Well-Posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation

We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation. Moreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data. Additionally, we provide results concerning exponential decay of solutions of the nonlinear equation.

The The most general of these popular models is Kuznetsov's equation for the acoustic velocity potential ψ, where c > 0 is the speed of sound, b ≥ 0 is the diffusivity of sound and B/A is the parameter of nonlinearity. Neglecting local nonlinear effects (in the sense that the expression c 2 |∇ψ| 2 − (ψ t ) 2 is sufficiently small) one arrives at the Westervelt equation The Kuznetsov equation in its turn can in some sense be regarded as a simplification (for a small ratio νPr −1 between the kinematic viscosity ν and the Prandtl number Pr) of the following fourth order in space equation This equation results from the following two general models from the original paper [5] (see equations (11), (13) there) We replace of ∆ψ in the last term of (1.4), (1.5) by 1 c 2 ψ tt , which can be justified by the main part of the differential operator (that corresponds to the wave equation ψ tt − c 2 ∆ψ = 0). Moreover, we consider potential diffusivity as appearing in (1.4). Therewith, equation (1.5)
The restriction on the dimension of Ω is imposed in order to be able to use various embedding theorems. Increasing the space dimension to d ≥ 4 would not be of relevance in applications anyway.
Throughout this paper we will assume b > 0 since this essential for several results we intend we prove (cf. Remarks 3.5 and 3.11).
We will prove local and global in time well-posedness and show that solutions decay exponentially with respect to certain norms.
In Section 2 we introduce the notation which will be used throughout this paper. We mention function spaces we will make use of, e.g., L p -spaces, Sobolev and Besov spaces and recall (respectively, refer to) embedding theorems needed for our proofs.
In Section 3 we consider the linearization of (1.6) in a general abstract form. We apply the theory of operator semigroups to (1.6) and show that the underlying semigroup is analytic on two different phase spaces which leads, together with certain spectral properties of the generator, to two exponentially decaying energy functionals. Moreover, we provide existence and uniqueness results for the solutions of the linear model.
In Section 4 we perform energy estimates as a preparation for the proof of global in time well-posedness.
Section 5 is devoted to the fully nonlinear equation (1.6). We prove local in time well-posedness by employing a fixed point argument. The space which we use in this fixed point argument is obtained by combining regularity results for the linearized Westervelt equation and the heat equation. We achieve local existence and uniqueness of solutions provided the given initial data are sufficiently small. Global in time well-posedness is obtained by using the energy estimates from Section 4 together with classical barrier's method which finally leads to an exponential decay result for the higher order energy functional introduced in Section 4.
Appendix A provides an overview of facts from the theory of operator semigroups (cf. [6], [21]) which are used in Section 3.
We denote by L p (Ω) the space of (classes of) Lebesgue integrable functions Ω → R with exponent p ∈ [1, ∞]. The norm of a function u ∈ L p (Ω) will be denoted by u L p (Ω) . In the special case p = 2 we simply write u := u L 2 (Ω) for the norm of a function u ∈ L 2 (Ω) and u, v := u, v L 2 (Ω) for the inner product of u, v ∈ L 2 (Ω).
More generally, we will always write u X for the X-norm of a function u ∈ X and C X֒→Y for the norm of the embedding X ֒→ Y .
The space C k (0, T ; X) consists of all k-times continuously differentiable functions u : By H we denote a (separable) Hilbert space equipped with the inner product ., .. H and the induced norm . H . In applications, we always think of H = L 2 (Ω).
Moreover, by A : D(A) → H we denote a self-adjoint, strictly positive and closed operator whose domain of definition D(A) is dense in H and always keep in mind that in applications we think of A = −∆ being the negative Dirichlet Laplacian defined on D(A) = H 2 (Ω) ∩ H 1 0 (Ω). Remark 2.1. As A was assumed to be strictly positive, we conclude that the spectrum of A, σ(A) ⊂ R + , is bounded from below and may therefore set µ := min σ(A) > 0. Moreover, on the strength of positivity, fractional powers A Θ , Θ ≥ 0 of A are well-defined and A Θ is again a strictly positive self-adjoint operator with domain of definition D(A Θ ). Note that D(A Θ1 ) ⊂ D(A Θ2 ) for Θ 1 > Θ 2 . Furthermore, as D(A) was assumed to be dense in H, we also have that D(A Θ ) is dense in H for 0 < Θ < 1.

