WELL-POSEDNESS OF THE WESTERVELT AND THE KUZNETSOV EQUATION WITH NONHOMOGENEOUS NEUMANN BOUNDARY CONDITIONS

In this paper we show wellposedness of two equations of nonlinear 
acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider 
Neumann boundary conditions which are of particular practical interest in 
applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations with potential degeneration and strong damping. We 
prove local in time well-posedness as well as global existence and exponential 
decay for a slightly modi ed model. A key step of the proof is an appropriate 
extension of the Neumann boundary data to the interior along with exploitation of singular estimates associated with the analytic semigroup generated by 
the strongly damped wave equation.

where p ∼ denotes the acoustic pressure fluctuations, v the acoustic particle velocity, c is the speed of sound, b the diffusivity of the sound, 0 the mass density, B/A the parameter of nonlinearity, and the identity 0 v t = −∇p ∼ holds.All parameters b, c, 0 , B/A are positive real numbers.A slightly simplified model resulting from neglecting local nonlinear effects modelled by the quadratic velocity term, is the Westervelt equation with β a = 1 + B/(2A).For a detailed derivation of the PDE we refer to [6], [13], [14], [20].These two equations can be equivalently rewritten as: respectively, where u = p, k = β a /( c 2 ) and denotes the antiderivative operator.Here the space dependent function U 0 ∈ H 1 0 (Ω) is to be chosen appropriately, see Theorem 1.2 below.The parameter σ allows to formally switch between both equations: σ = 2 0 in case of the Kuznetsov equation, and σ = 0 in case of the (simpler) Westervelt equation.From formulation (3) it gets obvious that these are quasilinear strongly damped wave equations with potential degeneracy.
After having established well-posedness results for these equations with Dirichlet boundary conditions in [8], [9], [10], [11], our aim is to cover also the Neumann boundary case, which is of particular practical relevance in view of an appropriate modelling of boundary excitation, e.g., by electromagnetomechanical or piezoelectric devices, see, e.g.[13].On the other hand, the mathematical treatment of Neumann boundary conditions presents us with new challenges resulting from a nontrivial nullspace associated with the linear generator as well as a different semigroup framework owning to the variational character of the Neumann problem.
Our particular emphasis is on problems that have nonhomogenuous boundary conditions.It is known that the analysis of non-homogenuous on the boundary problems can be quite subtle due to intrinsic incompatibility between fractional powers of generators and the spaces where solutions reside.This is particularly pronounced in the case of strongly damped dynamics (as considered in this paper) where the said incompatibility also involves time derivatives.Recent years have witnessed rapid development of PDE analysis enabling effective treatment of inhomogenity on the boundary (also with limited regularity) so that the generated results can be also applied in the context of nonlinear problems [2,3,16,12].The methods used rely on a combination of semigroup theory and analytic properties of the semigroup generated by homogenous on the boundary dynamics.These allow to build appropriate boundary kernels whose singularity can be controlled see [16,3] and references therein.The approach taken in this paper, and also in [10,11], is rooted in these methods.
For the model (3) with boundary conditions (5), due to the zero eigenvalue of the Laplace operator with Neumann boundary conditions, only local in time wellposedness can be shown.Global existence and exponential decay will be shown for the slightly modified model with initial data (u(0), u t (0)) = (u 0 , u 1 ).Throughout the remainder of this paper, all parameters are real and part of them satisfy sign conditions b, c > 0 , d, e, h ≥ 0 ( (the original model corresponding to d = e = h = 0) whereas the signs of k, σ may be arbitrary.For global existence and exponential decay we will additionally assume that min{d, e} + h > 0 .
The parameter b > 0 means that we assume strong damping.The following compatibility conditions will be imposed throughout this paper: In order to formulate our results we introduce the following energy functions: where |u| ≡ |u| L2(Ω) .For t = 0, E u,1 (0) Our first result pertains to local existence and uniqueness of solution.For this purpose we define, for some time interval I, Theorem 1.1.Let T > 0 be arbitrary and I ≡ (0, T ).There exist ρ T , ρT > 0 such that if E u,0 (0) + E u,1 (0) ≤ ρ T , and g ∈ X (0,T ) , g 2 X (0,T ) ≤ ρT with the compatibility conditions (11) then there exists a unique solution (u, u t ) solving the weak form for all φ ∈ H 1 (Ω) and all t ∈ (0, T ), of (7), (8) with initial conditions (6) and such that The said solution is unique and depends continuously (with respect to the topology generated by E u,1 ) on the initial data.
