The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

Abstract

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

1 Introduction

The Kuznetsov [1] and Blackstock [2] equations are classical models of nonlinear acoustics derived from the governing equations of fluid dynamics. In [3], nonlocal-in-time analogues of these equations were derived by assuming a class of general heat exchange laws in the acoustic medium, so-called Gutin–Pipkin heat flux laws [4].

In this work, we extend the analysis in [3] in which a fractionally damped Westervelt equation (a cousin of Blackstock’s and Kuznetsov’s equations) in pressure form was studied. Here, we intend to cover the more involved case of quadratic gradient nonlinearities and as a special case retrieve a nonlocal Westervelt equation in potential form. We analyze, in a smooth bounded domain $\Omega$ with homogeneous Dirichlet boundary conditions, the acoustic wave equation

$$\begin{aligned} (1+2k\psi ^\varepsilon _t)\psi ^\varepsilon _{tt}-c^2 \Delta \psi ^\varepsilon {- \mathfrak {K}_\varepsilon * \Delta \psi ^\varepsilon _{t}} + 2\ell \; \nabla \psi ^\varepsilon \cdot \nabla \psi ^\varepsilon _t=0, \end{aligned}$$

(1.1)

which we refer to as of Kuznetsov type. By setting $\ell =0$ and adjusting the medium-dependent constant $k$, we recover a nonlocal Westervelt equation in potential form.

We are also interested in the analysis of the Blackstock-type equation:

$$\begin{aligned} \psi ^\varepsilon _{tt}-c^2 (1-2k\psi ^\varepsilon _t)\Delta \psi ^\varepsilon - \mathfrak {K}_\varepsilon * \Delta \psi ^\varepsilon _{t} + 2\ell \; \nabla \psi ^\varepsilon \cdot \nabla \psi ^\varepsilon _t=0. \end{aligned}$$

(1.2)

Above, $c>0$ is the speed of sound and $k$, a nonlinearity parameter. Typically, $\ell \in \{0,1\}$ depending on the nonlinear acoustic model used, however, for the analysis, we can view it as a real number. The prime examples of the involved memory kernel $\mathfrak {K}_\varepsilon$ will be the Abel kernel:

$$\begin{aligned} \delta \tau _{\theta }^{-\alpha }\frac{1}{\Gamma (\alpha )} t^{\alpha -1}, \quad \alpha \in (0,1) \end{aligned}$$

and the Mittag–Leffler kernel

$$\begin{aligned} \delta \left( \frac{\tau _{\theta }}{\tau }\right) ^{a-b}\frac{1}{\tau ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\tau }\right) ^a\right) , \end{aligned}$$

where $E_{a,b}$ is the two-parametric Mittag–Leffler function (see also (2.3) below) with $0<a, b\le 1$. The constant $\tau _{\theta }>0$ in the kernels above serves as a scaling factor to adjust for the dimensional inhomogeneity introduced by the convolutions. The physical parameters $\delta$ and $\tau$ are the sound diffusivity and thermal relaxtion time, respectively, and are typically small. Thus, $\varepsilon$ on which the memory kernel $\mathfrak {K}_\varepsilon$ depends, and that we aim to send to zero, will be $\varepsilon =\delta$ in the case of the Abel kernel, whereas for the Mittag–Leffler kernels, we will have the two cases $\varepsilon =\delta$ or $\varepsilon = \tau$ (while the respective other small parameter is positive fixed).

The difference between (1.1) and (1.2), which resides in the position of the quasilinearity, leads to distinct assumptions and analyses when it comes to the uniform-in-$\varepsilon$ well-posedness, therefore, we will study them separately in Sects. 3 and 4.

1.1 State of the Art

To the best of our knowledge, together with [3], this is the first body of rigorous work dealing with the analysis and singular limits of quasilinear wave equations with damping of nonlocal/time-fractional type.

The analysis of these equations is challenging because of the combination of the nonlinear evolution and the nonrestrictive assumptions imposed on the memory kernel. To show well-posedness of Kuznetsov’s and Blackstock’s equations, one must ensure that the quasilinear coefficient $1 + 2k\psi _{t}^\varepsilon$ (respectively $1 - 2k\psi _{t}^\varepsilon$) does not degenerate, uniformly in $\varepsilon$. Note that the local well-posedness of the strongly damped Blackstock equation can be performed without the need of a nondegeneracy condition by leaning on the parabolic structure of the equation; see e.g., [5, 6]. However, since our goal in the analysis is not to rely on the strong damping so as to be able to generalize the results to weaker types of damping, i.e., fractional and to be able to take limits as the sound diffusivity parameter vanishes, we will require the coefficient $1 - 2k\psi _{t}^\varepsilon$ not to degenerate.

We note that the well-posedness analysis of the nonlocal Westervelt equation in pressure form

$$\begin{aligned} ((1+2{\tilde{k}}u^\varepsilon )u_t^\varepsilon )_t-c^2 \Delta u^\varepsilon - \mathfrak {K}_\varepsilon * {\Delta u_t^\varepsilon } = 0 \end{aligned}$$

(1.3)

with fractional kernels can be found in [7, 8] and in a more general framework in [3]. The unknown $u^\varepsilon$ is the acoustic pressure and ${\tilde{k}}$ is a medium dependent nonlinearity parameter related to $k$. Compared to the analysis in [3, 7, 8], the presence of a time-derivative ($\psi _{t}^\varepsilon$ instead of $u^\varepsilon$) in the nonlinear coefficient in (1.1) and (1.2) puts an additional strain on the analysis, as it requires an $\varepsilon$-uniform bound on $\Vert \psi _{t}^\varepsilon \Vert _{L^\infty (L^\infty (\Omega ))}$ and guaranteeing its smallness. We also need, in the case of the Kuznetsov and Blackstock settings, to extract enough regularity to absorb the quadratic gradient nonlinearity (that is, $2\ell \nabla \psi ^\varepsilon \cdot \nabla \psi _{t}^\varepsilon$). To achieve this goal, we will need to rely on an additional assumption on the kernel compared to [3] (see Assumption (${\textbf{A}}_{2}$). The well-posedness analysis will rely on considering a time-differentiated linearized PDE, and using a Banach’s fixed-point argument.

The local-in-time counterparts of the acoustic models considered in the present work are by now well-studied; we refer to, e.g., [6, 9,10,11,12,13] and the references contained therein.

1.2 Main Results

Our contributions are twofold. First, we establish the well-posedness of a family of nonlocal Blackstock and Kuznetsov equations supplemented with initial and homogeneous Dirichlet boundary conditions. Sufficient smoothness of initial data will be required, namely

$$\begin{aligned} \psi _0, \psi _1 \in \{v \in H^4(\Omega ) \cap H_0^1(\Omega ): \Delta v \vert _{\partial \Omega }=0 \}. \end{aligned}$$

Furthermore, smallness of data and short final time will be needed to ensure $\varepsilon$-uniform well-posedness of the nonlinear problem. However, smallness of the initial conditions will be imposed in a lower-order norm than that of their regularity space by relying on an Agmon’s interpolation inequality [14, Lemma 13.2]. Theorems 3.2 and 4.2 establish well-posedness of, among others, the fractionally damped Kuznetsov and Blackstock equations.

Second, we identify the assumptions on the memory kernel under which one can take the limit $\varepsilon \searrow 0$ in (1.1) or (1.2). Indeed, provided $\mathfrak {K}_\varepsilon$ verifies certain nonrestrictive assumptions and converges in the $\Vert \cdot *\ 1\Vert _{L^1(0,T)}$ norm to some measure $\mathfrak {K}_0$ as $\varepsilon \searrow 0$, we show that the solutions converge in the standard energy norm

$$\begin{aligned} \Vert \psi \Vert _{E }:= \left( \Vert \psi _t\Vert ^2_{L^\infty (L^2(\Omega ))}+ \Vert \psi \Vert ^2_{L^\infty (H^1(\Omega ))}\right) ^{1/2} \end{aligned}$$

(1.4)

to the solution of the models with $\mathfrak {K}_0$ as a memory kernel. To establish the rate of convergence, we need to consider specific examples. We will focus then on the aforementioned Abel and Mittag–Leffler kernels. The vanishing limits are detailed in Sects. 3.4 and 4.3, where the limiting memory kernel $\mathfrak {K}_0$ is additionally rigorously justified. These results significantly generalize [10, Theorem 7.1], where the vanishing sound diffusivity limit of the strongly damped Kuznetsov equation (obtained here by setting $\mathfrak {K}_\varepsilon =\varepsilon \delta _0$ in (1.1), where $\delta _0$ is the Dirac measure) has been studied.

The conditions on the kernels for the Blackstock case are weaker than those for the Kuznetsov case (cf. (${\textbf{A}}^{\textbf {K}}_{3}$) versus (${\textbf{A}}^{{\textbf {B}}}_{3}$)).

1.3 Organization of the Paper

The rest of the present paper is organized as follows. In Sect. 2, we motivate this study by recalling Kuznetsov’s and Blackstock’s equations with Gurtin–Pipkin heat flux law as derived in [3], and, to fix ideas, recall relevant classes of memory kernels in the context of acoustics. Section 3, will be dedicated to studying Kuznetsov’s equation. We first show uniform-in-$\varepsilon$ well-posedness in Theorem 3.2, which will enable us to state the main result of the section relating to the continuity of the solution with respect to the parameter $\varepsilon$; see Theorem 3.3. From there, one can extract the limiting behavior of the equation as the parameter vanishes; see Corollary 3.4 and Proposition 3.5. Section 4 follows a similar organization and is dedicated to extending the results to the Blackstock setting. So as not to burden the presentation, we mainly focus in Sect. 4 on the differences in the analysis between the Kuznetsov and Blackstock settings.

1.4 Notational Conventions

Below, we often use the notation $A\lesssim B$ for $A\le C\, B$ with a constant $C>0$ that may depend on final time T and the spatial domain $\Omega$, but never on the small parameter $\varepsilon$.

We assume throughout that $\Omega \subset \mathbb {R}^n$, where $n \in \{1, 2, 3\}$, is a bounded $C^4$-regular domain and introduce the following functional Sobolev spaces which are of interest in the analysis:

$$\begin{aligned} {H_\diamondsuit ^2(\Omega )}:= & {} \,H_0^1(\Omega )\cap H^2(\Omega ), \ {H_\diamondsuit ^3(\Omega )}:=\, \left\{ u\in H^3(\Omega ):\, u|_{\partial \Omega } = 0, \ \Delta u|_{\partial \Omega } = 0\right\} ,\\ {H_\diamondsuit ^4(\Omega )}:= & {} \,{H^4 (\Omega ) \cap H_\diamond ^3 (\Omega )}. \end{aligned}$$

Given a final time $T>0$ and $p, q \in [1, \infty ]$, we use $\Vert \cdot \Vert _{L^p (L^q(\Omega ))}$ to denote the norm on $L^p(0,T;L^q(\Omega ))$ and $\Vert \cdot \Vert _{L^p_t (L^q(\Omega ))}$ to denote the norm on $L^p(0,t;L^q(\Omega ))$ for $t \in (0,T)$. We use $(\cdot , \cdot )_{L^2}$ for the scalar product on $L^2(\Omega )$.

