the assumptions in subsection 1.1 be satisfied. For r ≥ 0 and θ ≥ 0, where C depends on a 2 , a 3 , a 4 , and θ.
Proof. Begin by taking the L 2 x (R N ) inner product of ∂ t u tt + u t − ∇ · a∇u = 0 and 2u tt to obtain Similarly we take the L 2 Now, define the functions x . Next, define the continuously differentiable function Observe that the left-hand sides of (16) and (17) To estimate the RHS of (18) from above, we estimate Z 1 (t), Z 2 (t), and Z 3 (t). First, notice that by assumption (A2) and 2a |∇u| |∇u t | ≤ a t+1 |∇u| 2 + (t + 1)a |∇u t | 2 . Next, assumption (A2) gives Observe via (t+1) 2 a 2 by assumptions (A2) and (A3). Now use (19) - (21) to estimate the RHS of (18) from above by The following estimates hold: The proof of (23) follows from the definitions of Z 1 (t) and E 1 (t; u). To show (24), The proof of (25) is similar to the proof of (24). Apply (23) and (25) to bound (22), and hence the RHS of (18), from above by Replace the RHS of (18) with (26) to obtain recalling that a∇u t , ∇u t L 2 x + u tt 2 L 2 x = 2E 2 (t; u). Multiply both sides of (27) by the integrating factor (t + 1) θ+2 and integrate in t, from 0 to r. Then apply the improved decay Propositions 1 and 2 to the integrals involving E 1 (t; u) and u t 2 L 2 x , respectively. This way, we get the last term on the RHS of (15). Thus we only need to bound the integrals involving the first two terms and the last term of (27) by the initial data.
The integral involving the first two terms of (27) is bounded via inequalities (24) and (25), i.e., observe that . To bound the integral involving the last term of (27), define T 0 := max {0, 2C (a 3 , θ) − 1}. Then for all r ≥ 0, To complete the proof, apply assumption (A1) and then the energy inequality Lemma 2.1 to the RHS of (28), obtaining for all r ≥ 0.
Proposition 4. Let u(x, t) be the solution to (1), and let the assumptions in subsection 1.1 be satisfied. For r ≥ 0 and θ ≥ 0, where C depends on a 2 , a 3 , a 4 , and θ.
The following observations will be employed later: Note that −W t ≤ W 2 t+1 because m ≤ e m for all m ∈ R. Definition 3.2. For i = 1, 2, respectively, we define the first and second weighted energies: The first weighted energy estimate is given by the following proposition.
Proposition 5. Let u(x, t) be the solution to (1), and let the assumptions in subsection 1.1 be satisfied. Assume γ in (34) is such that 0 < γ ≤ 1 2a2 . Then for t ≥ 0, Proof. For W (x, t) as in (34), define the functions Z 1 (t) := 2∇ · au t W ∇u and Z 2 (t) := a W t |∇u| 2 − 2W u 2 t − 2u t a ∇W · ∇u. Also define the function Y (t) := W u 2 t + aW |∇u| 2 , and note that Y (t) is continuously differentiable as an L 1 x R N function by Lemma 1.1. Next, multiply equation (1) by 2W u t to get Note that Z 2 (t) ≤ 0; to see this, observe that via Young's inequality. Then recall that −W t ≥ 0 by (35), and notice that 2aγ ≤ 1 since a ≤ a 2 and 0 < γ ≤ 1 2a2 . Now using Z 2 (t) ≤ 0, assumption (A2) and W t ≤ 0, we refine (38) and get , and integrating the former inequality with respect to x, in R N gives Now, R N Z 1 (t)dx = 0 since u has compact support in x, and we thus have To complete the proof, multiply both side of this inequality by the integrating factor (t + 1) −a3 , and then integrate with respect to t, on [0, r].
The following proposition gives the second weighted energy estimate.
Also define the function Y (t) := W u 2 tt + aW |∇u t | 2 + 2a t W ∇u · ∇u t , and note that Y (t) is continuously differentiable as an L 1 x R N function by Lemma 1.1. Next, multiply ∂ t u tt + u t − ∇ · a∇u = 0 by 2W u tt to get As in the proof of the weighted energy estimate Proposition 5, we get Z 2 (t) ≤ 0. Consequently, Z 2 (t) ≤ 0, assumption (A2) and W t ≤ 0 give To refine (40), apply the following inequalities, which are proved via Young's inequality: The refinement is Recall that m ≤ e m for all m ∈ R.

Now integrate this inequality with respect to t, on [0, r] and get
Observe that for r ≥ 0. Hence, we estimate the second term on the RHS of (45). Notice that Young's inequality and assumption (A2) give Thus, by the weighted energy estimate Proposition 5, To complete the proof, apply (45) and (46) to estimate the LHS and RHS of (44) from below and above, respectively. We obtain The next proposition shows that derivatives of the solution to (1) decay exponentially outside of a ball.
Proposition 7. (Exponential decay) Let u(x, t) be the solution to (1), and let the assumptions in subsection 1.1 be satisfied. For δ > 0 and for some k > 0, where n = 0, 1 and m = 1 − n, 2 − n. The constant C depends on a 1 , a 2 , a 3 , a 4 , R 0 , and δ.
4. The representation of the difference between solutions of (1) and (2) in terms of the fundamental solution of the parabolic problem (2). The differential equation in (2) has a pointwise, classical fundamental solution Γ(x, t; ξ, s) for x, ξ ∈ R N and 0 ≤ s < t. The fundamental solution allows the transfer of decay from the solution of (2) to the solution of (1). The properties of Γ(x, t; ξ, s) are in Friedman [2, Chapter 1]. Importantly, Friedman [2, Chapter 1, (6.12)] and [2, Chapter 1, (8.14) and Theorem 15] give the following lemma.
