Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions

The Blackstock-Crighton equation models nonlinear acoustic wave propagation in thermo-viscous fluids. In the present work we investigate the associated inhomogeneous Dirichlet and Neumann boundary value problems in a bounded domain and prove long-time well-posedness and exponential stability for sufficiently small data. The solution depends analytically on the data. In the Dirichlet case, the solution decays to zero and the same holds for Neumann conditions if the data have zero mean. We choose an optimal $L_p$-setting, where the regularity of the initial and boundary data are necessary and sufficient for existence, uniqueness and regularity of the solution. The linearized model with homogeneous boundary conditions is represented as an abstract evolution equation for which we show maximal $L_p$-regularity. In order to eliminate inhomogeneous boundary conditions, we establish a general higher regularity result for the heat equation. We conclude that the linearized model induces a topological linear isomorphism and then solve the nonlinear problem by means of the implicit function theorem.


Introduction
An acoustic wave propagates through a medium as a local pressure change. Nonlinear effects typically occur in case of acoustic waves of high amplitude which are used for several medical and industrial purposes such as lithotripsy, thermotherapy, ultrasound cleaning or welding and sonochemistry. Research on mathematical aspects of nonlinear acoustic wave propagation is therefore not only interesting from a mathematicians point of view. In fact, in case of medical applications, enhancement of the mathematical understanding of the underlying models should lead to a considerable reduction of complication risks.
The present work aims to provide a mathematical analysis of the Blackstock-Crighton-Kuznetsov equation 2A (u t ) 2 + |∇u| 2 tt and the Blackstock-Crighton-Westervelt equation tt for the acoustic velocity potential u, where c is the speed of sound, b is the diffusivity of sound and a is the heat conductivity of the fluid. Note that a = νPr, where ν is the is kinematic viscosity and Pr denotes the Prandtl number. Alternatively, (1.1) and (1.2) can be expressed in terms of the acoustic pressure p via the pressure density relation ρu t = p, where ρ denotes the mass density. The quantity B/A is known as the parameter of nonlinearity and is proportional to the ratio of the coefficients of the quadratic and linear terms in the Taylor series expansion of the variations of the pressure in a medium in terms of variations of the density. Note that (1.2) is obtained from (1.1) by neglecting local nonlinear effects in the sense that the expression c 2 |∇u| 2 − (u t ) 2 is sufficiently small. For a detailed introduction to the theory and applications of nonlinear acoustics we refer to [HB98]. Equations (1.1) and (1.2) result from two evolution equations of fourth order governing finite-amplitude sound in thermoviscous relaxing fluids, namely −c 2 a∆ 2 u + (a + b) ∆u tt + c 2 ∆u − u ttt = |∇u| 2 which have been derived by Blackstock [Bla63] from the basic equations describing the general motion of thermally relaxing, viscous fluids (continuity equation, momentum equation, entropy equation and an arbitrary equation of state) and also appear as equations (11) and (13) in Crighton's work [Cri79] on nonlinear acoustic models. We replace ∆u in the last term of (1.3) and (1.4) by 1 c 2 u tt , which can be justified by the main part of the differential operator corresponding to the wave equation u tt − c 2 ∆u = 0. Moreover, in (1.4) we consider potential diffusivity as in (1.3). Therewith, we arrive at equation (1.1) for which in [Bru15] the name Blackstock-Crighton-Kuznetsov equation has been introduced. For a more rigorous derivation of (1.1) we refer to Section 2 in [Bru15].
While (1.1) and (1.2) are enhanced models in nonlinear acoustics, the Kuznetsov (1.6) u tt − b∆u t − c 2 ∆u = 1 c 2 1 + B 2A (u t ) 2 t , are classical, well-accepted and widely used models governing sound propagation in fluids. As (1.1) and (1.2), they are derived from the basic equations in fluid mechanics. The Kuznetsov equation is the more general one of these classical models, in particular the Westervelt equation is obtained from the Kuznetsov equation by neglecting local nonlinear effects. Moreover, for a small ratio of ν and Pr, that is, for small heat conductivity, (1.5) and (1.6) can be regarded as simplifications of (1.1) and (1.2), respectively.
The classical models (1.5) and (1.6) have recently been extensively investigated. In particular, results on well-posedness for the Kuznetsov and the Westervelt equation with homogeneous Dirichlet [KL09] and inhomogeneous Dirichlet [KLV11], [KL12] and Neumann [KL11] boundary conditions have recently been shown in an L 2 (Ω)-setting on spatial domains Ω ⊂ R n of dimension n ∈ {1, 2, 3}. Moreover, there are results on optimal regularity and long-time behavior of solutions for the Westervelt equation with homogeneous Dirichlet [MW11] and for the Kuznetsov equation with inhomogeneous Dirichlet [MW13] boundary conditions in L p (Ω)-spaces where the spatial domain Ω ⊂ R n is of arbitrary dimension.