We denote by $*$ the Laplace convolution: $(f*g)(t)=\int _0^t f(t-s)g(s)\, d s$. Thus, by defining the Abel kernel:

$$\begin{aligned} g_\alpha (t):= \frac{1}{\Gamma (\alpha )} t^{\alpha -1}, \quad \alpha \in (0,1) \end{aligned}$$

(1.5)

and introducing the notational convention

$$\begin{aligned} g_0:= \delta _0, \end{aligned}$$

where $\delta _0$ is the Dirac delta distribution, we may define the Djrbashian–Caputo fractional derivative, for $w \in W^{1,1}(0,t)$, by

$$\begin{aligned} D _t^{\eta }w(t)=g_{\lceil \eta \rceil - \eta } *D _t^{\lceil \eta \rceil } w, \qquad -1<\eta <1, \end{aligned}$$

where $\lceil \eta \rceil$ is the smallest integer greater than or equal to $\eta$; see, for example, [15, §1] and [16, §2.4.1]. When $\eta <0$, $D _t^\eta$ is interpreted as a fractional integral.

2 Modeling and Relevant Classes of Kernels

Nonlinear acoustic equations with fractional dissipation arise as models of sound propagation through media with anomalous diffusion which can be described by a Gurtin–Pipkin flux law [4]. These laws are nonlocal-in-time relations between the heat flux $\varvec{q}$ and the gradient of the temperature of the medium $\theta$:

$$\begin{aligned} \varvec{q}= \mathfrak {K}_{\delta ,\tau } *\nabla \theta , \end{aligned}$$

where $\mathfrak {K}_{\delta ,\tau }$ may depend on the thermal conductivity, $\kappa$ (and, ultimately, on the sound diffusivity, $\delta >0$), and the thermal relaxation characteristic time, $\tau$. We discuss below two classes of kernels that are important in the context of nonlinear acoustics.

2.1 (I) Abel Kernels

The Abel memory kernel is given by

$$\begin{aligned} \mathfrak {K}_{\delta } = \delta \tau _{\theta }^{-\alpha } g_{\alpha }, \end{aligned}$$

(2.1)

with $g_\alpha$ defined in (1.5). The constant $\tau _{\theta }>0$ in (2.1) serves as a scaling factor to adjust for the dimensional inhomogeneity introduced by the fractional integral, in the way done in [17, Appendix B.4.1.2].

2.2 (II) Mittag–Leffler-Type Kernels

These kernels depend on the thermal relaxation time $\tau<<1$ and are a generalization of the widely studied exponential kernel. They are given by

$$\begin{aligned} \mathfrak {K}_{\tau } = \delta \left( \frac{\tau _{\theta }}{\tau }\right) ^{a-b}\frac{1}{\tau ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\tau }\right) ^{a}\right) , \end{aligned}$$

(2.2)

where the scaling $\tau _{\theta }$ ensures dimensional homogeneity. We recall that the generalized Mittag–Leffler function is given by

$$\begin{aligned} E_{a, b}(t) = \sum _{k=0}^\infty \frac{t^k}{\Gamma (a k + b)}, \qquad a > 0,\ t,\, b \in \mathbb {R}; \end{aligned}$$

(2.3)

see, e.g., [15, Ch. 2]. The case $a=b=1$ leads to the exponential kernel

$$\begin{aligned} \mathfrak {K}_\tau (t) = \frac{\delta }{\tau } \exp \left( -\frac{t}{\tau }\right) . \end{aligned}$$

(2.4)

As noted by [18], the Mittag–Leffler functions allow one to recast, into a Gurtin–Pipkin form, the Compte–Metzler heat flux laws [19], given by:

$$\begin{aligned} \text {(GFE I)} \qquad \qquad (1+\tau ^\alpha {D }_{t}^{\alpha })\varvec{q}(t) =&\, -\kappa {\tau _{\theta }^{1-\alpha }}{D }_{t}^{1-\alpha }\nabla \theta ;\\ \text {(GFE II)} \qquad \qquad (1+\tau ^\alpha {D }_{t}^{\alpha })\varvec{q}(t) =&\, -\kappa {\tau _{\theta }^{\alpha -1}}D _t^{\alpha -1} \nabla \theta ;\\ \text {(GFE III)} \qquad \qquad (1+\tau \partial _t)\varvec{q}(t) =&\, -\kappa {\tau _{\theta }^{1-\alpha }}{D }_{t}^{1-\alpha }\nabla \theta ; \\ \text { (GFE)}\qquad \qquad (1+\tau ^\alpha {D }_{t}^{\alpha })\varvec{q}(t) =&-\kappa \nabla \theta , \end{aligned}$$

where $\kappa$ has the usual dimension of thermal conductivity. In particular, the parameters (a, b) of the Mittag–Leffler kernel should be chosen as in Table 1. Here, $\alpha \in (0,1)$ ($\alpha \in (1/2,1)$ for GFE I [20]) is a fractional differentiation parameter.

Table 1 Parameters for the Mittag–Leffler kernels motivated by the Compte–Metzler laws

We refer the reader to, e.g, [3, 21, 22] for details on how the assumed heat flux affects the derivation of acoustic models.

2.3 Properties of the Mittag–Leffler Functions

When $1 \ge b \ge a > 0$, the Mittag–Leffler kernel (2.2) is completely monotone; see [23, Corollary 3.2]. Further, we will rely on the asymptotic behavior of Mittag–Leffler functions when establishing the convergence rates of the studied equation in Sect. 3.4. We recall a useful result to this end:

$$\begin{aligned} E_{a,b}(-x)\sim \frac{1}{\Gamma (b-a)\, x} { \text{ as } x\rightarrow \infty } \qquad \text { where } b\ge a>0; \end{aligned}$$

see, e.g., [23, Theorem 3.2].

2.4 Unifying the Physical Parameters

We wish to investigate the limiting behavior in $\delta$ and in $\tau$ of the resulting nonlocal Kuznetsov and Blackstock equations. Since the uniform well-posedness analysis in both cases is qualitatively the same, we unify the physical parameters $\delta$ and $\tau$ into one parameter $\varepsilon$. Thus, setting

$$\begin{aligned} \mathfrak {K}_\varepsilon = \varepsilon \mathfrak {K}\quad \text {with } \, \mathfrak {K}(t)= \left\{ \begin{aligned}&\tau _{\theta }^{-\alpha } g_{\alpha }(t)\\&\text {or} \\&\left( \frac{\tau _{\theta }}{\tau }\right) ^{a-b}\frac{1}{\tau ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\tau }\right) ^a\right) \, \text {with } \tau \text { fixed} \end{aligned} \right. \end{aligned}$$

will allow us to cover Abel and Mittag–Leffler kernels and study their vanishing sound diffusivity limit, whereas choosing

$$\begin{aligned} \mathfrak {K}_\varepsilon =\, \delta \left( \frac{\tau _{\theta }}{\varepsilon }\right) ^{a-b} \frac{1}{\varepsilon }\mathfrak {K}\left( \frac{t}{\varepsilon }\right) \quad \text {with } \ \mathfrak {K}(t)= t^{b-1} E_{a,b}(-t^a) \end{aligned}$$

covers the setting of Mittag–Leffler kernels and let the thermal relaxation tend to zero there.

Using the heat flux kernels introduced above, we can state the quasilinear wave equations of interest, namely, the nonlocal wave equation of Kuznetsov type [1]

$$\begin{aligned} (1+2k\psi _t)\psi _{tt}-c^2 \Delta \psi - \mathfrak {K}_\varepsilon * \Delta \psi _{t}+ \ell \partial _t |\nabla \psi |^2=0 \end{aligned}$$

(2.5)

and the nonlocal wave equation of Blackstock type [2, p. 20]

$$\begin{aligned} \psi _{tt}-c^2 (1-2k\psi _t)\Delta \psi - \mathfrak {K}_\varepsilon * \Delta \psi _{t}+ \ell \partial _t |\nabla \psi |^2=0. \end{aligned}$$

(2.6)

For their derivation, we refer to [3, Section 2]. Above, $k$, $\ell$ are real constants. The equations are expressed in terms of the acoustic velocity potential $\psi =\psi (x,t)$, which is related to the acoustic pressure u by

$$\begin{aligned} u = \varrho \psi _t, \end{aligned}$$

where $\varrho$ is the medium density. As discussed in [18], different choices of the kernel $\mathfrak {K}_\varepsilon$ lead to a rich family of flux laws that have appeared in the literature.

2.5 Assumptions on the Memory Kernel for Both Models

In the considered equations, (2.5) and (2.6), we refer to the nonlinearity in the leading term as Kuznetsov nonlinearity and in the second term as Blackstock nonlinearity. The type of nonlinearity present in the equation naturally plays a role in the uniform well-posedness (and thus limiting) analysis. More precisely, Kuznetsov and Blackstock nonlinearities will require different coercivity assumptions on the kernel; we refer to Sects. 3 and 4 for details.

However, to treat both equations, we need a uniform boundedness assumption on the kernel, which we state next. Note that because we intend to study the limiting behavior of the models as $\varepsilon \searrow 0$, we may restrict our attention in the analysis on an interval $(0, \bar{\varepsilon })$ for some fixed $\bar{\varepsilon }>0$ without loss of generality. Let $T>0$ and $\bar{\varepsilon }>0$.

We use $\Vert \cdot \Vert _{\mathcal {M}(0,T)}$ to denote the total variation norm, which in our context should be understood as:

$$\begin{aligned} \Vert \mathfrak {K}_\varepsilon \Vert _{\mathcal {M}(0,T)}={\left\{ \begin{array}{ll} \varepsilon &{}\text { if }\mathfrak {K}_\varepsilon = \varepsilon \delta _0,\\ \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)}&{}\text { if }\mathfrak {K}_\varepsilon \in L^1(0,T). \end{array}\right. } \end{aligned}$$

Furthermore, in order to extract sufficient regularity for the fixed-point proof we perform a bootstrap argument; see, e.g, estimate (3.21) below. The following assumption will be needed to control the arising convolution term.

3 Analysis of the Nonlocal Kuznetsov Equation

In this section, we focus on the Kuznetsov equation (2.5). We present the specific kernel assumptions needed in the well-posedness and the limiting analysis pertaining to this equation. In Sect. 4, we point out the differences and subtleties of treating the Blackstock equation.

