Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}–Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q$$\end{document}-Theory for a Quasilinear Non-isothermal Westervelt Equation

We investigate a quasilinear system consisting of the Westervelt equation from nonlinear acoustics and Pennes bioheat equation, subject to Dirichlet or Neumann boundary conditions. The concept of maximal regularity of type Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}–Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q$$\end{document} is applied to prove local and global well-posedness. Moreover, we show by a parameter trick that the solutions regularize instantaneously. Finally, we compute the equilibria of the system and investigate the long-time behaviour of solutions starting close to equilibria.


Introduction
Thermo-acoustic lensing describes the effect of how the speed of acoustic waves and the pressure of a region are influenced by the temperature of the underlying tissue. A meanwhile well-accepted model which takes care of this effect consists of the Westervelt equation [27] u tt − c 2 (θ ) u − b(θ ) u t = k(θ )(u 2 ) tt , (1.1) describing the propagation of sound in fluidic media, coupled with the so-called bioheat equation proposed by Pennes [20] ρ a C a θ t − κ a θ + ρ b C b W (θ − θ a ) = Q(u t ). In (1.1), the function u = u(t, x) denotes the acoustic pressure fluctuation from an ambient value at time t and position x. Furthermore, c(θ ) > 0 denotes the speed of sound, b(θ ) > 0 the diffusivity of sound and k(θ ) > 0 the parameter of nonlinearity. The physical meaning of the parameters in (1.2) are as follows: ρ a > 0 and κ a > 0 denote the ambient density and thermal conductivity, respectively. C a > 0 is the ambient heat capacity and θ a > 0 stands for the constant ambient temperature, ρ b > 0 is the density of blood, C b > 0 is the heat capacity of blood and W denotes the perfusion rate (cooling by blood flow).
The nonlinear function Q models the acoustic energy being absorbed by the surrounding tissue and Q is typically of quadratic type, see Remark 1.2.
Considering (1.1)-(1.2) in a bounded framework, we have to equip these equations with suitable boundary conditions. In this article, we propose either Dirichlet or Neumann boundary conditions on u and θ . Altogether, we end up with the following system and u 0 , u 1 , θ 0 denote the initial conditions for u, u t , θ at t = 0. We observe that as long as b(θ ) > 0, the term b(θ ) u t renders (1.1) into a strongly damped wave equation which is of parabolic type. Since (u 2 ) tt = 2u tt u + 2(u t ) 2 , we see that parabolicity is preserved as long as |u| is sufficiently close to zero. It follows that (1.3) represents a quasilinear parabolic system for the variables (u, u t , θ). Therefore, it is reasonable to apply L p -L q -theory in order to solve (1.3).
The Westervelt equation (with constant temperature) has been subject to a variety of articles over the last decades, see e.g. [4, 9-12, 14, 15, 25], which is just a selection.
To the best knowledge of the author, there is only the article [17] which provides analytical results for (1.3) in case of homogeneous Dirichlet boundary conditions for both u and θ and provided that the diffusivity of sound b does not depend on θ . The analysis in [17] is based on L 2 -theory and some (higher-order) energy estimates. To this end, the authors have to equip the initial data with more regularity than is actually needed.
Within the present article, we are interested in the existence and uniqueness of strong solutions to (1.3) having maximal regularity of type L p -L q . In particular, we present optimal conditions on the initial data (u 0 , u 1 , θ 0 ) and the boundary data (g j , h ), thereby improving the assumptions on (u 0 , u 1 , θ 0 ) in [17] (for details, see below). Additionally, we investigate the temporal regularity of the solutions to (1.3) as well as their long-time behaviour.
Our article is structured as follows. In Sect. 2 we consider a suitable linearization of (1.3) and we prove optimal regularity results of type L p -L q for the resulting parabolic problems. Section 3 is devoted to the proof of the following main-result concerning well-posedness of (1.3) under optimal conditions on the data (u 0 , u 1 , θ 0 , g j , h ).
there exists a unique solution of (1.3). Moreover, the solution (u, θ) is C 1 with respect to the data (g j , u 0 , u 1 , h , θ 0 ).

Remark 1.2
The nonlinear function Q can for instance be modeled by for some constant C > 0, see e.g. [6,7,19]. In these cases it can be readily checked that Q(0) = 0 and provided that For the proof of Theorem 1.1 we employ the implicit function theorem and the results on optimal regularity of the linearization from Sect. 2. In order to compare our results in Theorem 1.1 with [17, Theorem 4.1], we consider the very special case d ∈ {1, 2, 3}, p = q = s = 2 and g j = h = 0 in Theorem 1.1.
there exists a unique solution Let us compare the well-posedness result [17, Theorem 4.1] concerning (1.3) with homogeneous Dirichlet boundary conditions with our result. In [17], the authors assume that (plus compatibility conditions on ∂ ). Since for any r ≥ 2, we were able to reduce the regularity of the initial data (u 0 , u 1 , θ 0 ). Moreover, a crucial assumption in [17] is that the mapping [τ → b(τ )] is constant and furthermore, only homogeneous Dirichlet boundary conditions for u and θ are considered in [17]. In summary, Theorem 1.1 generalizes [17, Theorem 4.1] considerably.
In Sect. 4 we study the regularity of the solution with respect to the temporal variable t. We use a parameter trick which goes back to Angenent [3], combined with the implicit function theorem to prove that the solution enjoys higher regularity with respect to t as soon as t > 0, see Theorem 4.1. This result reflects the parabolic regularization effect.
Finally, in Sect. 5, we compute the equilibria of the system (1.3) if g j = 0 and h = (1 − )θ a and investigate the long-time behaviour of solutions starting close to equilibria. For the case of Dirichlet boundary conditions for u, we prove in Theorem 5.1 that the corresponding equilibria are exponentially stable. Since our assumptions on the initial data (u 0 , u 1 , θ 0 ) as well as on the nonlinearities are less restrictive compared to [18], Theorem 5.1 may be understood of a generalization of [18, Theorems 2.2 and 2.3].
The definitions and basic properties of the functions spaces being used in the analysis of (1.3) are provided in the Appendix A.