On the contrary, mathematical research on higher order partial differential equations arising in nonlinear acoustics is still in an early stage. Well-posedness and exponential decay results for the homogeneous Dirichlet boundary value problems associated with (1.1) and (1.2) in an L 2 (Ω)-setting where Ω ⊂ R n , n ∈ {1, 2, 3}, have been shown in [Bru15] and [BK14], respectively. In the present work we consider (1.1) and (1.2) with inhomogeneous Dirichlet and Neumann boundary conditions in L p (Ω)-spaces where the spatial domain Ω is of dimension n ∈ N. We show global well-posedness and long-time behavior of solutions in an optimal functional analytic setting in the sense that the regularity of the solution is necessary and sufficient for the regularity of the initial and boundary data. While in [Bru15] and [BK14] the results were proved by means of appropriate energy estimates and the Banach fixed-point theorem, the techniques used in the present paper are based on maximal L p -regularity for parabolic problems and the implicit function theorem in Banach spaces.
Our strategy for solving (1.7) and (1.8) is to prove that their linearizations induce isomorphisms between suitable Banach spaces and to apply the implicit function theorem. In some sense these linearizations can be considered as a composition of a heat problem and another linearized problem for the Westervelt equation. While the linearized Westervelt equation can be handled similar as in [MW11,MW13], the heat equation has to be solved with higher regularity conditions. The paper is organized as follows. The purpose of Section 2 is to recall several facts we need on our way to global well-posedness and exponential stability of (1.7) and (1.8). In particular, we mention all function spaces we use, provide facts about the homogeneous Dirichlet and Neumann Laplace operator and list some important embeddings and traces. We also give a short review of the concept of maximal L p -regularity for parabolic problems. Furthermore, we recall respectively prove optimal regularity results for the heat equation and the linearized Westervelt equation with inhomogeneous Dirichlet and Neumann boundary conditions. Section 3 is devoted to the inhomogeneous Dirichlet boundary value problem (1.7). First of all we consider the corresponding linear problem and represent it as an abstract parabolic evolution equation for which we show maximal L p -regularity. This gives us optimal regularity for the homogeneous linear version of (1.7). Based on the optimal regularity results for the heat and the linearized Westervelt equation from Section 2 we prove optimal regularity for the linear inhomogeneous Dirichlet boundary value problem in Proposition 3.5. The main result in this section is Theorem 3.6 which states global well-posedness of (1.7) and immediately implies exponential stability (Theorem 3.7).
In Section 4 we treat the inhomogeneous Neumann boundary value problem (1.8). Here, we proceed analogously to Section 3. Theorem 4.8 provides local well-posedness for (1.8). In the Neumann case, global well-posedness is shown for data having zero mean (Theorem 4.9). Moreover, Theorem 4.11 states long-time behavior of solutions.
In Appendix A we collect several facts about the homogeneous Neumann Laplacian. We outline how one finds a realization of the Laplacian with homogeneous Neumann boundary conditions such that its spectrum is contained in the positive half-line.
In Appendix B we first study the temporal trace operator acting on a class of anisotropic Sobolev spaces. We present its mapping properties, provide a right-inverse and thus obtain its precise range space. Moreover, we construct functions with prescribed higher-order initial data. Second, we prove some so-called mixed derivative embeddings which are often used for checking the continuity of differential operators acting on anisotropic spaces.
In Appendix C we prove some higher regularity results for the heat equation with inhomogeneous Dirichlet or Neumann boundary conditions and inhomogeneous initial conditions in a far more general framework than needed in the main text. In particular, we state explicitly all necessary compatibility conditions between initial and boundary data and show how they are used to contruct a solution with high regularity.

Preliminaries
The purpose of this section is to introduce the notation we are going to use throughout the paper and to recall several important facts and results we need to prove global well-posedness and long-time behavior of solutions for (1.7) and (1.8). As already mentioned in Section 1, we always assume that the spatial domain Ω ⊂ R n , n ∈ N, is bounded and has smooth boundary Γ = ∂Ω. We write J for a time interval and consider either J = (0, T ) for some finite time horizon T > 0 or J = R + = (0, ∞).
2.1. Function spaces, operators, embeddings and traces. The space BU C k (Ω) contains all k-times Fréchet differentiable functions Ω → R, whose derivatives up to order k are bounded and uniformly continuous. For p ∈ (1, ∞), let L p (Ω) denote the space of (equivalence classes of) Lebesgue measurable p-integrable functions Ω → R. We write W m p (Ω) for the Sobolev-Slobodeckij space and H m p (Ω) for the Bessel potential space of order m ∈ [0, ∞), where we have W m p (Ω) = H m p (Ω) if m ∈ N 0 . Moreover, W m p (Ω; X) and H m p (Ω; X) denote the vector valued versions and 0 W 1 p (J; X) denotes the space of all functions u ∈ W 1 p (J; X) with u(0) = 0. For p ∈ [1, ∞), q ∈ [1, ∞], s ∈ R + , the Besov space B s p,q (Ω) is defined as (L p (Ω), W m p (Ω)) s/m,q where m = ⌈s⌉ and (·, ·) s/m,q indicates real interpolation. It holds that where 0 ≤ k < s < m and s = (1 − Θ)k + Θm.