3.1 Kuznetsov-Specific Assumption on the Memory Kernel

To treat the Kuznetsov nonlinearity, we need to be able to extract some regularity from the $\mathfrak {K}_\varepsilon$ term after testing. More precisely, we require the following:

3.2 Uniform Well-Posedness of the Kuznetsov Equation with Fractional-Type Dissipation

We consider the following initial-boundary value problem:

$$\begin{aligned} \left\{ \begin{aligned}&(1+2k\psi ^\varepsilon )\psi _{tt}^\varepsilon -c^2 \Delta \psi ^\varepsilon - \mathfrak {K}_\varepsilon * {\Delta \psi _{t}^\varepsilon }+ 2 \ell \,\nabla \psi ^\varepsilon \cdot \nabla \psi _{t}^\varepsilon = 0\quad{} & {} \text {in } \Omega \times (0,T), \\&\psi ^\varepsilon =0 \quad{} & {} \text {on } \partial \Omega \times (0,T),\\&(\psi ^\varepsilon , \psi _{t}^\varepsilon )=(\psi _0, \psi _1), \quad{} & {} \text {in } \Omega \times \{0\}. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(3.1)

Our aim, initially, is to establish the well-posedness of (3.1), uniformly in $\varepsilon$. This result will be the basis for the subsequent study of the limiting behavior. We introduce the mapping $\mathcal {T}:\phi \mapsto \psi ^\varepsilon$, where $\phi$ will belong to a ball in a suitable Bochner space and $\psi ^\varepsilon$ will solve the linearized problem

$$\begin{aligned} \begin{aligned} \mathfrak {m}\psi ^\varepsilon _{tt}-c^2 \Delta \psi ^\varepsilon - \mathfrak {K}_\varepsilon * \Delta \psi ^\varepsilon _{t}+ \nabla \mathfrak {l}\ \cdot \nabla \psi ^\varepsilon _t=0 \ \text {in }\Omega \times (0,T), \end{aligned} \end{aligned}$$

(3.2a)

with variable coefficients

$$\begin{aligned} \mathfrak {m}=1+2k\phi _t, \qquad \mathfrak {l}= 2 \ell \phi , \qquad k, \ell \in \mathbb {R}, \end{aligned}$$

(3.2b)

supplemented by the initial and boundary conditions:

$$\begin{aligned} \begin{aligned} (\psi ^\varepsilon , \psi ^\varepsilon _t) \vert _{t=0}=\,(\psi _0, \psi _1), \qquad \psi ^\varepsilon \vert _{\partial \Omega }=0, \end{aligned} \end{aligned}$$

(3.2c)

the idea being that a fixed-point of this mapping ($\phi =\psi ^\varepsilon$) would solve the nonlinear problem.

3.2.1 Uniform Well-Posedness of a Linear Problem with Variable Coefficients

The well-definedness of the mapping and the fixed-point argument rest on the uniform well-posedness of the linear problem, which we therefore consider first. This linear analysis is conducted using an energy method with a smooth semi-discretization in space and weak compactness arguments. In particular, the uniform analysis goes through by testing the time-differentiated PDE with $\Delta ^2 \psi _{tt}^\varepsilon$ in combination with a bootstrap strategy.

Proposition 3.1

Let assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$) on the memory kernel hold. Let $\varepsilon \in (0, \bar{\varepsilon })$, $k$, $\ell \in \mathbb {R}$, $T>0$, and let

$$\begin{aligned} \phi \in X_\phi :=W^{2,\infty }(0,T;{H_\diamondsuit ^2(\Omega )}) \cap W^{1,\infty }(0,T;{H_\diamondsuit ^3(\Omega )}) \cap L^2(0,T;{H_\diamondsuit ^4(\Omega )}).\nonumber \\ \end{aligned}$$

(3.3)

Assume that there exist $\overline{\mathfrak {m}}$ and $\underline{\mathfrak {m}}$, independent of $\varepsilon$, such that the nondegeneracy condition

$$\begin{aligned} 0<\underline{\mathfrak {m}}\le \mathfrak {m}(\phi )=1+2k\phi _t(x,t)\le \overline{\mathfrak {m}}\quad \text {a.e. in } \ \Omega \times (0,T), \end{aligned}$$

(3.4)

holds. Furthermore, assume that the initial conditions satisfy

$$\begin{aligned} (\psi _0, \psi _1) \in {\left\{ \begin{array}{ll} {H_\diamondsuit ^4(\Omega )}\times {H_\diamondsuit ^3(\Omega )}\ \text {if } \mathfrak {K}_\varepsilon = \varepsilon \delta _0 \ \text {or } \mathfrak {K}_\varepsilon \equiv 0, \\ {H_\diamondsuit ^4(\Omega )}\times {H_\diamondsuit ^4(\Omega )}\ \text {if } \mathfrak {K}_\varepsilon \not \equiv 0 \in L^1(0,T). \end{array}\right. } \end{aligned}$$

Then there exists $\Xi >0$, independent of $\varepsilon$, such that if

$$\begin{aligned} \Vert \phi \Vert _{L^1(H^4(\Omega ))} + \Vert \phi _{t}\Vert _{L^1(H^3(\Omega ))} + \Vert \phi _{tt}\Vert _{L^1(H^2(\Omega ))} \le \Xi , \end{aligned}$$

(3.5)

and if the final time $T=T(\Vert \phi \Vert _{X_\phi })$ is small enough compared to $C_{{\textbf {A}}_3}$, then there is a unique solution $\psi ^\varepsilon$ of (3.2) in

$$\begin{aligned} X_{\psi }= & {} \, W^{3,1}(0,T;H_0^1(\Omega )) \cap W^{2,\infty }(0,T;{H_\diamondsuit ^2(\Omega )}) \nonumber \\{} & {} \cap W^{1,\infty }(0,T;{H_\diamondsuit ^3(\Omega )}) \cap L^2(0,T;{H_\diamondsuit ^4(\Omega )}). \end{aligned}$$

(3.6)

This solution satisfies the estimate

$$\begin{aligned}{} & {} \Vert \psi ^\varepsilon \Vert ^2_{X_\psi }+ \int _0^{T} \Vert (\mathfrak {K}_\varepsilon * \nabla \Delta \psi _{tt}^\varepsilon )(s)\Vert ^2_{L^2(\Omega )} \, d s \nonumber \\{} & {} \quad {\le C_{lin }(T)\Bigl (}\, \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )}+ \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)} ^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )} {\Bigr )}, \end{aligned}$$

(3.7)

where $\eta = 0$ if $\mathfrak {K}_\varepsilon \in \{ 0, \varepsilon \delta _0 \}$ and $\eta =1$ otherwise. The constant $C_{lin }(T)$ does not depend on $\varepsilon$.

3.3 Discussion of the Statement

Before proceeding to the proof, let us discuss the statement made above.

• Proposition 3.1 ensures well-posedness of the linearized problem (3.2) under the condition that the final time $T=T(\Vert \phi \Vert _{X_\phi })$ is small enough compared to $C_{{\textbf {A}}_3}$. The high order in space of the testing strategy combined with the time-fractional evolution lead to such a condition.

  While the relationship between T and $C_{{\textbf {A}}_3}$ is made more precise below (see inequality (3.14)), we want to point out that one of the ways to extend the well-posedness final time T is to make $C_{{\textbf {A}}_3}$ larger which can be ensured if we have a smaller $\delta$-parameter (see discussion page 19 for explanations on the relationship between $\delta$ and $C_{{\textbf {A}}_3}$).

• It is noteworthy that we need more regular data when $\mathfrak {K}_\varepsilon \not \equiv 0 \in L^1(0,T)$. This is primarily due to needing to work with the time-differentiated equation in the analysis (see (3.8)) in order to extract sufficient regularity of the solution to treat the nonlinearities. For that we will use the differentiation rule

  $$\begin{aligned} \left( \int _0^t \mathfrak {K}_\varepsilon (t-s) \Delta \psi _{t}^\varepsilon (s)\, d s \right) _t = \int _0^t \mathfrak {K}_\varepsilon (t-s) \Delta \psi _{tt}^\varepsilon (s)\, d s + \mathfrak {K}_\varepsilon (t) \Delta \psi ^\varepsilon _{t}(0), \end{aligned}$$

  which invokes a regularity assumption on $\Delta \psi _{t}^\varepsilon (0)$.

Proof

We conduct the analysis based on energy arguments performed on a Galerkin semi-discretization in space of the problem; see, for example, [3, 10] for similar arguments and the books [24, 25] for details on the Galerkin procedure. We emphasize that compared to [3], where nonlocal Westervelt equation is analyzed, the presence of the quadratic gradient nonlinearity ($2\ell \,\nabla \psi ^\varepsilon \cdot \nabla \psi _{t}^\varepsilon )$ combined with the time-derivative in the quasilinear coefficient ($1+2k\psi _{t}^\varepsilon$) in equation (3.1) will necessitate extracting more regularity from the linearized equation’s solution, in turn leading to a higher-order testing.

The existence of a unique approximate solution follows by reducing the semi-discrete problem to a system of Volterra integral equations of the second kind. The details are provided in Appendix A. For notational simplicity, we drop the superscript discretization parameter in the estimates below when denoting the approximate solution.

We follow the approach of [10], where the local-in-time Kuznetsov equation is studied, and consider a time-differentiated semi-discrete problem. We set $p=\psi ^\varepsilon _t$, where the dependence of p on $\varepsilon$ is omitted to simplify the notation. The time-differentiated semi-discrete equation is then given by

$$\begin{aligned} \mathfrak {m}p_{tt}-c^2\Delta p - \mathfrak {K}_\varepsilon * \Delta p_{t}= F \end{aligned}$$

(3.8)

with the right-hand side

$$\begin{aligned} F(t)= -\nabla \mathfrak {l}_t \cdot \nabla p-\nabla \mathfrak {l}\cdot \nabla p_t -\mathfrak {m}_t p_t + \eta \mathfrak {K}_\varepsilon (t) \Delta p(0), \end{aligned}$$

(3.9)

where $\eta$ is as in the statement of the proposition and $p(0)= \psi _1$. We intend to multiply (3.8) with $(-\Delta )^2p_t$ and integrate over space and (0, t). When integrating by parts in the estimates below, we rely on the fact that $p_{tt}=\Delta p=\Delta p_t=0$ on the boundary for sufficiently smooth Galerkin approximations.

