Global well-posedness and exponential stability for Kuznetsov's equation in L_p-spaces

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher&Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal L_p-regularity for parabolic equations and the implicit function theorem.


Introduction
We are concerned with Kuznetsov's quasilinear equation which is a well-accepted equation in nonlinear acoustics and describes the propagation of sound in fluidic media. The function u(t, x) denotes the acoustic pressure fluctuation from an ambient value at time t and position x. Furthermore, c > 0 denotes the velocity of sound, b > 0 the diffusivity of sound and k > 0 the parameter of nonlinearity. The velocity fluctuation v(t, x) is related to the pressure fluctuation by means of an acoustic potential ψ(t, x), such that u = ρ 0 ψ t , v = −∇ψ, with ambient mass density ρ 0 .
Kuznetsov's equation can be derived from the balances of mass and momentum (the compressible Navier-Stokes equations for a Newtonian fluid) and a state equation for the pressure-dependent density of the fluid, where terms of third or higher order in the fluctuations are neglected. We refer to the monograph of M. Kaltenbacher [11] and Kuznetsov's article [12] for the derivation. Observe that Kuznetsov's equation degenerates if 1 − 2ku = 0, since (u 2 ) tt = 2uu tt + 2(u t ) 2 , but for |u| < (2k) −1 the equation is parabolic.
Global well-posedness for the corresponding Dirichlet boundary value problem in an L 2 -setting for the spatial dimension n ∈ {1, 2, 3} was established by B.
Kaltenbacher & I. Lasiecka [10] by means of appropriate energy estimates and Banach's fixed point theorem. The purpose of the present paper is to extend these results to an L p -setting for arbitrary dimensions and to provide shorter and more elegant proofs in an optimal functional analytic setting, in the sense that the regularity conditions on the initial and boundary data are both necessary and sufficient for the regularity of the solution. We also impose appropriate smallness conditions to avoid the above mentioned degeneracy and to make use of the theory for quasilinear parabolic equations.
Let Ω ⊂ R n be a bounded domain with smooth boundary Γ = ∂Ω, let J = (0, T ) or J = R + = (0, ∞), and let u 0 , u 1 , v 0 and g be given functions on Ω and J × Γ, respectively. We consider the following initial-boundary value problem. (1.1) Here u : J ×Ω → R is the unknown pressure fluctuation, (u 0 , u 1 , v 0 ) : Ω → R 2+n are the given initial data and g : J × Γ → R is a given inhomogeneity on the boundary. In this paper we prove existence and uniqueness of strong solutions in suitable subspaces of the Lebesgue space L p (J × Ω) with exponent p > max{1, n/2}. In the case J = R + we are interested in solutions with exponential decay and therefore consider the functions t → e ωt u(t), t → e ωt v t (t) with ω ≥ 0, which we abbreviate as e ωt u, e ωt v t , respectively. We say that (u, v) is a strong solution to ( and all equations in (1.1) are satisfied pointwise almost everywhere. Here the symbols W s p and H s p denote the Sobolev-Slobodeckij and Bessel potential spaces of order s and exponent p, respectively, and BU C k denotes the space of functions having bounded and uniformly continuous derivatives up to order k. The condition e ωt u ∈ E u (J) will be written equivalently as u ∈ e −ω E u (J). In the case of a finite interval J = (0, T ), the exponential factor e ωt can be dropped and we have e −ω E u (J) = E u (J), for instance. Our main result is the following. Theorem 1. Let n ∈ N, p > max{1, n/2}, p = 3/2, J = (0, T ) or J = R + and define ω 0 := min{bλ 0 /2, c 2 /b} > 0, where λ 0 > 0 denotes the smallest eigenvalue of the negative Dirichlet-Laplacian in L p (Ω). Then for every ω ∈ (0, ω 0 ), there is ρ > 0 such that problem (1.1) admits a unique solution if the data (g, u 0 , u 1 , v 0 ) satisfy the regularity and compatibility conditions 4) and the smallness condition This theorem extends the results of Kaltenbacher & Lasiecka [10, Theorem 1.1 and Theorem 1.2] in several ways. First, they only consider the case p = 2 and n ∈ {1, 2, 3}. Second, the solution space in the case J = (0, T ) is given by and the initial data (u 0 , u 1 ) must satisfy the condition u 0 W 2 2 + u 1 W 2 2 < ρ for some (small) ρ > 0, as a consequence of the assumptions on (u 0 , u 1 , g). Most notably, the condition u 1 ∈ W 2 2 (Ω) is not necessary for (1.5). In the case p = 2 we only require that u 0 W 2 2 + u 1 W 1 2 < ρ, where the condition u 1 ∈ W 1 2 (Ω) is necessary, since, by the properties of the temporal trace (see Section 2.2), our solution u must satisfy which is also not necessary in view of u| Γ = g. We can simplify and extend it to g ∈ W 7/4 2 (J; L 2 (Γ)) ∩ W 1 2 (J; W 3/2 2 (Γ)), which is necessary for u ∈ E u (J) by the properties of the spatial trace (see Section 2.2). Finally, Kaltenbacher & Lasiecka use that the velocity is given by This implies that v 0 = −ρ −1 0 ∇U 0 depends on u 0 , u 1 and is small in W 1 2 (Ω). We are able to remove the dependence of v 0 on the initial values u 0 and u 1 .
Besides these extensions we point out that in the case J = R + we always impose some exponential decay on g whereas Kaltenbacher & Lasiecka impose smallness conditions on the primitive of g instead.
Compared to [10, Theorem 1.3], Theorem 1 does not imply that u tt (t) Lp → 0 exponentially as t → ∞, since E u (R + ) is not contained in BU C 2 (R + ; L p (Ω)). To obtain such higher order differentiability, we impose additional regularity conditions on g in the following result.
Theorem 2. Suppose that the conditions of Theorem 1 are satisfied for J = R + and assume in addition that there exists ω g > ω such that Finally, we are interested in the case where the inhomogeneity g vanishes except for a finite time interval (0, T ). Then the solution becomes smooth with respect to time on every interval (T + δ, ∞), δ > 0, and all temporal derivatives decay exponentially: Corollary 3. Suppose that the conditions of Theorem 1 are satisfied for J = R + and assume in addition that g(t) = 0 for all t > T with some T > 0. Then for every ω ∈ (0, ω 0 ) there existsρ ≤ ρ such that for ) for all j ∈ N 0 and every δ > 0 and there exist C j ≥ 0 such that This paper is organized as follows. In Section 2 we introduce the notation and we provide some results concerning maximal L p -regularity, trace-theory and analytic mappings in Banach spaces. In Section 3 we study a linearized version of (1.1) and prove an optimal regularity result for this linear problem in Lemma 4. The proof of Theorem 1 is given in Section 4. We employ the implicit function theorem combined with the results in Section 3. Finally, in Section 5 we prove Theorem 2 by applying a parameter trick which goes back to Angenent [4] together with the implicit function theorem.