We always write X ֒→ Y if the Banach space X is continuously embedded into the Banach space Y . Moreover, let L(X, Y ) be the space of all bounded linear operators between X and Y . A linear operator A : X → Y is called an isomorphism if it is bounded and bijective. Then the closed graph theorem implies that A −1 : Y → X is also bounded and therefore A : X → Y is a homeomorphism. Now, let X and X be Banach spaces such that X ֒→ L 1,loc (J; X) where L 1,loc (J; X) is the space of locally integrable functions J → X. For any ω ∈ R we define the exponentially weighted space We recall that the spectrum σ(−∆ D ) is a discrete subset of (0, ∞) consisting only of eigenvalues λ D n = λ n (−∆ D ), n ∈ N 0 with finite multiplicity. We write λ We shall use the embeddings W 1 p (J) ֒→ BU C(J) and W s p (Ω) ֒→ W t p (Ω) for s ≥ t. We always write γ D = ·| Γ and γ N = ∂ ν · | Γ = ν · (∇·)| Γ for the Dirichlet and the Neumann trace, respectively. Moreover, γ t = · | t=0 denotes the temporal trace. Let B ∈ {D, N }, j D = 0, j N = 1. For p ∈ (1, ∞), k ∈ N and l ∈ N 0 the spatial trace is bounded, see Appendix B. Furthermore, the trace is bounded for every s ∈ (j B + 1/p, ∞), cf. [Tri83, Theorem 3.3.3]. The temporal trace is bounded for α ∈ (1/p, 1], s ∈ [0, ∞) and s + 2α / ∈ N for α < 1. The same holds when the domain Ω is replaced by its boundary Γ. Finally, we mention that for p ∈ (1, ∞), t, s ∈ N 0 , τ, σ ∈ N, θ ∈ (0, 1) we have the mixed derivative embedding where again Ω can be replaced by Γ. For more general embeddings of this form we refer to Appendix B.
2.2. Maximal L p -regularity. Let J = (0, T ) or J = R + = (0, ∞) and assume p ∈ (1, ∞). We say that a closed linear operator A : D(A) → X with dense domain D(A) in a Banach space X admits maximal L p -regularity on J if for each F ∈ L p (J; X) the abstract Cauchy problem admits a unique solution u ∈ E(J) = W 1 p (J; X) ∩ L p (J; D(A)) for v 0 = 0. Furthermore, the abstract inhomogeneous Cauchy problem (2.6) is said to admit maximal is a homeomorphism. Then its inverse is the solution map If A : D(A) → X has maximal L p -regularity on J, then the abstract Cauchy problem (2.6) has maximal L p -regularity on J, cf. Section III.1.5 in [Ama95]. The following result is very useful and will be used several times throughout this paper.
is a homeomorphism. Then is a homeomorphism.
2.3. Optimal regularity results. In order to prove our results on optimal regularity for the linearized versions of (1.7) and (1.8), we need optimal regularity results for the heat equation and the linearized Westervelt equation. We always let a, b, c ∈ (0, ∞).
2.3.1. Dirichlet boundary conditions. Recall that λ D 0 > 0 always denotes the smallest eigenvalue of the negative Dirichlet Laplacian in L p (Ω).
Proof. We first let ω = 0. By [DHP03, Theorem 8.2], Lemma A.5 and [Dor93, Theorem 2.4], the Neumann Laplacian in L p,0 (Ω) with domain D(∆ N,0 ) = D(∆ N ) ∩ L p,0 (Ω) has maximal regularity on R + . We therefore obtain a unique solution u 3 ∈ H u,0 of the problem for every given f 3 ∈ L p (R + ; L p,0 (Ω)). Furthermore, problem (2.11) admits at most one solution. Indeed, let us construct it as u = u 1 + u 2 + u 3 where we first solve for some sufficiently large µ > 0 with [DHP07, Theorem 2.1]. Next, we let u 2 solve the ordinary differential equation Finally, with f 3 = f −f +µ(u 1 −ū 1 ), we obtain u 3 as above. It is easy to check that v = u 1 +u 3 and w = u 2 satisfy the assertion. The case ω > 0 can be reduced to the previous one by multiplying the functions u, f , g with t → e ωt and using that the spectrum of −a∆ N,0 + ω is contained in (0, ∞).
Finally we arrive at our optimal regularity result for the linearized Westervelt equation with inhomogeneous Neumann boundary conditions. Lemma 2.7. Let p ∈ (1, ∞) \ {3} and set ω 0 = min{bλ N 1 /2, c 2 /b}. Then for every ω ∈ (0, ω 0 ) the linear initial boundary value problem if and only if the data satisfy the following conditions: Proof. From Lemma 2.6 we obtain uniqueness. In order to show necessity of (i)-(iv) one proceeds as in the proof of Lemma 2.5. It therefore remains to show sufficiency. Let δ > ω.
has a unique solution θ ∈ e −ω W u,0 . Furthermore, we define θ w as the solution of the ordinary differential equation Then v = ϕ v + θ v and w = ϕ w + θ w satisfy the assertion and we are done.
2.4. Analysis in Banach spaces. For later use in the proof of global well-posedness of (1.7) and (1.8) we will now recall the concept of analytic mappings in Banach spaces and the analytic version of the implicit function theorem. The remainder of this section is collected from Section 15.1 in [Dei85].
Let X and Y be Banach spaces over the same field K = R or K = C and let U ⊂ X be open. Then F : U → Y is called analytic at x 0 ∈ U if there is some r > 0 and continuous symmetric k-linear operators F k : In particular, every bounded linear map F : X → Y is analytic.