We note that

$$\begin{aligned}{} & {} (\mathfrak {m}p_{tt}, (-\Delta )^2 p_{t})_{L^2} \\{} & {} \quad =\, \frac{1}{2}\frac{d }{d t}( \mathfrak {m}\Delta p_{t}, \Delta p_t)_{L^2}-\frac{1}{2}(\mathfrak {m}_t\Delta p_{t}, \Delta p_t)_{L^2}+(p_{tt}\, \Delta \mathfrak {m}+2\nabla p_{tt}\cdot \nabla \mathfrak {m}, \Delta p_t)_{L^2}. \end{aligned}$$

Additionally,

$$\begin{aligned} -c^2( \Delta p, (-\Delta )^2 p_{t})_{L^2} =\, \frac{c^2}{2}\frac{d }{d t}( \nabla \Delta p, \nabla \Delta p)_{L^2}. \end{aligned}$$

(3.10)

By the time-differentiated semi-discrete PDE (3.8), we have $F=0$ on the boundary and thus

$$\begin{aligned} (F, (-\Delta )^2p_t)_{L^2}=(\Delta F, \Delta p_t)_{L^2}. \end{aligned}$$

Therefore, testing (3.8) with $(-\Delta )^2 p_t$ and using assumption (${\textbf{A}}^{\textbf {K}}_{3}$) yields the inequality

$$\begin{aligned}{} & {} \frac{1}{2} \Vert \sqrt{\mathfrak {m}}\Delta p_t\Vert _{L^2(\Omega )}^2 \big \vert _0^t +\frac{c^2}{2} \Vert \nabla \Delta p\Vert _{L^2(\Omega )}^2\big \vert _0^t + C_{{\textbf {A}}_3}\int _0^{t} \Vert (\mathfrak {K}_\varepsilon * \nabla \Delta p_t)(s)\Vert ^2_{L^2(\Omega )} \, d s \nonumber \\{} & {} \quad \le \, \int _0^t\Bigl ( \frac{1}{2}(\mathfrak {m}_t\Delta p_t,\Delta p_t)_{L^2} -( p_{tt}\, \Delta \mathfrak {m}+2\nabla p_{tt}\cdot \nabla \mathfrak {m},\Delta p_t)_{L^2} \Bigr )\, d s \nonumber \\{} & {} \qquad +\int _0^t (\Delta F,\Delta p_t)_{L^2} \, d s . \end{aligned}$$

(3.11)

We note that the value of $\Vert \Delta p_t (0)\Vert _{L^2(\Omega )}=\Vert \Delta \psi _{tt} (0)\Vert _{L^2(\Omega )}$, can be bounded by testing (3.1) at $t=0$ by $\Delta ^2 p_t(0)$ to obtain:

$$\begin{aligned} \Vert \Delta p_t (0) \Vert _{L^2(\Omega )} \le \left\| \Delta \left[ \frac{1}{\mathfrak {m}}\left( c^2\Delta \psi _0 - 2\ell \nabla \psi _0 \cdot \nabla \psi _1 \right) \right] \right\| _{L^2(\Omega )}. \end{aligned}$$

The $\mathfrak {m}$ terms on the right-hand side of (3.11) can be estimated as follows. We have

$$\begin{aligned} \int _0^t\frac{1}{2}(\mathfrak {m}_t\Delta p_t,\Delta p_t)_{L^2}\, d s\le & {} \,\frac{1}{2}\Vert \mathfrak {m}_t\Vert _{L^1(L^\infty (\Omega ))} \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2 \\\le & {} \, \Vert \phi _{tt}\Vert _{L^1(L^\infty (\Omega ))} \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2 \lesssim \Xi \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2, \end{aligned}$$

where in the last inequality, we have used the embedding $H^2(\Omega )\hookrightarrow L^\infty (\Omega )$ combined with (3.5).

Secondly, by Hölder and Poincaré–Friedrichs as well as Young’s inequalities, and the embedding $H^1(\Omega ) \hookrightarrow L^6(\Omega )$, we have

$$\begin{aligned}{} & {} \int _0^t -(p_{tt}\, \Delta \mathfrak {m}+ \nabla p_{tt}\cdot \nabla \mathfrak {m},\Delta p_t)_{L^2} \, d s \nonumber \\{} & {} \quad \lesssim \, \Vert \mathfrak {m}\Vert _{L^\infty (H^3(\Omega ))}\left( \Vert \nabla p_{tt}\Vert _{L^1_t(L^2(\Omega ))}^2+\gamma \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2 \right) \nonumber \\{} & {} \quad \lesssim \, \Vert \phi \Vert _{X_\phi }\left( \Vert \nabla p_{tt}\Vert _{L^1_t(L^2(\Omega ))}^2+\gamma \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2 \right) \end{aligned}$$

(3.12)

for any $\gamma >0$. The second term on the right-hand side above will be absorbed by the left-hand side for sufficiently small $\gamma$. To estimate $\Vert \phi \Vert _{X_\phi }\Vert \nabla p_{tt}\Vert _{L^1_t(L^2(\Omega ))}^2$, we use the time-differentiated PDE to express $\nabla p_{tt}$:

$$\begin{aligned} \nabla p_{tt}= \nabla \left[ \frac{1}{\mathfrak {m}} \left( c^2 \mathfrak {n}\Delta p +\mathfrak {K}_\varepsilon * \Delta p_{t}+ F \right) \right] . \end{aligned}$$

From here and the uniform boundedness of $\mathfrak {m}$ in (3.4), we have

$$\begin{aligned}{} & {} \Vert \nabla p_{tt}\Vert _{L^1_t(L^2(\Omega ))} \nonumber \\{} & {} \quad \lesssim \, \Vert \nabla \mathfrak {m}\Vert _{L^\infty (L^\infty (\Omega ))} \left\{ \Vert \Delta p\Vert _{L^1_t(L^2(\Omega ))} + \left\| \mathfrak {K}_\varepsilon *\Delta p_{t}\right\| _{L^1_t(L^2(\Omega ))}+\Vert F\Vert _{L^1_t(L^2(\Omega ))} \right\} \nonumber \\{} & {} \qquad +\Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))} + \left\| \mathfrak {K}_\varepsilon * \nabla \Delta p_{t}\right\| _{L^1(L^2(\Omega ))}+\Vert \nabla F\Vert _{L^1_t(L^2(\Omega ))}. \end{aligned}$$

(3.13)

Recall that in view of (3.12), we need an estimate of $\Vert \phi \Vert _{X_\phi }\Vert \nabla p_{tt}\Vert _{L^1(L^2(\Omega ))}^2$. After squaring (3.13), the corresponding kernel terms can be absorbed by the left-hand side of (3.11) by relying on the smallness of $T\Vert \phi \Vert _{X_\phi }$ in front of them (relative to $C_{{\textbf {A}}_3}$):

$$\begin{aligned}{} & {} \Vert \phi \Vert _{X_\phi } \left( {\Vert \nabla \mathfrak {m}\Vert ^2_{L^\infty (L^\infty (\Omega ))}}\left\| \mathfrak {K}_\varepsilon * \Delta p_{t}\right\| _{L^1_t(L^2(\Omega ))}^2 +\left\| \mathfrak {K}_\varepsilon *\nabla \Delta p_{t} \right\| _{L^1_t(L^2(\Omega ))}^2\right) \nonumber \\{} & {} \quad \lesssim \, (1+\Vert \phi \Vert ^2_{X_\phi })\Vert \phi \Vert _{X_\phi }\left\| \mathfrak {K}_\varepsilon *\nabla \Delta p_{t} \right\| _{L^1_t(L^2(\Omega ))}^2 \nonumber \\{} & {} \quad \lesssim \, (1+\Vert \phi \Vert ^2_{X_\phi })\Vert \phi \Vert _{X_\phi }T\left\| \mathfrak {K}_\varepsilon *\nabla \Delta p_{t} \right\| _{L^2_t(L^2(\Omega ))}^2. \end{aligned}$$

(3.14)

It remains to estimate the F and $\nabla F$ terms in (3.13), as well as $\Delta F$ term in (3.11). Since $F \vert _{\partial \Omega }=0$, we have

$$\begin{aligned} \Vert F\Vert _{L^1_t(H^2(\Omega ))} \lesssim \Vert \Delta F\Vert _{L^1_t(L^2(\Omega ))} \end{aligned}$$

and it is thus sufficient to bound $\Delta F$. Recalling how F is defined in (3.9), we have

$$\begin{aligned} \Delta F= & {} -\Delta \mathfrak {m}_t p_t-2\nabla \mathfrak {m}_t\cdot \nabla p_t-\mathfrak {m}_t \Delta p_t +\Delta (-\nabla \mathfrak {l}_t \cdot \nabla p-\nabla \mathfrak {l}\cdot \nabla p_{t} )\\{} & {} + \eta \mathfrak {K}_\varepsilon (s) \Delta ^2 p(0), \end{aligned}$$

with further

$$\begin{aligned}{} & {} \Delta (-\nabla \mathfrak {l}_t \cdot \nabla p-\nabla \mathfrak {l}\cdot \nabla p_{t} )\nonumber \\{} & {} \quad =\, -\nabla \Delta \mathfrak {l}_t \cdot \nabla p-2D^2 \mathfrak {l}_t: D^2 p-\nabla \mathfrak {l}_t \cdot \nabla \Delta p\nonumber \\{} & {} \qquad -\nabla \Delta \mathfrak {l}\cdot \nabla p_t-2D^2 \mathfrak {l}: D^2 p_t-{\nabla \mathfrak {l}\cdot }\nabla \Delta p_t, \end{aligned}$$

(3.15)

where $D^2 v=(\partial _{x_i} \partial _{x_j} v)_{ij}$ denotes the Hessian. Since the last term in (3.15) is problematic to estimate in the $L^1(0,t; L^2(\Omega ))$ norm, given the regularity we can expect from $\nabla \Delta p_t$ based on the left-hand side of (3.11), we split it in the following manner:

$$\begin{aligned} \int _0^t (\Delta F,\Delta p_t)_{L^2} \, d s \le&\, \Vert \Delta F-g\Vert _{L^1(L^2(\Omega ))} \Vert \Delta p_t\Vert _{L^{\infty }_t(L^2(\Omega ))} +\left| \int _0^t (g,\Delta p_t)_{L^2(\Omega )} \, d s \right| , \end{aligned}$$

where $g=\nabla \mathfrak {l}\cdot \nabla \Delta p_t$. Then, since $\Delta p_t=0$ on $\partial \Omega$,

$$\begin{aligned} \left| \int _0^t (g,\Delta p_t)_{L^2} \, d s \right| =\frac{1}{2}\left| \int _0^t (\Delta \mathfrak {l},(\Delta p_t)^2)_{L^2} \, d s \right|\lesssim & {} \, \Vert \Delta \mathfrak {l}\Vert _{L^1(L^\infty (\Omega ))} \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2\nonumber \\\lesssim & {} \, \Xi \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2 \end{aligned}$$