The linear problem -semigroup framework
Before turning to the nonlinear analysis, we consider the linearization of (1.6) in a generalized abstract form.
We investigate the abstract linear partial differential equation defined on the Hilbert space H with initial conditions 3.1. Semigroup methods for the homogeneous equation. We are first going to treat the homogeneous version of (3.1), i.e.
The partial differential equation (3.3) can be written as a first-order system of the form and (3.13) Operators of the type 0 I −c 2 A −bA have been extensively investigated, e.g., in [2], [3] and [19] and we will modify some arguments used therein in order to show that A given by (3.12) generates an analytic semigroup on H 1 .
Proof. We proceed analogously to [2]. Let yields the desired result. The resolvent R(λ,Ã 1 ) ofÃ 1 is explicitly given by Lemma 3.3. We have the following uniform bounds for all λ with Re(λ) > 0: Proposition 3.4. The operatorÃ 1 generates an analytic semigroup on H 1 .
Proof. As we already know thatÃ 1 generates a strongly continuous semigroup of contractions on H 1 , it remains to show that there exists a constant M > 0 such that Recall the explicit representation of the resolvent (3.16). Let λ ∈ C with Re(λ) > 0 and x = (x 1 , x 2 , x 3 ) T ∈ H 1 . By Lemma 3.3 and (2.1) we have that Remark 3.5. Note that the assumption b > 0 is essential to establish the uniform bound (3.19). In case b = 0 the strongly continuous semigroup of contractions generated byÃ 1 on H 1 is not analytic (see also Remark 3.11). Proof. Note that B 1 from (3.14) is a bounded operator on H 1 . Therefore, the result follows at once from the perturbation theorem for analytic semigroups.
Lemma 3.7. The linear operatorÃ is the generator of a bounded analytic semigroup on H 2 .
Proof. Note thatÃ is self-adjoint on H 2 and its spectrum is given by The result follows at once by using Lemma A.6.
3.1.3. Exponential decay for the homogeneous equation. The previous results enable us now to show exponential decay of solutions in the homogeneous case.
Notation. We introduce the energy functionals Our aim is to show exponential decay for the energies E 1 [ψ] and E 2 [ψ].
Proof. The spectrum of A is given by Combining the upper bounds for Re(κ n (µ i )), n ∈ {1, 2, 3} leads to the desired spectral bound.
Remark 3.11. Note that for Lemma 3.10 the assumption b > 0 is essential. If b = 0, the spectrum of A is given by σ(A) = {−aµ i , ±icµ i : µ i ∈ σ(A)}. Hence, in this case, A is not a sectorial operator and can thus not be the generator of an analytic semigroup (cf. Theorem A.5).

Solutions of the homogeneous equation.
We now consider the homogeneous initial boundary value problem which we represented as (3.10). Recall that, if A is the infinitesimal generator of an analytic semigroup, the initial value problem (3.25) has a unique solution for every x ∈ X ([21, Corollary 4.1.5]).

Semigroup methods for the inhomogeneous equation.
In this section we consider the inhomogeneous initial boundary value problem where f : Ω × (0, T ] → R is given. We represent it as an inhomogeneous abstract ordinary differential equation of the form with the initial conditions (3.2), where A and Ψ are given by (3.12) and (3.13), respectively and F (t) = (0, 0, f (t)) T .
Proof. The result follows analogously to the one Corollary 3.14 by applying [21, Corollary 4.3.3].

Energy estimates
In this section, we derive energy estimates which will enable us to prove global existence of solutions in Section 5. Again, we consider the equation or, equivalently, Remark 4.1. In order to interchange the order of differentiation, we need to assume that the following estimates only hold for sufficiently smooth solutions. But in fact, by using density arguments, this restriction can finally be removed.