Theorem 2.9 (Implicit Function Theorem, cf. [Dei85, 15.1]). Let X, Y and Z be Banach spaces over the same field K = R and K = C. Assume U ⊂ X and V ⊂ Y are neighborhoods of x 0 ∈ X and y 0 ∈ Y , respectively. Furthermore, suppose

The Dirichlet boundary value problem
In this section we prove global well-posedness and exponential stability for (1.7). First of all, we consider the linearized version of the inhomogeneous Dirichlet boundary value problem and represent it as an abstract evolution equation. We show that this abstract equation admits maximal L p -regularity and derive an optimal regularity result for the linearized equation associated with (1.7). Then we use the implicit function theorem to construct a solution of the nonlinear problem (1.7). Exponential decay of this solution is an immediate consequence.
3.1. Maximal L p -regularity for the linearized equation. Suppose J = (0, T ) or J = R + . For f ∈ L p (J × Ω) we consider the initial boundary value problem where u 0 , u 1 , u 2 : Ω → R and g, h : J × Γ → R are the given initial and boundary data, respectively. In order to address the problem of maximal L p -regularity for the linearized equation, we represent (3.1) with g = h = 0 as an abstract Cauchy problem This motivates us to consider the Banach space and the densely defined linear operator A D : D(A D ) → X D given by Therewith, we may write (3.1) as an abstract evolution equation First of all we will treat the issue of maximal L p -regularity of First we show that A D 1 : D(A D ) → X D has maximal L p -regularity. To this end, we consider the Cauchy Since we know from Lemma 2.2 that the homogeneous heat equation admits maximal L pregularity, we obtain that for all f 3 ∈ L p (R + × Ω) there exists a unique solution v 3 ∈ W 1 p (R + ; L p (Ω)) ∩ L p (R + ; D(∆ D )). Moreover, as f 2 + v 3 ∈ L p (R + ; D(∆ D )), Lemma C.5 implies that there is a unique solution v 2 ∈ W 1 p (R + ; D(∆ D )) ∩ L p (R + ; D((∆ D ) 2 )) Now, note that for α > 0 the operator Moreover, by the fact that A D 2 : X D → X D is a bounded linear operator, Proposition 4.3 and Theorem 4.4 in [DHP03] imply that there exists some µ > 0 such that µ + A D 1 + A D 2 has maximal regularity which concludes the proof.
In order to show maximal regularity for the operator A D : D(A D ) → X D , we need the following result on its spectrum.
is an isomorphism.
Proof. We follow the proof of Theorem 2.5 in [MW11]. From Proposition 3.1 we know µ + A D admits maximal regularity on R + for some µ > 0. Multiplying v D t + A D v D = F by e −µt shows that A D has maximal L p -regularity on bounded intervals J = (0, T ). Lemma 3.2 tells us that spectral bound admits maximal L p -regularity in the sense that there exists a unique solution Proof. Based on the choices of X D and D(A D ) in (3.2) and (3.3), it is straightforward to check that the condition v D ∈ e −ω (W 1 p (R + ; X D ) ∩ L p (R + ; D(A D ))) where v D is given by (3.4) implies u ∈ e −ω (E u ∩ W 2 p (R + ; W 2 p (Ω))). Since the mixed derivative embedding gives us E u ֒→ W 2 p (R + ; W 2 p (Ω)), we arrive at u ∈ e −ω E u . Next, we determine (X, D(A)) 1−1/p,p . It is trivial that (D((∆ D ) 2 ), D((∆ D ) 2 )) 1−1/p,p = D((∆ D ) 2 ), i. e. we have u 0 ∈ W 4 p (Ω) with u| Γ = ∆u| Γ = 0. Moreover, since for p ∈ (1, ∞) we have 2/p ∈ R \ N unless p = 2, (2.1) gives us We now arrive at the final result for this section and prove optimal regularity for the linear initial boundary value problem (3.1) in the sense that the regularity of the data f , g, h, u 0 , u 1 and u 2 are necessary and sufficient for the existence of a unique solution u ∈ e −ω E u .

3.2.
Global well-posedness and exponential stability. Based on Proposition 3.5, we now show that there exists a unique global solution of the nonlinear initial boundary value problem (1.7) which depends continuously (in fact, even analytically) on the (sufficiently small) initial and boundary data. Moreover, we prove that the equilibrium u = 0 is exponentially stable.
Step 1: The implicit function theorem applies. First of all, we will now verify the assumptions of the implicit function theorem (Theorem 2.9).
By setting w = u • + u ⋆ , f = e ωt u • and g = e ωt u ⋆ we are done. Altogether, we have that Step The map D u• G(0, 0) : e −ω L p (R + × Ω) → e −ω E u,h is an isomorphism since, according to Corollary 3.4, for every f ∈ e −ω L p (R + × Ω) the equation (a∆ − ∂ t )(u tt − c 2 ∆u − b∆u t ) = f admits a unique solution u ∈ e −ω E u,h .