(3.16)

with $\Xi$ as in (3.5). We then estimate the remaining $F-g$ terms as follows:

$$\begin{aligned}\begin{aligned}&\Vert \Delta F-g\Vert _{L^1(L^2(\Omega ))}\\&\quad \lesssim \, \Vert \Delta \mathfrak {m}_t\Vert _{L^1(L^2(\Omega ))} \Vert p_t\Vert _{L^\infty _t(L^\infty (\Omega ))} +\Vert \nabla \mathfrak {m}_t\Vert _{L^1(L^3(\Omega ))} \Vert \nabla p_t\Vert _{L^\infty _t(L^6(\Omega ))} \\&\qquad +\Vert \mathfrak {m}_t\Vert _{L^1(L^\infty (\Omega ))} \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))} \\&\qquad +\Vert \nabla \Delta \mathfrak {l}_t \cdot \nabla p \Vert _{L^1(L^2(\Omega ))}+\Vert \nabla \mathfrak {l}_t \cdot \nabla \Delta p \Vert _{L^1_t(L^2(\Omega ))}+\Vert D^2 \mathfrak {l}_t: D^2 p \Vert _{L^1_t(L^2(\Omega ))}\\&\qquad +\Vert \nabla \Delta \mathfrak {l}\cdot \nabla p_t \Vert _{L^1_t(L^2(\Omega ))} +\Vert D^2 \mathfrak {l}: D^2 p_t \Vert _{L^1_t(L^2(\Omega ))} + \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,t)} \Vert \Delta ^2 \psi _1\Vert _{L^2(\Omega )}, \end{aligned} \end{aligned}$$

where we can further bound the $\mathfrak {l}$ terms above:

$$\begin{aligned}{} & {} \Vert \nabla \Delta \mathfrak {l}_t \cdot \nabla p \Vert _{L^1_t(L^2(\Omega ))} {+\Vert \nabla \mathfrak {l}_t \cdot \nabla \Delta p \Vert _{L^1_t(L^2(\Omega ))}} +\Vert \nabla \Delta \mathfrak {l}\cdot \nabla p_t \Vert _{L^1_t(L^2(\Omega ))}\nonumber \\{} & {} \quad \lesssim \, \Xi (\Vert \nabla p \Vert _{L^\infty _t(H^2(\Omega ))}+\Vert \Delta p_t \Vert _{L^\infty _t(L^2(\Omega ))}). \end{aligned}$$

(3.17)

By elliptic regularity, the Hessian satisfies

$$\begin{aligned} \Vert D^2 v\Vert _{L^p(\Omega )}\le C_{H } \Vert \Delta v\Vert _{L^p(\Omega )} \end{aligned}$$

for all $v\in H_0^1(\Omega )\cap W^{2,p}(\Omega )$ where $p\in (1,6]$; see, e.g., [26, Theorem 2.4.2.5]. Therefore, together with the embedding $H^1(\Omega ) \hookrightarrow L^6(\Omega )$ and Poincaré–Friedrichs inequality, we have the bound

$$\begin{aligned} \Vert D^2 \mathfrak {l}_t: D^2 p \Vert _{L^1(L^2(\Omega ))} \lesssim \Vert D^2 \mathfrak {l}_t \Vert _{L^1(L^3(\Omega ))}\Vert D^2 p \Vert _{L^\infty _t(L^6(\Omega ))} \lesssim \, \Xi \Vert \nabla \Delta p \Vert _{L^\infty _t(L^2(\Omega ))}\nonumber \\ \end{aligned}$$

(3.18)

and, similarly,

$$\begin{aligned} \Vert D^2 \mathfrak {l}: D^2 p_t \Vert _{L^1(L^2(\Omega ))} \lesssim \Vert D^2 \mathfrak {l}\Vert _{L^1(L^\infty (\Omega ))} \Vert \Delta p_t \Vert _{L^\infty _t(L^2(\Omega ))} \lesssim \Xi \Vert \Delta p_t \Vert _{L^\infty _t(L^2(\Omega ))}.\nonumber \\ \end{aligned}$$

(3.19)

Altogether, we can write

$$\begin{aligned} \begin{aligned} \Vert \Delta F -g \Vert _{L^1_t(L^2(\Omega ))} \lesssim&\, \Xi \left( \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}+\Vert \nabla \Delta p \Vert _{L^\infty _t(L^2(\Omega ))} \right) \\&+ \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,t)} \Vert \Delta ^2 \psi _1\Vert _{L^2(\Omega )}. \end{aligned} \end{aligned}$$

Thus, combining the derived bounds for sufficiently small $\Xi$ allows us to absorb all right-hand side terms by the left-hand side in (3.11) and thus, combined with (3.13), leads to

$$\begin{aligned}{} & {} \Vert \nabla p_{tt}\Vert _{L^1_t(L^2(\Omega ))}^2 + \Vert \Delta p_t\Vert _{L^2(\Omega )}^2 +\Vert \nabla \Delta p\Vert _{L^2(\Omega )}^2 + \int _0^{t} \Vert (\mathfrak {K}_\varepsilon * \nabla \Delta p_t)(s)\Vert ^2_{L^2(\Omega )} \, d s \nonumber \\{} & {} \quad \le \, { C_{lin }(T)\Bigl (} \Vert \psi _0\Vert ^2_{H^4(\Omega )}+ \Vert \psi _1\Vert ^2_{H^3(\Omega )}+ \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)} ^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )} {\Bigr )}. \end{aligned}$$

(3.20)

Bootstrap argument for $\psi ^\varepsilon \in L^2(0,T;{H_\diamondsuit ^4(\Omega )})$. If $\ell \not =0$, then, due to inequality (3.19) and the perspective of the fixed-point argument, we have to estimate the term $\Delta ^2\psi ^{\varepsilon }$. To this end, we can employ the additional regularity to be achieved by estimating ${\Vert \Delta ^2 \psi ^\varepsilon \Vert _{L^2(L^2(\Omega ))}}$ via a bootstrap argument analogously to [10, p. 22]. Multiplying the identity

$$\begin{aligned} {[}\mathfrak {K}_\varepsilon *\partial _t+ c^2]\Delta \psi ^\varepsilon =r:=\mathfrak {m}p_t+\nabla \mathfrak {l}\cdot \nabla p \end{aligned}$$

with $\Delta ^3\psi ^\varepsilon$ and integrating by parts, due to assumption (${\textbf{A}}_{2}$), we have

$$\begin{aligned}{} & {} c^2\Vert \Delta ^2\psi ^\varepsilon \Vert _{L^2_t(L^2(\Omega ))}^2\nonumber \\{} & {} \quad \le \, \int _0^t(\Delta r,\Delta ^2\psi ^\varepsilon )_{L^2}\, d s + C_{{\textbf {A}}_2}\Vert \Delta ^2 \psi _0\Vert ^2_{L^2(\Omega )}\nonumber \\{} & {} \quad \le \Vert \Delta ^2\psi ^\varepsilon \Vert _{L^2_t(L^2(\Omega ))} \left\| \Delta [\mathfrak {m}p_t-\nabla \mathfrak {l}\cdot \nabla p]\right\| _{L^2_t(L^2(\Omega ))} + C_{{\textbf {A}}_2}\Vert \psi _0\Vert ^2_{H^4(\Omega )},\qquad \end{aligned}$$

(3.21)

where we have used the fact that r and $\Delta ^2 \psi ^\varepsilon$ vanish on the boundary. Thanks to (3.20) and the fact that $\phi \in X_\phi$, we readily obtain that $\Delta [\mathfrak {m}p_t-\nabla \mathfrak {l}\cdot \nabla p] \in L^2(0,T;L^2(\Omega )).$ Thus, combining (3.20) and (3.21), leads to estimate (3.7), at first for the semi-discrete solution.

Limiting procedure as the discretization parameter vanishes. Thanks to the the uniform (with respect to the spatial discretization parameter) energy bound, one can extract a weakly($-*$) converging subsequence, which can be shown to converge to an exact solution $\psi ^\varepsilon$ of the problem. Uniqueness can be argued by contradiction. As these arguments are completely analogous to [10, Proposition 3.1], we omit their details here. $\square$

For the nonlinear analysis, we intend to use Agmon’s inequality with the goal of imposing smallness of data in a less restrictive space; see (3.27) below. Agmon’s inequality is given by

$$\begin{aligned} \Vert u\Vert _{L^\infty (\Omega )} \le C_{A }\Vert u\Vert _{L^2(\Omega )}^{1-n/4}\Vert u\Vert ^{n/4}_{H^2(\Omega )} \end{aligned}$$

for some arbitrary $u\in H^2(\Omega )$ and $n \le 3$; see [14, Lemma 13.2]. It is thus helpful to also have a uniform lower-order bound for the solution of (3.2). Under the assumptions of Proposition 3.1, testing (3.2) with $\psi _{t}^\varepsilon$ and integrating over space and time yields, after usual manipulations,

$$\begin{aligned}{} & {} \frac{1}{2} \left\{ \Vert \sqrt{\mathfrak {m}(t)}\psi _{t}^\varepsilon (s)\Vert ^2_{L^2(\Omega )}+c^2\Vert \nabla \psi ^\varepsilon (s)\Vert ^2_{L^2(\Omega )} \right\} \Big \vert _0^t+C_{{\textbf {A}}_3}\int _0^{t} \Vert (\mathfrak {K}_\varepsilon * \nabla \psi _{t}^\varepsilon )(s)\Vert ^2_{L^2(\Omega )} \, d s \\{} & {} \quad \le \, \frac{1}{2}\Vert \mathfrak {m}_t\Vert _{L^1(L^\infty (\Omega ))}\Vert \psi _{t}^\varepsilon \Vert ^2_{L^\infty _t(L^2(\Omega ))}- \int _0^t( \nabla \mathfrak {l}\cdot \nabla \psi _{t}^\varepsilon , \psi _{t}^\varepsilon )\, d s . \end{aligned}$$

To bound the last term on the right, we integrate by parts in space and use that $\psi _{t}^\varepsilon \vert _{\partial \Omega }=0$:

$$\begin{aligned} \left| \int _0^t \int _{\Omega } (\nabla \mathfrak {l}\cdot \nabla \psi _{t}^\varepsilon ) \psi _{t}^\varepsilon \, d xd s\right|= & {} \, \frac{1}{2} \left| \int _0^t \int _{\Omega } (- \Delta {\mathfrak {l}}) \psi _{t}^\varepsilon \psi _{t}^\varepsilon \, d xd s\right| \nonumber \\\lesssim & {} \, \Vert \Delta {\mathfrak {l}}\Vert _{L^1(L^\infty (\Omega ))}\Vert \psi _{t}^\varepsilon \Vert _{L^\infty _t(L^2(\Omega ))}^2. \end{aligned}$$

(3.22)

Thus

$$\begin{aligned} \frac{1}{2} \left\{ \Vert \sqrt{\mathfrak {m}(t)}\psi _{t}^\varepsilon (s)\Vert ^2_{L^2(\Omega )}+c^2\Vert \nabla \psi ^\varepsilon (s)\Vert ^2_{L^2(\Omega )} \right\} \Big \vert _0^t \le C(\Omega )\,\Xi \Vert \psi _{t}^\varepsilon \Vert ^2_{L_t^\infty (L^2(\Omega ))},\nonumber \\ \end{aligned}$$

(3.23)

where we recall that $\Xi$ is defined in (3.5). Utilizing the above estimates and the smallness of $\Xi$ according to condition (3.5), leads to the sought-after uniform-in-$\varepsilon$ lower order estimate:

$$\begin{aligned} \Vert \psi _{t}^\varepsilon (t)\Vert ^2_{L^2(\Omega )} + \Vert \nabla \psi ^\varepsilon (t)\Vert ^2_{L^2(\Omega )} \lesssim \Vert \psi _1\Vert ^2_{L^2(\Omega )} + \Vert \nabla \psi _0\Vert ^2_{L^2(\Omega )}. \end{aligned}$$

(3.24)

We are now ready to state the uniform well-posedness result for the initial-boundary value problem (3.1) for the nonlinear Kuznetsov equation.