Definition 4.2. We use the notations: for scalar f (ξ, s) and vector g(ξ, s), with f (ξ, s), |g(ξ, s)| ∈ L p ξ for any 1 ≤ p ≤ ∞. The following lemma makes use of Lemma 4.1 to get bounds for the operators Γ t,s x and (a∇ ξ Γ) t,s x . Lemma 4.3. (Diffusion operator estimates) Let Γ(x, t; ξ, s) be the fundamental solution of (2). Then for f (x, ·), |g(x, ·)| ∈ L 1 (R N ) ∩ L 2 (R N ) and 0 ≤ s < t, the following properties hold: Proof. We prove (i) and (ii). By Lemma 4.1(i), Take the L 2 x norm of both sides of this inequality. To get (i), apply Young's convolution inequality h * x k L 2 t−s and k = |f (x, s)|. To get (ii), let h = |f (x, s)| and k = exp −C |x| 2 t−s . To prove (iii) and (iv), repeat the proof of (i) and (ii), except use Lemma 4.1(ii) instead of (i).
by Lemma 4.3(i) and the continuity of Γ(x, t; ξ, s). The next proposition precisely determines the difference between the solutions of (1) and (2) in terms of the fundamental solution of (2).
Write the above identity as u t − ∇ · (a∇u ) = −u tt + f and consider this as a nonhomogeneous version of (2). Then by the presentation and uniqueness theorems in Friedman [2, Chapter 1, Theorems 12 and 16], Apply integration by parts in s to t/2 0 Γ t,s x u ss ds, moving the derivative from u ss to Γ t,s x , and get For the last term on the RHS of (55), use the fact that Γ(x, t; ξ, s) is a classical solution to the backwards problem, i.e., use where x, ξ ∈ R N and 0 ≤ s < t. Then apply the divergence theorem to the last term on the RHS of (55) and get t/2 0 Γ t,s x u ss ds = Γ t,t/2 t,s x (∇ ξ u s ) ds.
Using this identity, rewrite (54) as where v (x, t) = Γ t,0 x (u + u t ). Take → 0. Using the regularity (4), the first three terms on the RHS of (56) converge, respectively, in L 2 x to the first three terms on the RHS of (53) because of the diffusion operator estimate Lemma 4.3(i). Similarly, the fourth term on the RHS of (56) converges in L 2 x to the fourth term on the RHS of (53) because of the diffusion operator estimate Lemma 4.3(iii). Therefore, we Note that the diffusion operator estimate Lemma 4.3(i) gives By using the regularity (4) and the boundedness of a and ∇a, we get: 5. The diffusion phenomenon and decay. In the proof of Theorem 1.2, the improved decay is used to extract decay from the second and third terms on the RHS of the integral identity (53), after using the diffusion operator estimate Lemma 4.3(i). Then the diffusion operator estimate Lemma 4.3(iv) is used to extract decay from the fourth term on the RHS of the integral identity. This comes at the price of having to estimate ∇u s (x, s) L 1 x , which is paid by the exponential decay Proposition 7, followed by another application of the improved decay.
Proof of Theorem 1.2. We consider the cases when t < 1 and t ≥ 1. First, assume that t < 1. Then by the energy inequality Lemma 2.1, and the diffusion operator estimate Lemma 4.
, and Theorem 1.2 is verified for t < 1. Now, assume that t ≥ 1. Define the functions Also define the function which has continuous derivative Y (t) = (t + 1) x ds via the regularity (4) and (52).
In the integral identity (53), subtract v(x, t) from both sides, and then apply · 2 L 2 x to obtain (t + 1) where , and We estimate each of I 1 , I 2 , and I 3 . For I 1 , the diffusion operator estimate Lemma Thus and using 2u s u ss ≤ u 2 s s+1 + (s + 1)u 2 ss gives x . Let θ = N −1 2 . Then by the improved decay Propositions 2 and 4, respectively, where the constant C depends on a 2 , a 3 , a 4 , and N . Now, proceed to estimate I 2 . Observe that the diffusion operator estimate Lemma 4.3(i) gives and the RHS is bounded from above by C (t + 1) t t/2 u ss (x, s) 2 L 2 x ds via Hölder's inequality. Thus x ds ≤ C Z 4 (t).

MONTGOMERY TAYLOR
Therefore, as above, use the improved decay Proposition 4 with θ = N −1 2 to get where the constant C depends on a 2 , a 3 , a 4 , and N . Now we address I 3 . Observe that the diffusion operator estimate Lemma 4.3(iv) gives since t ≥ 1. Thus by Hölder's inequality, Then the exponential decay Proposition 7 with δ = 1   2N and Combine this with the estimate ∇u s (x, s) where A(s) c is the ball B 0 (s + 1) (1+δ)/2 and |A(s) c | is the volume of the ball, and obtain the estimate Apply this estimate to the RHS of (62) and get (t + 1) Now use the improved decay Proposition 3 with θ = N −1 2 to obtain (t + 1) where the constant C depends on a 1 , a 2 , a 3 , a 4 , N, and R 0 . Next, apply (60), (61) and (63) to the RHS of (59) to get (t + 1) 5 2 Y (t) ≤ C Z 1 (t) + C (u 0 , u 1 ) where the constant C depends on a 1 , a 2 , a 3 , a 4 , N, and R 0 . Then by (58) and u 2 L 2 by the diffusion operator estimate Lemma 4.3(ii). Thus Therefore, estimates (64) and (66) give Multiply both sides of (67) by (t + 1) − 5 2 . Then use the integrating factor exp 2C 3 (t + 1) −3/2 to get Y (t) ≤ C (u 0 , u 1 ) 2 H 2 ×H 1 (R N ) . Therefore, the RHS of (67) is bounded by the initial data, giving , completing the proof.