Step 3: Dependence of the solution on the data. It remains to show that the solution u ∈ e −ω E u depends analytically on (g, h, u 0 , u 1 , u 2 ). To this end, we define the spaces From Proposition 3.5 with f = 0 we obtain that u ⋆ depends linearly and continuously and thus analytically on (g, h, u 0 , u 1 , u 2 ) ∈ D. Moreover, u ⋆ → u • = ϕ(u ⋆ ) is analytic on B ρ (0) and therefore u • ∈ e −ω E u,h depends analytically on the data (g, h, u 0 , u 1 , u 2 ) ∈ D. Altogether, u = u • + u ⋆ enjoys the same property which concludes the proof.
An immediate consequence of Theorem 3.6 is that the global solution u ∈ e −ω E u of (1.7) decays to zero at an exponential rate.
Theorem 3.7 (Exponential stability -the Dirichlet case). Under the same assumptions as in Theorem 3.6, the solution u decays exponentially fast to zero as t → ∞, in the sense that for some C ≥ 0 depending on the boundary and initial data g, h, u 0 , u 1 and u 2 .

The Neumann boundary value problem
In this section we treat the inhomogeneous Neumann boundary value problem (1.8). We proceed analogously to the Dirichlet case, that is, we first consider the linearized equation and then construct a solution of the nonlinear problem (1.8) by means of the implicit function theorem.
Note that, in the Dirichlet case, the fact that the operator −A D : D(A D ) → X D defined by (3.3) has a strictly negative spectral bound (Lemma 3.2) was crucial in order to show that the linearized equation (3.1) admits maximal regularity on R + , see the proof of Theorem 3.3. In the Neumann case, due to the zero eigenvalue of −∆ N : D(∆ N ) → L p (Ω) with D(∆ N ) = {u ∈ W 2 p (Ω) : ∂ ν u = 0 on Γ}, we cannot expect to obtain maximal regularity on R + . For this reason we consider −∆ N,0 : D(∆ N,0 ) → L p,0 (Ω), where D(∆ N,0 ) = D(∆ N ) ∩ L p,0 (Ω). The spectrum of −∆ N,0 is contained in (0, ∞), therefore we can prove maximal regularity of the homogeneous linear Neumann boundary problem on R + analogously to the Dirichlet case. However, if we restrict ourselves to finite time intervals J = (0, T ), then we do not necessarily need to use the realization −∆ N,0 in L p,0 (Ω). In case of finite time intervals we use −∆ N . As a consequence, we will prove global well-posedness of (1.8) only if the data u 0 , u 1 , u 2 and g, h have zero mean whereas local well-posedness holds also for data with non-zero mean.
4.1. Maximal L p -regularity for the linearized equation. As in Section 3.1, let J = (0, T ) or J = R + and assume p ∈ (1, ∞). Here, for we f ∈ L p (J × Ω) we consider on J × Γ, (u, u t , u tt ) = (u 0 , u 1 , u 2 ) on {t = 0} × Ω, where u 0 , u 1 , u 2 : Ω → R and g, h : J × Γ → R are the given initial and boundary data, respectively. Analogously to the Dirichlet case we first represent (4.1) with g = h = 0 as an abstract evolution equation of the form introducing the Banach space and defining the coefficient operator On one hand, in the following we will show maximal regularity of A N on finite time intervals.
On the other hand, as already pointed out, we are going to use the realization −∆ N,0 of the homogeneous Neumann Laplacian. For this reason we introduce the Banach space  Proof. The result can be proved similarly to Proposition 3.1. For some α > 0 consider Clearly, the operator A N 2 : X N → X N is bounded. Moreover, A N 1 : D(A N 1 ) → X N has maximal L p -regularity on R + which is seen as in the proof of Proposition 3.1 by Let F ∈ L p (R + ; X N ). Now one solves stepwise the equations above, starting with the last one, to get a unique solution v 3 ∈ W 1 p (R + ; L p (Ω)) ∩ L p (R + ; D(∆ N )). Then f 2 + v 3 ∈ L p (R + ; D(∆ N )). Here, we need to employ Lemma C.5 in order to obtain a unique solution v 2 ∈ W 1 p (R + ; D(∆ N )) ∩ L p (R + ; D((∆ N ) 2 )). As in the Dirichlet case, the first equation gives us a unique solution v 1 ∈ W 1 p (R + ; D((∆ N ) 2 )). Altogether, since the condition F ∈ L p (R + ; X N ) implies existence of a unique solution v ∈ W 1 p (R + ; X N ) ∩ L p (R + ; D(A N )) we conclude that A N 1 : D(A N ) → X N admits maximal L p -regularity on R + . Finally, as in the proof of Proposition 3.1, a perturbation argument implies that there exists some ν > 0 such that ν + A N 1 + A N 2 = ν + A N has the property of maximal L p -regularity on R + . Maximal L p -regularity of ν + A N,0 follows analogously by considering the operators A N,0 1 and A N,0 2 which are equal to A N 1 and A N 2 upon replacement of ∆ N by ∆ N,0 and proceeding as above.