Theorem 3.2

Let $\varepsilon \in (0, \bar{\varepsilon })$ and k, $\ell \in \mathbb {R}$. Furthermore, let $(\psi _0, \psi _1)$ be as in Proposition 3.1 and such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )} \le r^2, \qquad \text {if } \mathfrak {K}_\varepsilon = \varepsilon \delta _0 \ \text {or } \mathfrak {K}_\varepsilon \equiv 0, \\ \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^4(\Omega )} \le r^2, \qquad \text {if } \mathfrak {K}_\varepsilon \not \equiv 0 \in L^1(0,T). \end{array}\right. } \end{aligned}$$

where r does not depend on $\varepsilon$. Let assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$) on the kernel $\mathfrak {K}_\varepsilon$ hold. Then, there exist a data size $r_0=r_0(r)>0$ and final time $T=T(r)>0$, both independent of $\varepsilon$, such that if

$$\begin{aligned} \Vert \psi _0\Vert ^2_{H^1(\Omega )}+\Vert \psi _1\Vert ^2_{L^2(\Omega )} \le r_0^2, \end{aligned}$$

(3.25)

then there is a unique solution $\psi ^\varepsilon$ in $X_\psi$ (where $X_\psi$ is defined in (3.6)) of (3.1), which satisfies the following estimate:

$$\begin{aligned}{} & {} \Vert \psi ^\varepsilon \Vert ^2_{X_\psi }+ \int _0^{T} \Vert (\mathfrak {K}_\varepsilon * \nabla \Delta \psi _{tt}^\varepsilon )(s)\Vert ^2_{L^2(\Omega )} \, d s \\{} & {} \quad \le \, C_{nonlin }\left( \, \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )}+\eta \,\Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)} ^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )}\right) , \end{aligned}$$

where $\eta = 0$ if $\mathfrak {K}_\varepsilon \in \{ 0, \varepsilon \delta _0 \}$ and $\eta =1$ otherwise. Here, $C_{nonlin }=C_{nonlin }(\Omega ,\overline{\phi }, T)$ does not depend on the parameter $\varepsilon$.

Proof

The proof follows using the Banach fixed-point theorem in the general spirit of [10, Theorem 6.1] which considers $\varepsilon$-uniform local-in-time analysis of the Kuznetsov equation. A similar idea was also used in the study of the uniform-in-$\varepsilon$ Westervelt equation in pressure form [3, Theorem 3.1]. The mapping $\mathcal {T}:\phi \mapsto \psi$ is introduced on

$$\begin{aligned} \mathcal {B}= & {} \left\{ \phi \in X_\psi \,: \, (\phi ,\phi _{t})|_{t=0}= (\psi _0, \psi _1), \right. \nonumber \\{} & {} \left. \Vert \phi \Vert _{L^1(H^4(\Omega ))} + \Vert \phi _{t}\Vert _{L^1(H^3(\Omega ))} + \Vert \phi _{tt}\Vert _{L^1(H^2(\Omega ))} \le \Xi , \right. \nonumber \\{} & {} \qquad \left. {4|k|}\Vert \phi _t\Vert _{L^\infty (L^\infty (\Omega ))} \le 1, \ \Vert \phi \Vert _{X_{\psi }} \le R \ \right\} . \end{aligned}$$

(3.26)

The radius $R>0$ large enough as well as $\Xi >0$ small enough will be determined below. Note that the set $\mathcal {B}$ is non-empty as the solution of the linear problem with $k=\ell =0$ belongs to it if R is sufficiently large. Furthermore, since $4 |k|\Vert \phi _t\Vert _{L^\infty (L^\infty (\Omega ))} \le 1$, $\mathfrak {m}$ does not degenerate:

$$\begin{aligned} \frac{1}{2}= \underline{\mathfrak {m}} \le \mathfrak {m}\le \overline{\mathfrak {m}}=\frac{3}{2}. \end{aligned}$$

Moreover $\phi \in X_\psi$ implies that $\phi \in X_\phi$, and the smallness condition on $T=T(\Vert \phi \Vert _{X_\phi })$ follows for $\phi \in \mathcal {B}$ if we choose the final time T relative to R and $C_{{\textbf {A}}_3}$ according to the assumptions of the linear well-posedness theory given in Proposition 3.1.

• The self-mapping property. Using the linear well-posedness theory from Proposition 3.1, we ensure that the solution of the linear problem satisfies

  $$\begin{aligned} \Vert \psi ^\varepsilon \Vert ^2_{X_\psi } \le \, C_{lin }(T)\left( \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )}+ \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)} ^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )} \right) . \end{aligned}$$

Provided that R is chosen large enough so that

$$\begin{aligned} C_{lin }(T) \left( \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )}+ \eta C_{{\textbf {A}}_1}^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )} \right) \le R^2, \end{aligned}$$

the solution $\psi$ satisfies the R bound in (3.26). Next, we have

$$\begin{aligned} \Vert \phi \Vert _{L^1(H^4(\Omega ))} + \Vert \phi _{t}\Vert _{L^1(H^3(\Omega ))} + \Vert \phi _{tt}\Vert _{L^1(H^2(\Omega ))} \le (T+\sqrt{T}) R, \end{aligned}$$

thus, the condition on $\Vert \phi \Vert _{L^1(H^4(\Omega ))} + \Vert \phi _{t}\Vert _{L^1(H^3(\Omega ))} + \Vert \phi _{tt}\Vert _{L^1(H^2(\Omega ))} \le \Xi$ will be fulfilled if we choose the final time $T>0$ small enough so that

$$\begin{aligned} (T+\sqrt{T}) R \le \Xi . \end{aligned}$$

To finish proving the self-mapping property, it remains to show that the bound

$$\begin{aligned} {4|k|}\Vert \psi _t\Vert _{L^\infty (L^\infty (\Omega ))} \le 1 \end{aligned}$$

in (3.26) holds. Using Agmon’s interpolation inequality, we write

$$\begin{aligned} \Vert \psi _{t}^\varepsilon (t)\Vert _{L^\infty (\Omega )} \le C_{A }\Vert \psi _{t}^\varepsilon (t)\Vert _{L^2(\Omega )}^{1-n/4}\Vert \psi _{t}^\varepsilon (t)\Vert ^{n/4}_{H^2(\Omega )}. \end{aligned}$$

We combine this with a uniform lower-order estimate in $\varepsilon$ for the linearized problem established in (3.23):

$$\begin{aligned} \Vert \psi ^\varepsilon \Vert ^2_{E } \lesssim \Vert \psi _0\Vert ^2_{H^1(\Omega )}+\Vert \psi _1\Vert ^2_{L^2(\Omega )}. \end{aligned}$$

This approach yields

$$\begin{aligned} \Vert \psi _{t}^\varepsilon \Vert _{L^\infty (L^\infty (\Omega ))} \le C_{A }\left\{ C_lower (\Xi )(\Vert \psi _0\Vert ^2_{H^1(\Omega )}+\Vert \psi _1\Vert ^2_{L^2(\Omega )})\right\} ^{1/2-{n}/8}R^{{n}/4}, \end{aligned}$$

(3.27)

where $C_lower (\Xi )$ is the hidden constant in (3.24). We choose $r_0>0$ in (3.25) small enough, so that

$$\begin{aligned} 4|k| C_{A }C_lower (\Xi )^{1/2-{n}/8} r_0^{1-{n}/4}\,R^{{n}/4} \le 1, \end{aligned}$$

which via (3.27) implies $4|k| \Vert \psi _{t}^\varepsilon \Vert _{L^\infty (L^\infty (\Omega ))} \le 1$. Altogether, we conclude that $\mathcal {T}(\mathcal {B}) \subset \mathcal {B}$.

• Contractivity. We next prove that $\mathcal {T}$ is strictly contractive with respect to the energy norm (1.4). Take $\phi ^{(1)}$ and $\phi ^{(2)}$ in $\mathcal {B}$, and denote their difference by $\overline{\phi }= \phi ^{(1)} -\phi ^{(2)}$. Let $\psi ^{\varepsilon ,(1)}=\mathcal {T}(\phi ^{(1)})$ and $\psi ^{\varepsilon ,(2)}=\mathcal {T}(\phi ^{(2)})$. Their difference $\overline{\psi }^\varepsilon =\psi ^{\varepsilon ,(1)} -\psi ^{\varepsilon ,(2)} \in \mathcal {B}$ then solves

  $$\begin{aligned}{} & {} (1+ 2 k\phi _t^{(1)})\overline{\psi }^\varepsilon _{tt}-c^2 \Delta \overline{\psi }^\varepsilon - \mathfrak {K}_\varepsilon * \Delta \overline{\psi }^\varepsilon _{t}+2 \ell \nabla \phi ^{(1)} \cdot \nabla \overline{\psi }^\varepsilon _t\\{} & {} \quad = -2 k\overline{\phi }_t \psi ^{\varepsilon ,(2)} _{tt}- 2\ell \nabla \overline{\phi }\cdot \nabla \psi ^{\varepsilon ,(2)}_t, \end{aligned}$$

with zero initial and boundary data. We will next test this equation with $\overline{\psi }_t^\varepsilon$ and integrate over space and time. As we are in a similar setting to when deriving the lower-order bound in (3.23) with now $\mathfrak {m}=1+2 k\phi ^{(1)}_t$, and a non-zero source term, we only discuss how to estimate the terms resulting from the right-hand side contributions. The resulting $k$ term on the right can be estimated as follows:

$$\begin{aligned} \left| -2 k\int _0^t \int _{\Omega }\overline{\phi }_t \psi _{tt}^{\varepsilon ,(2)} \overline{\psi }^\varepsilon _t \, d xd s\right| \lesssim \, \Vert \psi _{tt}^{\varepsilon ,(2)}\Vert _{L^1(L^\infty (\Omega ))}\left( \Vert \overline{\psi }^\varepsilon _t\Vert _{L^\infty _t(L^2(\Omega ))}^2+\Vert \overline{\phi }_t\Vert _{L^\infty _t(L^2(\Omega ))}^2\right) . \end{aligned}$$