Notation and preliminaries
In this paper, the symbol Ω always denotes a bounded domain in R n with smooth boundary Γ = ∂Ω and J denotes the interval (0, T ) or R + = (0, ∞). BU C k (Ω) is the space of k-times Fréchet-differentiable functions whose derivatives are bounded and uniformly continuous in Ω and C ∞ c (R n ) denotes the space of smooth functions φ with compact support supp φ ⊂ R n . Moreover, L p (Ω) denotes the Lebesgue space We refer to the monographs [1,16] for a detailed treatment of these spaces. We mention that, we will use the Sobolev embeddings W 1 has maximal L p -regularity in the sense that there exists a unique solution if and only if the given data f, g, u 0 satisfy the regularity conditions and the compatibility condition u 0 | Γ = g| t=0 in the sense of traces. By Banach's closed graph theorem, this implies that a solution satisfies the a priori estimate Here the symbol indicates that the left-hand side u E1(J) can be estimated by a constant C ≥ 0 times the right-hand side, where C does not depend on u, f, g, u 0 .

Trace theory.
To verify the necessity of the conditions on the data, we use Lemma 3.5 and Lemma 3.7 in [8], which imply that the spatial and temporal traces Therefore we obtain g(0) = u 0 | Γ in the sense of B 2−3/p pp (Γ) for p > 3/2. In the case p < 3/2 these traces do not exist (see e.g. [8]). The case p = 3/2 is excluded, since the trace space looks more complicated in this case [17,Theorem 4 2.3. Exponentially weighted spaces. Let Y , X(J) be Banach spaces such that X(J) ֒→ L 1,loc (J; Y ) and let ω ∈ R. To describe exponential decay of solutions we employ the Banach space e −ω X(J) := {u ∈ L 1,loc (J; Y ) : (t → e ωt u(t)) ∈ X(J)}, equipped with the norm u e −ω X(J) := e ω· u X(J) , where we write e ω· u or e ωt u instead of (t → e ωt u(t)) for the sake of brevity.

2.4.
Analytic mappings between Banach spaces. Let X, Y be Banach spaces over the same scalar field R or C and let U ⊂ X be open. We say that a mapping F : U ⊂ X → Y is analytic at u ∈ U , if there exists r > 0 and bounded symmetric k-linear operators F k : X k = X × · · · × X → Y , k ≥ 0, such that every F (u + h) for h ∈ X, h < r, can be represented as where F k denotes the norm of the k-linear operator F k , i. e. the smallest number We refer to the monographs [5,18] for more information on analytic mappings.

The linearized problem
In this section we establish maximal regularity for the linearization of problem (1.1) in the following sense.