Since ν + A N has maximal L p -regularity on R + , multiplication of (4.2) with e −νt shows that A N has maximal L p -regularity on bounded intervals J = (0, T ).  ∈ (1, ∞). Then A N : D(A N ) → X N has maximal L p -regularity on J and therefore Next, observe that −A N,0 has a strictly negative spectral bound. This can be shown likewise to Lemma 3.2 since the spectrum of the negative Neumann Laplacian in L p,0 (Ω) is contained in (0, ∞) and consists only of eigenvalues of finite multiplicity.  Theorems 4.2 and 4.4 immediately yield optimal regularity for (4.1) with homogeneous boundary conditions, i. e. g = h = 0.  , then (4.4) admits optimal regularity in the sense that there exists a unique solution if and only if f ∈ L p (J × Ω), u 0 ∈ W 4 p (Ω), u 1 ∈ W 4−2/p p (Ω), u 2 ∈ W 2−2/p p (Ω) and the initial and boundary data are compatible, that is, we have ∂ ν u 0 | Γ = ∂ ν ∆u 0 | Γ = ∂ ν u 1 | Γ = 0 and, if p > 3, also ∂ ν ∆u 1 | Γ = ∂ ν u 2 | Γ = 0 in the sense of traces.
Proof. Assertion (i) follows immediately from Theorem 4.2. Analogously to the proof of Corollary 3.4 one verifies that v N ∈ W 1 p (J; X N ) ∩ L p (J; D(A N )), F ∈ L p (J; X N ) and v N 0 ∈ (X N , D(A N )) 1−1/p,p imply u ∈ E u (J), f ∈ L p (J × Ω) and the desired regularity of the initial values, respectively. Based on Theorem 4.4, the second claim follows analogously.
Finally we arrive at our global optimal regularity result for (4.1). As in the Dirichlet case, sufficiency is shown by a combination of an optimal regularity result for the linearized Westervelt equation and a higher regularity result for the heat equation.
Proposition 4.6. Let p ∈ (1, ∞) \ {3} and define ω 0 = min{aλ N 1 , bλ N 1 /2, c 2 /b}. Then for every ω ∈ (0, ω 0 ) the linear initial boundary value problem has a unique solution of the form u(t, if and only if the data satisfy the conditions Moreover, the solution fulfills the estimate Proof. It is not surprising that the proof of necessity can be done similarly to Proposition 3.5. Assume that u(t, x) = v(t, x) + w(t, x) is a solution of (4.5) with v ∈ e −ω E u and ∂ 3 t w ∈ e −ω L p (R + ). Since, apart from having zero mean, v has the same regularity as u in Proposition 3.5, we are conclude that e ωt f = −e ωt (v + w) ttt − ab∆ 2 (e ωt v t ) − ac 2 ∆(e ωt v) + (a + b)∆(e ωt v tt ) + c 2 ∆ 2 (e ωt v t ) ∈ L p (R + × Ω) and (i) is readily checked. Moreover, w, w t and w tt are just time-dependent and thus constant at t = 0, hence the regularity of the initial values (ii) can be shown as in the Dirichlet case. Moreover, the regularity of the boundary data (iii) is obtained from the spatial trace (2.2) with the same choices of k and l as in the proof of Proposition 3.5 and setting j B = 1. Concerning (iv) it is straightforward to show Next, we show that conditions (i)-(iv) are sufficient for the existence of a unique solution u(t, x) = v(t, x) + w(t) of (4.5) such that v ∈ e −ω E u and w ttt ∈ e −ω L p (R + ). As in the Dirichlet case, we interchange the order of differentiation on the left-hand side and consider the subproblems From condition (ii) we obtain a∆u 0 −u 1 ∈ W 2 p (Ω) and a∆u 1 −u 2 ∈ W 2−2/p p (Ω). Furthermore, (iii) implies ah − g t ∈ e −ω W ν . On the strength of Lemma 2.7 we obtain that (4.6) admits a unique solution of the form ϕ(t, x) = ϕ 1 (t, x) + ϕ 2 (t) with ϕ 1 ∈ e −ω W u,0 and ∂ 2 t ϕ 2 ∈ e −ω L p (R + ). We now make the ansatz u(x, t) = v(x, t) + w(t) such thatv(·, t) = 0. Applying |Ω| −1 Ω to a∆u − u t = ϕ we deduce that w solves the ordinary differential equation w t = −ϕ 2 + a|Γ||Ω| −1ḡ with w(0) =ū 0 . Hence ∂ 3 t w ∈ e −ω L p (R + ). Moreover, v is a solution of (4.8) a∆v In order to apply Corollary C.4, we first note that the right-hand side ϕ 1 + a|Γ||Ω| −1ḡ belongs to e −ω W u sinceḡ only depends on time and belongs to e −ω W 2 p (R + ). The rescaled function v a (t, x) = av(t/a, x) should solve the system ∆v a − ∂ t v a = ϕ 1 + a|Γ||Ω| −1ḡ in Ω, ∂ ν v a = ag on Γ, v a (0) = au 0 − aū 0 . (4.9) Hence the compatibility condition (C.7) becomes and is clearly satisfied. Therefore Corollary C.4 yields a unique solution v a ∈ e −ω E u,0 of problem (4.9) and thus u = v + w solves problem (4.5).