We can estimate the $\ell$ term similarly:

$$\begin{aligned}{} & {} \left| -2\ell \int _0^t \int _{\Omega } \nabla \overline{\phi }\cdot \nabla \psi ^{\varepsilon ,(2)}_t \, \overline{\psi }_t^\varepsilon \, d xd s\right| \\{} & {} \quad \lesssim \Vert \nabla \psi ^{\varepsilon ,(2)}_t\Vert _{L^1(L^\infty (\Omega ))} \left( \Vert \overline{\psi }_t^\varepsilon \Vert ^2_{L^\infty _t(L^2(\Omega ))}+\Vert \nabla \overline{\phi }\Vert _{L^\infty _t(L^2(\Omega ))}^2\right) . \end{aligned}$$

Thus using that $\Vert \psi ^{\varepsilon ,(i)}\Vert _{X_{\psi }}$, $\Vert \phi ^{\varepsilon ,(i)}\Vert _{X_{\psi }} \le R$ (with $i=1,2$), we obtain

$$\begin{aligned} \Vert \overline{\psi }^\varepsilon \Vert ^2_{E } \le C(\Omega ) {{T}}R \left( \Vert \overline{\psi }^\varepsilon \Vert ^2_{E } + \Vert \overline{\phi }\Vert ^2_{E } \right) . \end{aligned}$$

Therefore by reducing T if needed (independently of $\varepsilon$), we can absorb the energy term $\Vert \overline{\psi }^\varepsilon \Vert ^2_{E }$ on the right-hand-side of the inequality and ensure that the mapping $\mathcal {T}$ is strictly contractive with respect to the energy norm (1.4). The arguments showing that $\mathcal {B}$ is closed with respect to this norm are analogous to those of [10, Theorem 4.1]. By the Banach fixed-point theorem, we therefore obtain a unique fixed-point $\psi ^\varepsilon =\mathcal {T}(\psi ^\varepsilon )$ in $\mathcal {B}$, which solves the nonlinear problem. $\square$

3.4 Verifying the Assumptions for Relevant Classes of Kernels

In Sect. 2, we saw that Abel and Mittag–Leffler kernels are of particular interest in the context of nonlinear acoustic modeling. We next discuss to which extent they satisfy the assumptions imposed in the preceding analysis. The verification of assumptions (${\textbf{A}}_{1}$) and (${\textbf{A}}^{\textbf {K}}_{3}$) on these kernels (with the choice $\varepsilon =\tau$, $\delta >0$ fixed for the Mittag–Leffler kernels) follows from [3, Section 5]; we compile them for convenience of the reader in Table 2.

The verification in [3, Section 5] of (${\textbf{A}}^{\textbf {K}}_{3}$) uses two different techniques: Fourier analysis and the theory of Volterra integrodifferential equations for completely monotone kernels. As per the discussion of [3, Section 5.2], assumption (${\textbf{A}}^{\textbf {K}}_{3}$) is verified for the Abel kernel (2.1), and by fixing $\delta = 1$ for the Mittag–Leffler kernels (2.2) when $a\le b$ (which covers the exponential, GFE II, and GFE kernels). When $a>b$ (e.g., for kernels GFE I and GFE III), we need the ratio $\dfrac{\tau _{\theta }}{\varepsilon }$ to be a constant.

3.4.1 Influence of the Sound Diffusivity $\delta >0$ fixed ($\varepsilon =\tau$) on the Coercivity Constant $C_{{\textbf {A}}_3}$

As mentioned, the verification of (${\textbf{A}}^{\textbf {K}}_{3}$) in [3, Section 5.2] for the Mittag–Leffler kernels was performed by fixing $\delta = 1$. Let $y\in L^2(0, T; L^2(\Omega ))$ and $t\in (0,T)$. Then, by [3, Section 5.2], we have for those kernels

$$\begin{aligned} \int _0^{t} \int _{\Omega }\left( \frac{1}{\delta }\mathfrak {K}_\varepsilon * y \right) (s) \,y(s)\, d xd s\ge \check{c} \int _0^{t} \Vert (\frac{1}{\delta }\mathfrak {K}_\varepsilon * y)(s)\Vert ^2_{L^2(\Omega )} \, d s , \end{aligned}$$

where $\check{c}>0$ depends neither on $\varepsilon \in (0,\bar{\varepsilon })$ nor on $\delta$. Thus, by straightforward manipulations, we can prove that (${\textbf{A}}^{\textbf {K}}_{3}$) is verified for $\mathfrak {K}_\varepsilon$ with $C_{{\textbf {A}}_3}=\dfrac{\check{c}}{\delta }$, and smaller values of $\delta$ will thus lead to larger values of $C_{{\textbf {A}}_3}$.

It remains to discuss the verification of (${\textbf{A}}_{2}$). Assumption (${\textbf{A}}_{2}$) can be checked by means of [3, Lemma 5.1]. Given a Hilbert space H, a function $w \in W^{1,1}(0,T; H)$, and a memory kernel $\mathfrak {K}\in L^1(0,T)$ verifying that for all $t_0>0$, $\mathfrak {K}\in W^{1,1}(t_0,T)$, $\mathfrak {K}\ge 0$ on (0, T), and $\mathfrak {K}'\vert _{[t_0,T]}\le 0$ a.e., [3, Lemma 5.1] ensures that

$$\begin{aligned} \int _0^T ( {w}(t), \mathfrak {K}* {w}_t(t))_{H}\, d t\ge & {} \, \frac{1}{2}(\mathfrak {K}* \Vert {w}\Vert _H^2)(T)-\frac{1}{2}\int _0^T\mathfrak {K}(t)\, d t\, \Vert {w}(0)\Vert _H^2 \nonumber \\\ge & {} \, -\frac{1}{2} \Vert \mathfrak {K}\Vert _{L^1(0,T)} \Vert {w}(0)\Vert _H^2. \end{aligned}$$

(3.28)

Thus, the kernel assumptions in [3, Lemma 5.1] are satisfied up to an $\varepsilon$-dependent final time for the Mittag–Leffler kernels when $a>b$ (with the choice $\varepsilon =\tau$, $\delta >0$ fixed) in the sense that

$$\begin{aligned} \int _0^t\bigl ((\mathfrak {K}_\varepsilon *y_t)(s), y(s)\bigr )_{L^2(\Omega )}\, d s \ge - \frac{1}{2} \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,t)} \Vert y(0)\Vert ^2_{L^2(\Omega )}, \quad y\in W^{1,1}(0,t;L^2(\Omega )) \end{aligned}$$

holds for all

$$\begin{aligned} 0 \le t \le t_f = t_f(\varepsilon ). \end{aligned}$$

The $\varepsilon$-dependence in the case $a>b$ is owed to the fact that positivity and decreasing monotonicity can only be verified up to such a final time. This dependency is marked by the symbol in Table 2.

As for the Abel and Mittag–Leffler kernel with $a\le b$, they are completely monotone and thus verify the assumptions of (3.28) for all final times $T \in (0,\infty )$. The constant $C_{{\textbf {A}}_2}$ in (${\textbf{A}}_{2}$) can then be estimated by $\frac{1}{2}\Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,\infty )}$ which is indeed independent of $\varepsilon$; see [3, Section 5.1].

3.4.2 Influence of a Variable Sound Diffusivity $\varepsilon = \delta \in (0,\bar{\varepsilon })$ (with $\tau >0$ fixed) in the Mittag–Leffler Kernels

For the Mittag–Leffler kernels (2.2) (including the exponential kernel (2.4)), we had a choice between setting $\varepsilon = \delta$ (with $\tau$ fixed) or $\varepsilon =\tau$ (with $\delta$ fixed). Note that the case of $\varepsilon = \delta$ ($\tau$ fixed) is somewhat simpler to discuss as the small parameter $\varepsilon$ appears as a multiplication factor in the kernels. Thus, considering a neighbourhod $(0,\bar{\varepsilon })$, it is clear that we can use the upper bound $\bar{\varepsilon }$ to bound the assumption constants uniformly in $\varepsilon$.

To showcase the statement above, consider a generic kernel $\mathfrak {K}$ (independent of $\varepsilon$) verifying assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$) with constants $\check{c}_1$, $\check{c}_2$, and $\check{c}_3$, respectively. Then, straightforward manipulations show that $\mathfrak {K}_\varepsilon = \varepsilon \mathfrak {K}$ also verifies (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$) with constants $C_{{\textbf {A}}_1}= \bar{\varepsilon }\check{c}_1$, $C_{{\textbf {A}}_2}= \dfrac{1}{\bar{\varepsilon }}\check{c}_2$, and $C_{{\textbf {A}}_3}= \dfrac{1}{\bar{\varepsilon }}\check{c}_3$, respectively.

Furthermore, by fixing $\tau$, note that the final time of verifying (${\textbf{A}}_{2}$) no longer depends on $\varepsilon$ for the Mittag–Leffler kernels with $a>b$.

Table 2 Kernels for flux laws discussed in Sect. 2 and the assumptions they satisfy

3.5 Limiting Behavior of the Kuznetsov Equation for Relevant Classes of Kernels

We next wish to determine the limiting behavior of solutions to (3.1) as $\varepsilon \searrow 0$. The key result towards this is proving the continuity of the solution with respect to the memory kernel.

Theorem 3.3

Let $\varepsilon _1, \varepsilon _2 \in (0, \bar{\varepsilon })$. Under the assumptions of Theorem 3.2, for sufficiently small T (independent of $\varepsilon$), the following estimate holds:

$$\begin{aligned} \Vert \psi ^{\varepsilon _1}-\psi ^{\varepsilon _2}\Vert _{E }\lesssim \Vert (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*1\Vert _{L^1(0,T)}, \end{aligned}$$

where $\Vert \cdot \Vert _{E }$ is defined in (1.4).