2)
if and only if g ∈ e −ω F g , g(0) = 0, and in the case p > 3/2 also g t (0) = 0. The solution w satisfies the estimate e ω w Eu e ω g Fg .
Proof. We first prove sufficiency. Using maximal regularity [13, Proposition 8 and formula (52)], we obtain a unique solution v ∈ e −ω E 1 of the parabolic problem Indeed, the operator ∂ t : F g → F Γ is bounded and thus e ωt ∂ t g FΓ = ∂ t (e ωt g) − ωe ωt g FΓ e ωt g Fg . Next we define Using ω > 0, we infer from w t = v, from the identity and Young's inequality, that w ∈ e −ω E u . Moreover it holds that and w t (0) = v(0) = 0. It follows that b∆w(0) = w t (0) = 0 and w(0)| Γ = g(0) = 0, hence w(0) = 0 in Ω. Therefore w is a solution to (3.2). The necessity follows from the spacial trace theorem applied to w, w t ∈ e −ω E 1 . To obtain uniqueness, let w be a solution to (3.2) with g = 0. Since w t solves a heat problem with homogeneous data, we obtain w t = 0 and therefore also w = 0 by the initial condition w(0) = 0. The estimate follows from the closed graph theorem.
Proof of Lemma 4. We obtain uniqueness from our previous maximal regularity result [14,Theorem 2.5] for (3.1) for the case g = 0. To verify the necessity of the conditions on the data, suppose that u is a solution to (3.1). Then e ωt u and (e ωt u) t = ωe ωt u + e ωt u t belong to E 1 and (i) is readily checked. Taking the spatial trace yields e ωt g, (e ωt g) t ∈ F Γ which implies (ii). The exponential weight e ωt does not affect the initial regularity and therefore (iii) follows by taking the temporal trace and using the embedding W 1 p (J; W 2 p (Ω)) ֒→ BU C(J; W 2 p (Ω)). Using that and applying the spatial trace to u(0), u t (0) and the temporal trace to (g, g t ), we see that g(0) = u 0 | Γ is valid in the sense of W 2−1/p p (Γ) for all p and ∂ t g(0) = u 1 | Γ is valid in the sense of B 2−3/p pp (Γ) if p > 3/2. It remains to prove sufficiency of the conditions. First we reduce the problem to the case u 0 = 0, u 1 = 0, f = 0. This cannot be done by just solving the problem with g = 0, due to the compatibility conditions. Therefore we extend u 0 , u 1 and f to some functionsũ 0 ∈ W 2 p (R n ),ũ 1 ∈ W 2−2/p p (R n ) andf ∈ e −ω L p (R + × R n ). By means of a cut-off function φ ∈ C ∞ c (R n ) such that φ(x) = 1 for x ∈ Ω and φ(x) = 0 for x / ∈ B R := {y ∈ R n : |y| < R} for some R > 0, we define new dataû 0 :=ũ 0 φ, u 1 :=ũ 1 φ andf :=f φ and consider the problem Using maximal regularity [14, Theorem 2.5], we obtain a unique solution Letū denote the restriction ofû to Ω and letḡ := g −ū| Γ . Then the final solution u will be given by u = v +ū, where v solves the problem From Lemma 5 we obtain a unique solutionv ∈ e −ω E u of the problem Then the function w : which has a unique solution w ∈ e −ω E u by [14,Theorem 2.5]. The function u := w +v +ū is the desired solution of (3.1) and the estimate follows from the closed graph theorem. This concludes the proof of Lemma 4.

The nonlinear problem
In this section we construct a solution to the nonlinear problem (1.1) of the form (u+u * , v). Here u * solves the linearized problem (3.1) for the data (f = 0, g, u 0 , u 1 ) and u satisfies homogeneous boundary and initial conditions. The (small) deviation (u, v) from (u * , 0) will be found by the implicit function theorem.
For p > max{1, n/2}, we employ the Banach function spaces Observe that now E v is a somewhat larger space compared to (1.2).
In a similar way, we can check that is analytic, where we make use of the continuity of the embeddings (Ω)) (valid for p ≥ 1/2) and the inequality To obtain the analyticity of , we use e ωt ∇w t = ∇(e ωt w) t − ωe ωt ∇w and the estimate v · e ωt ∇w t p ≤ v L∞(R+;L2p(Ω)) e ωt ∇w t Lp(R+;L2p(Ω)) v Ev e ωt w Eu , which is valid for p ≥ n/2 since W 1 p (Ω) ֒→ L 2p (Ω) is continuous in this case. The Fréchet derivative of H w. r. t. (u, v) at (0, 0, 0, 0) is given by We will now show that D This function belongs to E v , as can be seen from ∇ū ∈ e −ω W 1 p (R + ; W 1 p (Ω) n ) ֒→ e −ω BU C(R + ; W 1 p (Ω) n ) = e −ω E vt . Proof of Theorem 1. It suffices to consider the case J = R + , since the considered function spaces over J = (0, T ) can be identified with subspaces of the corresponding spaces over R + by means of extension and restriction, see [1,Theorem 5.19] for the scalar-valued case and [15, Lemma 2.5] for the vector-valued case.