Maximal Regularity of a Linearization
Let us consider the two linear problems For the linear problems (2.1) and (2.2) we have the following results.
Then there exists a unique solution Then there exists a unique solution

of (2.2) if and only if
(1) f 2 ∈ L p ((0, T ); L q ( )); is a solution of (2.2), then clearly f ∈ L p ((0, T ; L q ( )) by the assumptions on a j and by the first equation in (2.2). Furthermore, Concerning the boundary data g j , note that This readily implies and and We now prove that the conditions in Lemma 2.2 are also sufficient. To this end, we first consider the problem in . Then x) (by the compatibility condition on u 0 ) and Consequently, the function u is the unique solution of the problem Uniqueness can be seen as follows. If u 1 and u 2 are two solutions of (2.4), then Next, we consider the problem for givenf 2 ∈ L p ((0, T ); L q ( )). Note that for a sufficiently smooth solution, it holds that B j w t = 0 in (0, T ) × ∂ . We reformulate (2.5) as a first order system. To this end, let z = (z 1 , z 2 ) = (w, w t ) and F = (0,f 2 ). Then with the initial condition z(0) = 0 in and the boundary condition Then, we have of (2.5). Finally, we solve (2.4) to obtain a solutioñ Then, we solve (2.5) withf 2 := a 2 ũ ∈ L p ((0, T ); L q ( )) to obtain a solutioñ It is readily checked that the sum is the unique solution of (2.2).
Finally, let us consider the following coupled linear problem in . (2.7) is linear and bounded. Suppose furthermore that a 1 ,

Proof of Theorem 1.1
We will prove Theorem 1.1 by means of the implicit function theorem. To this end, for fixed but arbitrary T > 0, let us first introduce the function spaces Next, we define a function Note that for each u ∈ E u 1 . Since (by assumption) d/q < 2, it holds that for some constant C > 0. Leṫ Then,Ė u 1 → L 2 p ((0, T ); L 2q ( )) provided 1/ p + d/2q < 2, which is satisfied, since d/q < 2 and p > 1. Therefore for some constant C > 0. Finally, note that since (by assumption) 2/r + d/s < 2. It follows that for some constant C > 0, since b, k ∈ C(R). Similarly, we obtain In summary, the mapping is well-defined and by the assumptions on b, c, k and Q. Let (h * , θ * 0 ) ∈ Y θ be given and denote by θ * ∈ E θ 1 the unique solution of which exists thanks to Lemma 2.1. Then, obviously, (0, θ * , 0, 0, 0, h * , θ * 0 ) = 0 and where D (u,θ) denotes the total derivative of with respect to (u, θ). By Lemma 2.3, the linear operator is invertible. Hence, the implicit function theorem yields some δ > 0 and the existence of a C 1 -function This completes the proof of Theorem 1.1. (2) The nonlinearity (u 2 ) tt in (1.3) can be replaced by the more general formulation This kind of nonlinearity has been derived in [13]. If f (s) = 2s, we are in the situation of (1.3).

Moreover, there exists a constant C > 0 such that the estimate
For m ∈ N 0 and p ∈ [1, ∞], the Sobolev space W m p (R d ; X ) is defined to be the completion of S(R d ; X ) with respect to the norm We note that W 0 p (R d ; X ) = L p (R d ; X ) is the X -valued Lebesgue space. Choose a sequence (ϕ k ) k≥0 ⊂ S(R d ; R) with the propertieŝ ϕ 0 =φ,φ 1 (ξ ) =φ(ξ/2) −φ(ξ ),φ k (ξ ) =φ 1 (2 −k+1 ξ), k ≥ 2, and with a generating function ϕ ∈ S(R d ; R) satisfying For p, q ∈ [1, ∞], s ∈ R, the Besov space B s pq (R d ; X ) is defined to be the space of all f ∈ S (R d ; X ) such that For All these function spaces are Banach spaces with respect to the norms defined above, see [2, Chapter VII].
• If X is a UMD space, then L p (R d ; X ) is a UMD space for p ∈ (1, ∞).
• If p, q ∈ (1, ∞), s ∈ R and X = R, then the scalar versions H s p , B s pq , F s pq of the spaces introduced above are UMD spaces.
• Every UMD space is reflexive.
By [24,Remark 5], for s ∈ R and p ∈ (1, ∞), the identity holds if and only if X can be renormed as a Hilbert space. If this is the case, then for any k ∈ N 0 , p ∈ (1, ∞), since every Hilbert space X is of class UMD and in particular it follows that for any s ∈ R provided X is a Hilbert space.

Restricted Spaces
For open D ⊂ R d and F ∈ {B s pq , F s pq , H s p , W m p }, we define