Notation. We introduce the energy functionals
as well as the sum of (4.2) and (4.3), Proof. Differentiating D w w = f with respect to time, taking inner products in H with w ttt and then integrating with respect to time we get Performing integration by parts, we obtain Estimating the right hand side yields which, together with (4.6), implies the desired estimate (4.5).
holds withb sufficiently small and C sufficiently large.
Proof. We recall Proposition 3 in [11] with α ≡ 1, i.e. for the energy E 0 [w](t) in (4.1) we have the estimate forb sufficiently small andC sufficiently large. Next, we use (4.5) and multiply it with a sufficiently small constant λ, λ ≤b b which implies that, on the left hand side of (4.8), we have C 0 = 3b −2 c 0 and that in (4.9) we get (choice of λ) Adding (4.8) and (4.10) and using (2.1) for A −1/2 f H , we obtain (4.7) withb sufficiently small andČ sufficiently large. Now we use the following energy identity for the heat equation: Applying (4.11) to v = ψ ttt (i.e., D h v = w ttt ), to v = A 1/2 ψ tt (i.e., D h v = A 1/2 w tt ), and to v = A 1/2 ψ t (i.e., D h v = A 1/2 w t ) we obtain that the left hand side terms under the time integrals in (4.7) provide us with the estimates Inserting into (4.7) we end up with Lemma 4.4. We have the estimate withb sufficiently small andC sufficiently large.
It remains to estimate the (quadratic, hence small for small ψ) terms on the right-hand side in terms of the left-hand side. We have for τ ∈ (0, t) where we have used the embedding (2.2). Similarly we get holds. Inserting the latter into (4.12) and using the fact that under the integral on the left-hand side we have and where we abbreviated   We again use the differential operators where this time we inserted A = −∆. With this notation, the linearized version of (1.7) reads D h D w ψ = f.
(ii) It is clear that b∆ψ t = ψ tt − c 2 ∆ψ −f ∈ C(0, T ; L 2 (Ω)) and thus ψ ∈ C 1 (0, T ; H 2 (Ω) ∩ H 1 0 (Ω)). (iii) Lemma 4.2 with H = L 2 (Ω), A = −∆, f =f and t = T gives us and, invoking the assumptions and the fact that for any finite time horizon T we have C 1 (0, T ; H 2 (Ω)∩H 1 0 (Ω)) ⊂ H 1 (0, T ; H 2 (Ω)∩H 1 0 (Ω)), we infer T 0 ψ ttt 2 < ∞ and thus ψ ∈ H 3 (0, T ; L 2 (Ω)) as claimed. We consider the equation Our strategy in order to prove local existence of solutions of (1.7) is to apply Banach's Fixed Point Theorem to the map where ψ is a solution of (5.4) and the space W is given by : v V ≤m} wherem has to be suitably chosen later. Therefore, we need to show that T is a self-mapping, W is closed and furthermore, that T is a contraction.
Step 1. T : W → V is a self-mapping.
This can be achieved by taking sufficiently small initial data.
Step 3. T : W → W is a contraction.
Proof. As the map T is a self-mapping and a contraction on W and moreover, W is a closed subset of V, we conclude that on the strength of the Banach Fixed Point Theorem T : W → W has a unique fixed point, i.e. there exists a unique solution ψ ∈ W such that T (ψ) = ψ. Condition (5.8) comes from (5.7).

5.3.
Global well-posedness. Our strategy in order to prove global in-time wellposedness of equation (1.7) is to use the classical barrier method. Let ψ be a local in time solution of (1.7) according to Theorem 5.5 and let t < T , where T is the maximal existence time (possibly T = ∞). The barrier method now yields global in time existence for sufficiently small initial data.
Remark 5.7. The term r(τ ) is not required for the proof of global well-posedness, but it provides us with higher order regularity. In particular, as Λ(t) ≥ 0 and r(τ ) ≥ 0, we obtain