Remark 1.1.The boundary space X I is optimal with respect to parabolic maximal regularity displayed by the energy function E u,1 (t) for solutions with Neumann data.
Our next theoren deals with global wellposedness.
Finally, we shall present results on energy decays.
Theorem 1.3.Let the assumptions of Theorem 1.2 be satisfied.Fix M > 0 and correspondingly ρ, ρ according to Theorem 1.2.If the Cauchy data satisfy (11), ( 12), ( 13) and additionally for some C g > 0, ω g > 0, then u satisfies the exponential decay estimate Like in [10] local well-posedness will be established by an application of the Banach Fixed Point Theorem together with appropriate energy estimates.Global well-posedness will be shown by means of barrier's method, i.e., by assuming that after finite time degeneration occurs and deriving a contradiction by means of energy estimates.In order to take into account the inhomogeneous boundary in an appropriate way, we will consider its extension to the interior using an abstract strongly damped linear wave equation as well as a related variation of parameters formula as developed in [11] (see also [2], [15] for the purely parabolic case).
The main technical difference with respect to the Dirichlet case lies in the following facts (i) that we have to take care of a nontrivial nullspace of the Laplace operator with Neumann boundary conditions (this is critical when dealing with quasilinear problems that depend crucially on total dissipation) , (ii) the treatment of boundary extensions is different due to the variational nature associated with Neumann problems, (iii) the characterization of fractional powers of the operators associated with strongly damped wave operators subject to Neumann boundary conditions provides an additional tool that needs to be skillfully used in resolving existence questions of quasilinear dynamics.
Being able to proceed along the lines of the Dirichlet case [10], [11], we keep the present paper short by only pointing out the major differences of the Neumann as compared to the Dirichlet case.

Strongly damped abstract wave equation.
In what follows we recall results from [10], [11] on the following non-homogenuous and nonautonomus abstract strongly damped wave equation: with initial conditions Here is a positive selfadjoint operator, H is a suitable Hilbert space.We shall introduce the following notation We are interested in studying regularity properties of solutions u, u t due to the forcing f and initial conditions u 0 , u 1 .
To take into account inhomogeneous boundary conditions, we will decompose u = u 0 + ḡ with u 0 having to satisfy homogeneous Cauchy (boundary and initial) conditions and ḡ will denote an appropriate extension of the Cauchy data, i.e., For carrying out this extension, we will use results on the strongly damped abstract wave equation with α ≡ 1.
2.1.Extension of nonhomogeneous boundary data to the interior.We consider which can be rewritten as where we have β = h, γ = d c 2 , δ = e − bγ.Moroever, we define the harmonic extension operator with nonnegative parameters β, γ, where the elliptic boundary value problem is to be satisfied in the weak sense.For all s ∈ R the operator N ∆ is a bounded mapping where cf., e.g, Theorem 6.6 in [18]. 1 Using the definition of N ∆ , we can restate (17) and hence, setting A = A β,γ = −∆ + γid with zero Robin boundary conditions ∂ ∂ν + βid, H = L 2 (Ω), we write ( 17), (18) as the abstract second order ODE: where in splitting the brackets, we admit representation of the equation in the dual space to D(A).This procedure is standard by now [16], [3] and references therein.
In what follows we shall use regularity properties of the generator e At ∈ L(H) where It was shown in [5] that e At is an analytic, strongly continuous semigroup defined on H and that where In addition we have that [5,16] Denoting W (t) ≡ (w(t), w t (t)), we can rewrite (22) as and obtain the following "variation of parameter formula" representing weak solutions to the nonhomogenuous boundary value problem driven by the non-degenerate (α = 1) damped wave equation as introduced in [11]: ) and ∂Ω g ds = 0.