Finally, uniqueness follows by considering two solutions of (4.5), the difference u of which solves (4.4) with f = 0 and u 0 = u 1 = u 2 = 0, hence u = 0 and the proof is complete. Remark 4.10. In case of Theorem 4.8 we define u ⋆ to be the solution according to Proposition 4.7 which satisfies (4.1) for the data (f = 0, g, h, u 0 , u 1 , u 2 ) und suppose u • satisfies homogeneous boundary and initial conditions. The solution is then of the form u = u ⋆ + u • and u • is found by the implicit function theorem. The claim then follows likewise to the proof of Theorem 3.6.
However, if we want to prove global well-posedness, we need to use Proposition 4.6 for the linearized equation, where for given data (f = 0, g, h, u 0 , u 1 , u 2 ) according to (ii)-(iv) the solution is of the form u(t, x) = v(t, x) + w(t), where v ∈ e −ω E u,0 has zero mean and w is only time-dependent. If w = 0, the term ((u t ) 2 ) tt in the nonlinear right-hand side of (1.8) causes problems. Recall (3.12) and note that due to Proposition 4.6 we in fact have u ⋆ = v ⋆ +w ⋆ with v ⋆ ∈ e −ω E u,0 and ∂ 3 t w ⋆ ∈ e −ω L p (R + ). Then w ⋆ , ∂ t w ⋆ and ∂ 2 t w ⋆ are in general not contained in e −ω L p (R + ) and thus (3.13) fails. However, if we assume that the data (g, h, u 0 , u 1 , u 2 ) have zero mean then w ⋆ = 0 and, since u ⋆ = v ⋆ in this case, Theorem 4.9 follows analogously to the result on global well-posedness for the Dirichlet boundary value problem (Theorem 3.6).
Finally, provided the data have zero mean, we obtain the following result on exponential stability for the Neumann problem (1.8).
Theorem 4.11 (Exponential stability -the Neumann case). Under the same assumptions as in Theorem 4.9, the solution u decays exponentially fast to zero as t → ∞ in the sense that for some C ≥ 0 depending on the boundary and initial data g, h, u 0 , u 1 and u 2 .
Proof. Note that we have u ∈ e −ω E u,0 ֒→ e −ω E u , therefore the result follows likewise to Theorem 3.7.
Appendix A. The Neumann Laplace operator Let p ∈ (1, ∞) and assume, as always, that Ω ⊂ R n is a bounded domain with smooth boundary Γ = ∂Ω. The homogeneous Neumann-Laplacian is given by where D(∆ N ) = {u ∈ W 2 p (Ω) : ∂ ν u = 0 on Γ}. It is well-known that −∆ N has compact resolvent and that its spectrum σ(−∆ N ) is a discrete subset of [0, ∞) consisting only of eigenvalues (λ N n ) n≥0 with finite multiplicity. In particular, 0 = λ N 0 ∈ σ(−∆ N ) is an isolated eigenvalue of −∆ N . We seek for a realization of the Laplace operator with homogeneous Neumann boundary conditions such that the spectrum is contained in In order to remove the zero eigenvalue we will use several results from Appendix A in [Lun95]. In what follows, A : D(A) ⊂ X → X denotes a linear closed linear operator whose domain D(A) is dense in the real or complex Banach space X = {0}. We say that a subset σ 1 ⊂ σ(A) is a spectral set if both, σ 1 and σ(A) \ σ 1 are closed in C. Let σ 1 be a bounded spectral set and let σ 2 = σ(A) \ σ 1 . Since dist(σ 1 , σ 2 ) > 0, there exists a bounded open set O such that σ 1 ⊂ O and O ∩ σ 2 = ∅. We may assume that the boundary γ of O consists of a finite number of rectifiable closed Jordan curves, oriented counterclockwise and define a linear bounded operator P by The following result shows how find a realization A 2 of A such that σ(A 2 ) = σ(A) \ σ 1 .
. Let σ 1 be a bounded spectral set. Then the operator P is a projection and P (X) is contained in D(A n ) for every n ∈ N. Moreover, if we set X 1 = P (X), X 2 = (I − P )(X) and define the operators The crucial point is thus to determine the space X 2 . In case σ 1 = {λ 0 } where λ 0 is an isolated point of σ(A) and a pole of R(·, A) the following result helps to determine the spaces X 1 and X 2 .
We now apply the foregoing results to the strong Neumann-Laplacian −∆ N and set λ 0 = λ Proof. Since −∆ N is closed, densely defined and has compact resolvent, the result is an immediate consequence of Corollary IV.1.19 in [EN00].
We introduce the space K u = {u ∈ L p (Ω) : u is constant} and start with the following observation.
Proof. We consider the map P : It is straightforward to verify that for u ∈ L p (Ω) we indeed have Ω u − u Ω dx = 0 and hence P u ∈ L p,0 (Ω). Furthermore, P 2 u = P u − P u Ω = P u implies that P is a projection. Finally N (P ) = K u since u − u Ω = 0 if and only if u ∈ L p (Ω) is constant.
Therewith, by means of Proposition A.2 have determined the space X 2 = L p,0 (Ω) which gives us the following positive realization of the Neumann Laplacian.
Theorem A.7. The spectrum of the closed and densely defined operator

Appendix B. Traces and mixed derivatives
In this section we consider the temporal trace operator in some anisotropic fractional Sobolev spaces. Furthermore, we present some mixed derivative embeddings for such spaces which are needed for proving suitable mapping properties of differential operators.