1.The following representations hold: where In case a) with f = 0, and if additionally (14) for some C g , ω g > 0 holds for all t > 0, then there exists Ĉ > 0, ω > 0 such that for all 0 < s < T Proof.The derivation of "variation of parameters formula" follows from the same arguments as used in [16] (see also [4] for related problems in the context of strongly damped wave equations).The Dirichlet case considered in [10] is somewhat different, however many steps are conceptually the same.One of the main differences is that in the Neumann case we have A 1/2 N ∆ ∈ L(H −1/2 (Γ); L 2 (Ω)) which fact along with the analyticity of the semigroup e At allows to deal with formulas (26), (27) within the framework of the basic state space H (not valid in the Dirichlet case which requires a different handling [11]).In fact, in the formula (27) there is a cancellation of singularity in the first term on the right hand side which is due to assumed compatability conditions and the above mentioned regularity of the Neumann map.The estimates on w tt L∞(I;H) , A 1/2 w tt L2(I;H 1 (Ω)) , A 1/2 w t L2(I;H) , A 1/2 w t L∞(I;H) , directly follow from ( 26), (27): From (26) we get and from ( 27) The improved estimates for |w| H 2 (Ω) , |w t | H 2 (Ω) can be derived by multiplication of with A(w − N ∆ g) and with A(w t − N ∆ g t ), respectively: Firstly we get secondly, Here we note that w − N ∆ g ∈ D(A) is equivalent to the fact that both w and g have the right spatial regularity (w ∈ H 2 (Ω), g ∈ H −1/2 ((Γ)).Thus, the resulting energy estimates along with the apriori regularity of g give the needed spatial regularity of w.
For exponential decay, we use (25) to obtain from (27) (with f = 0) Considering the sum of 1 + 2 c 4 max{1, δ 2 C 2 0 } times (31) and the integral from s to T of the square of (30), we get for the energy Here C 0 is the constant in the continuous embedding estimate |z| ≤ C 0 |A 1/2 z|, (see also Assumption (3.1) below).Using a standard argument from [19] and N ∆ g(t) H 2 (Ω) ≤ C g e −ωgt we obtain from (32) . The decay estimate of A 1/2 w tt is obtained by differentiating (22) (with f = 0) wrt.time and multiplying with w tt : Considering (17) we thus define an extension (16) ḡ = w according to ( 17) with (33) 2.2.Energy estimates for the variable coefficient model.In this section we will consider the strongly damped abstract wave equation (15) with α(t) ∈ L ∞ (Ω) being a positive multiplier on Our assumptions on the initial and boundary data are the folllowing.
Assumptions 2.1, 2.2 will be imposed throughout the rest of this section.
Using the energy estimates from Lemma 2.2 and Remark 2.1, the local well-posedness results can be done exactly along the lines of the proof of Theorem 3.1 in [10] both for the original model (3), (5) and for (7), (5) with min{d, e} + h > 0. In this case the contractivity is achieved by calibrating the size of initial data with respect to the running time.One remark should be made regarding the term f as given above.It contains ḡtt .Since the estimates in Theorem 2.2 call for f t , one must be careful how to estimate ḡttt without involving three time derivatives of g.However, this can be done by using the method used in Proposition 3.5 in [10], which in turn relies on special properties of convolutions generated by analytic semigroups.Proof of global existence and exponential decay -results stated in Theorems 1.2, 1.3 -is more involved.It requires the construction of a set of smooth initial data that is invariant under the dynamics.This set is defined by inequalities ( 12), ( 13) in Theorem 1.2.The crux of the problem lies in proving a certain observability inequality (as in controllability problems) for smooth solutions.This can only be shown for the modified problem (7), (5) with min{d, e}+h > 0 by following the line of attack designed for the proofs of Theorems 3.2 and 3.3 in [10].The procedure amounts to: (i) proving observability estimates which depend on dissipation present in the equation, (ii) construct an appropriate set of smooth initial data that is invariant under dynamics, (ii) for the latter use of the estimates in Lemma 2.2 which incorporate time dependent coefficients.We shall briefly outline the procedure.The operator A in [10] is replaced by A = A β,γ , β = h, γ = min{ d c 2 , e b }.The observability estimate is derived for solution u 0 of (46) with the time-space variable coefficients given in (47).The estimates of Lemma 2.2 along with estimates for the coefficients α(t), f (t) (see Proposition 3.5 , Lemma 3.6 in [10] ) lead to the expression  H i (ρ, ρ) → 0 as ρ, ρ → 0 i = 1, 2, 3 .
The above inequality with s = 0 allows to tune the parameters ρ, ρ and the size of E 0 (0) in order to obtain the desired invariance.In particular, using the energy estimates from Lemma 2. for some b, C1 , C, ω > 0, which by a standard semigroup argument [19] yields exponential decay of E 0 (t) and therewith of E(t).
Note that, as expected, the required order of spatial differentiability of g is one order less for the Neumann as compared to the Dirichlet case and that there is no requirement for triple differentiation in time.

1. 1 .
Main Results.The main goal of this paper is to provide results on existence of solutions for the Kuznetsov and the Westervelt equation on a bounded domain Ω ⊆ R n , n = 1, 2, 3, with Lipschitz boundary and Neumann boundary conditions ∂u ∂ν = g on Γ = ∂Ω