A bounded linear operator r : X → Y between Banach spaces X and Y is called a retraction, if there is a bounded linear map r c : Y → X such that rr c = I Y . Thus r is surjective and r c is a bounded right-inverse for r. The map r c is called a co-retraction for r.
The  Then, for every α ∈ (1/p, 1], the trace operator ) is a co-retracton for γ t and the following embedding is continuous. 1/p, p)).
By using Vandermonde's matrix the numbers c ij are given by (c 0j , . . . , c lj ) T := V −1 e T j . Hence From Corollary B.2 we infer that S A j acts as a bounded linear operator for m ∈ N, m ≥ j + 1. Therefore the desired co-retraction is given by (vii) The corresponding result for B = N has the compatibility conditions Aiming at stability for Neumann boundary conditions, we will also prove the following result, where we consider the subspace L p,0 (Ω) := {f ∈ L p (Ω) : Ω f (x)dx = 0}.
Then problem (C.1) has a unique solution u ∈ E l,k 0 := E l,k ∩ L p (J; L p,0 (Ω)), if and only if the data (f, g, u 0 ) satisfy the regularity conditions (C.3) and the compatibility conditions (C.4) and Next, we study the original heat problem , l ∈ N 0 , k ∈ N, p ∈ (1, ∞) such that j B /2 + 3/2p = 1. Then problem (C.8) has a unique solution u ∈ e µ B E l,k if and only if the data (f, g, u 0 ) satisfy the regularity conditions and the compatibility conditions Then problem (C.8) has a unique solution u ∈ e µ N E l,k 0 if and only if the data (f, g, u 0 ) satisfy the regularity conditions (C.9) and the compatibility conditions (C.10), (C.7).
Proof. In problem (C.1) we multiply f , g with e µ B t , so that e µ B t f = e µ B t (∂ t + µ B − ∆)u = (∂ t − ∆)e µ B t u, e µ B t g = γ B e µ B t u.
This shows that e µ B t u solves (C.8) for (e µ B t f, e µ B t g, u 0 ) if and only if u solves the shifted problem (C.1) for (f, g, u 0 ). Hence Theorem C.1 and Corollary C.3 yield the assertions.
In order to obtain higher regularity results we consider the spaces These spaces can be easily characterized by X k B = {u ∈ W 2k p (Ω; E) : γ B ∆ j u = 0 for 0 ≤ j ≤ k − 1}. By commuting the operator µ B + ∂ t − ∆ B with (µ B − ∆ B ) k it follows that µ B − ∆ B has maximal regularity of type L p (R + ; X k B ) for every B ∈ {D, N }, k ∈ N 0 , that is, µ B + ∂ t − ∆ : 0 W 1 p (R + ; X k B ) ∩ L p (R + ; X k+1 B ) → L p (R + ; X k B ) is a topological linear isomorphism. Moreover, the map ǫ + ∂ t : 0 W l+1 p (R + ; E) → 0 W l p (R + ; E) is a topological linear isomorphism for every ǫ > 0 and every l ∈ N 0 , see e. g. [MS12]. Hence, by commuting µ B + ∂ t − ∆ B with ǫ + ∂ t , we obtain the following result.
Then the temporal trace operator γ t : W 1 p (R + ; X k B ) ∩ L p (R + ; X k+1 B ) → (X k B , X k+1 B ) 1−1/p,p is a bounded and surjective and therefore (µ B + ∂ t − ∆ B , γ t ) : W 1 p (R + ; X k B ) ∩ L p (R + ; X k+1 B ) → L p (R + ; X k B ) × (X k B , X k+1 B ) 1−1/p,p is also a topological linear isomorphism for B ∈ {D, N }, k ∈ N 0 . C.3. Boundary conditions. We will use the following result for constructing a function with prescribed boundary conditions (C.5).
By means of the chain rule, Hölder's inequality and the mixed derivative embeddings, it can be shown that the linear operators g → ϕ j g, Θ * j , M j and Θ j * act continuously in the relevant spaces and the properties of Θ j and ϕ j with respect to the normal direction imply that indeed B l,k B c l,k g = g. This concludes the proof of Lemma C.6.
C.4. Proof of Theorem C.1. We have already discussed the necessity of the regularity conditions and the compatibility conditions on (f, g, u 0 ). It remains to prove the uniqueness and existence of a solution u ∈ E l,k for given data (f, g, u 0 ) subject to these conditions. In order to prove uniqueness, it suffices to consider the most general case l = 0, k = 1, where E l,k = W 1 p (J; L p (Ω)) ∩ L p (J; W 2 p (Ω)) and (f, g, u 0 ) = 0. If further µ 0 is sufficiently large, then the general result of [DHP03] implies that µ 0 − ∆ has maximal regularity of type L p (R + ; L p (Ω)) and this yields u = 0 in case µ B ≥ µ 0 .
Next, we employ spectral theory to cover the case µ B ∈ (−λ B 0 , ∞), where λ B 0 = λ 0 (−∆ B ) ≥ 0 denotes the smallest eigenvalue of −∆ B . It is well known that, since D(∆ B ) is compactly embedded into L p (Ω), the spectrum of ∆ B is discrete and consists only of eigenvalues with