Proof

We note that the difference $\overline{\psi }=\psi ^{\varepsilon _1}-\psi ^{\varepsilon _2}$ solves the equation

$$\begin{aligned}{} & {} (1+ 2k\psi _t^{\varepsilon _1} )\overline{\psi }_{tt}-c^2 \Delta \overline{\psi }- \mathfrak {K}_{\varepsilon _1}* \Delta \overline{\psi }_{t}\nonumber \\{} & {} \quad =\, -2k\overline{\psi }_t \psi ^{\varepsilon _2} _{tt}- 2\ell \left( \nabla \psi ^{\varepsilon _1} \cdot \nabla \overline{\psi }_t +\nabla \overline{\psi }\cdot \nabla \psi ^{\varepsilon _2}_t \right) +(\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})* \Delta \psi _t^{\varepsilon _2}\nonumber \\ \end{aligned}$$

(3.29)

with zero boundary and initial conditions. We can test (3.29) by $\overline{\psi }_{t}$ and proceed similarly to the the derivation of the lower-order estimate in (3.23). We therefore mainly discuss the contribution of the right-hand side in (3.29). To this end, the resulting $k$ term after testing can be estimated as follows:

$$\begin{aligned}{} & {} \left| -2 k\int _0^t \int _{\Omega }\overline{\psi }_t^2 \psi _{tt}^{\varepsilon _2} \, d xd s\right| \\{} & {} \quad \lesssim \, \Vert \psi _{tt}^{\varepsilon _2}\Vert _{L^1(L^\infty (\Omega ))}\Vert \overline{\psi }_t\Vert _{L^\infty _t(L^2(\Omega ))}^2. \end{aligned}$$

We can estimate the $\ell$ term similarly:

$$\begin{aligned}{} & {} \left| -2\ell \int _0^t \int _{\Omega }\left( \nabla \psi ^{\varepsilon _1} \cdot \nabla \overline{\psi }_t +\nabla \overline{\psi }\cdot \nabla \psi ^{\varepsilon _2}_t \right) \overline{\psi }_t \, d xd s\right| \lesssim \Vert \nabla \psi ^{\varepsilon _1}_t\Vert _{L^1(L^\infty (\Omega ))} \Vert \nabla \overline{\psi }_t^\varepsilon \Vert ^2_{L^\infty _t(L^2(\Omega ))} \\{} & {} \quad + \Vert \nabla \psi ^{\varepsilon _2}_t\Vert _{L^1(L^\infty (\Omega ))} \left( \Vert \overline{\psi }_t\Vert ^2_{L^\infty _t(L^2(\Omega ))}+\Vert \nabla \overline{\psi }\Vert _{L^\infty _t(L^2(\Omega ))}^2\right) . \end{aligned}$$

The $(\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})$ term we first rewrite as

$$\begin{aligned}{} & {} \int _0^t \left( (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})* \Delta \psi _t^{\varepsilon _2}, \overline{\psi }_t\right) _{L^2(\Omega )}\, d s \nonumber \\{} & {} \quad =\, \int _0^t \big ((1*(\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*\Delta \psi _{tt}^{\varepsilon _2})+(1*(\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2}))\Delta \psi _1, \overline{\psi }_t(s)\big )_{L^2(\Omega )}\, d s . \end{aligned}$$

Then it can be controlled using Young’s convolution inequality as follows:

$$\begin{aligned}{} & {} \left| \int _0^t \left( (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})* \Delta \psi _t^{\varepsilon _2}, \overline{\psi }_t\right) _{L^2}\, d s \right| \\{} & {} \quad \le \, \Vert (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*1\Vert _{L^1(0,T)} \left\{ \Vert \Delta \psi _{tt}^{\varepsilon _2}\Vert _{L_t^1( L^2(\Omega ))}+ \Vert \Delta \psi _1\Vert _{L^2(\Omega )}\right\} \Vert \overline{\psi }_t\Vert _{L_t^\infty ( L^2(\Omega ))}. \\{} & {} \quad \le \, \Vert \Delta \psi _1\Vert _{L^2(\Omega )}\left( \frac{1}{4\gamma }\Vert (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*1\Vert _{L^1(0,T)}^2 + \gamma \Vert \overline{\psi }_t\Vert _{L_t^\infty ( L^2(\Omega ))}^2\right) \\{} & {} \qquad +\frac{1}{2} \Vert \Delta \psi _{tt}^{\varepsilon _2}\Vert _{L_t^1( L^2(\Omega ))}\left( \Vert (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*1\Vert _{L^1(0,T)}^2 + \Vert \overline{\psi }_t\Vert _{L_t^\infty ( L^2(\Omega ))}^2\right) \end{aligned}$$

for an arbitrary real number $\gamma >0$. In the proof of Theorem 3.2 we have shown that $\psi ^{\varepsilon _1}$, $\psi ^{\varepsilon _2}$ $\in \mathcal {B}$, with $\mathcal {B}$ defined in (3.26), which crucially implies uniform boundedness of $\Vert \Delta \psi _{tt}^{\varepsilon _2}\Vert _{L_t^1( L^2(\Omega ))}$. Therefore, we can use the properties of $\mathcal {B}$ to derive the following estimate:

$$\begin{aligned}{} & {} \frac{1}{2} \underline{\mathfrak {m}}\sup _{\sigma \in (0,t)}\Vert \overline{\psi }_t(\sigma )\Vert _{L^2(\Omega )}^2+\frac{c^2}{2}\sup _{\sigma \in (0,t)}\Vert \nabla \overline{\psi }(\sigma )\Vert _{L^2(\Omega )}^2\\{} & {} \quad \lesssim \,C(\Omega )(TR+\gamma \Vert \Delta \psi _1\Vert _{L^2(\Omega )})\left( \Vert \overline{\psi }_t\Vert _{L_t^\infty (L^2(\Omega ))}^2 + \Vert \nabla \overline{\psi }\Vert _{L_t^\infty ( L^2(\Omega ))}^2 \right) \\{} & {} \qquad + (TR+\frac{1}{4\gamma }\Vert \Delta \psi _1\Vert _{L^2(\Omega )}) \Vert (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*1\Vert _{L^1(0,T)}^2. \end{aligned}$$

By reducing the final time T and $\gamma$ if needed (independently of $\varepsilon$), we can absorb the energy term on the right-hand-side of the inequality which concludes the proof. $\square$

Note that using Theorem 3.3 to compute the limiting behavior of $\psi ^\varepsilon$ relies on the continuity of the kernel $\mathfrak {K}_\varepsilon$ with respect to the parameter $\varepsilon$ in a weaker norm compared to $\Vert \cdot \Vert _{L^1(0,T)}$. This setting allows us to consider $\delta _0$ as a possible limit as well as to establish rates of convergence for the Mittag–Leffler kernels by exploiting their completely monotone properties; see, e.g., Proposition 3.5 for the usefulness of this result in obtaining a convergence rate.

The convergence rate of the solutions will depend on the form of the kernel $\mathfrak {K}_\varepsilon$ and its specific dependence on $\varepsilon$. We therefore treat the vanishing sound diffusivity and relaxation time limits separately.

3.5.1 The Vanishing Sound Diffusivity Limit

We first discuss the setting $\mathfrak {K}_\varepsilon =\varepsilon \mathfrak {K}$. Recall that two important examples of this class of kernels (up to a constant) are

$$\begin{aligned} \mathfrak {K}_\varepsilon = \varepsilon \mathfrak {K}\quad \text {with } \ \mathfrak {K}(t)= \left\{ \begin{aligned}&\tau _{\theta }^{-\alpha } g_{\alpha }(t)\\&\text {or} \\&\left( \frac{\tau _{\theta }}{\tau }\right) ^{a-b}\frac{1}{\tau ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\tau }\right) ^a\right) \, \text {with } \tau \text { fixed} \end{aligned} \right. \end{aligned}$$

where the Abel kernel $g_\alpha$ is defined in (1.5) for $\alpha \in (0,1)$ and $g_0= \delta _0$. The Mittag–Leffler kernel with $0< a,b\le 1$ is defined in (2.2). Here $\varepsilon$ holds the place of the sound diffusivity $\delta$.

Corollary 3.4

Under the assumptions of Theorems 3.2 and 3.3 with the kernel

$$\begin{aligned} \mathfrak {K}_\varepsilon = \varepsilon \mathfrak {K}, \quad \varepsilon \in (0, \bar{\varepsilon }) \end{aligned}$$

satisfying assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$), the family of solutions $\{\psi ^\varepsilon \}_{\varepsilon \in (0, \bar{\varepsilon })}$ of (3.1) converges in the energy norm to the solution $\psi$ of the inviscid Kuznetsov equation

$$\begin{aligned} \left\{ \begin{aligned}&(1+2k\psi _t)\psi _{tt}-c^2 \Delta \psi + 2\ell \, \nabla \psi \cdot \nabla \psi _t = 0 \quad{} & {} \text {in } \Omega \times (0,T), \\&\psi =0 \quad{} & {} \text {on } \partial \Omega \times (0,T),\\&(\psi , \psi _t)=(\psi _0, \psi _1), \quad{} & {} \text {in } \Omega \times \{0\}, \end{aligned} \right. \end{aligned}$$

at a linear rate

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E }\lesssim \varepsilon \quad \text{ as } \ \varepsilon \searrow 0. \end{aligned}$$

Proof

In this setting, the limiting kernel is $\mathfrak {K}_0=0$, and it satisfies assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$). By Theorem 3.3, we then immediately have

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E } \le C \Vert \mathfrak {K}_\varepsilon *1\Vert _{L^1(0,T)} = C \varepsilon \Vert \mathfrak {K}*1\Vert _{L^1(0,T)}, \end{aligned}$$

for some $C>0$, independent of $\varepsilon$, which concludes the proof. $\square$

3.5.2 The Vanishing Thermal Relaxation Time Limit with Mittag–Leffler Kernels

We now turn our attention to the kernels that were motivated by the presence of thermal relaxation in the heat flux laws of the propagation medium, and have the form

$$\begin{aligned} \mathfrak {K}_\varepsilon (t)=\,\delta \left( \frac{\tau _{\theta }}{\varepsilon }\right) ^{a-b}\frac{1}{\varepsilon ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\varepsilon }\right) ^a\right) , \quad a,b \in (0,1]. \end{aligned}$$

In what follows, we intend to take the limit $\varepsilon \searrow 0$, while keeping $\tau _{\theta }>0$ fixed. Unlike our work in [3], on the fractionally damped Westervelt equation, we limit ourselves here to the case $a-b \le 0$ (since, to the best of our knowledge, (${\textbf{A}}_{2}$) does not hold uniformly in $\varepsilon$ when $a-b>0$). We will then prove that solutions $\psi ^\varepsilon$ of (3.1) converge to the solution $\psi$ of the following time-fractional equation:

$$\begin{aligned} (1+2{k} \psi _t)\psi _{tt}-c^2 \Delta \psi - \tau _{\theta }^{a-b} {D _t^{a-b+1}\Delta u} + 2\ell \,\nabla \psi \cdot \nabla \psi _t =f, \end{aligned}$$

supplemented by the same boundary and initial conditions as in (3.1). Recall that

$$\begin{aligned} D _t^{a-b+1}\Delta \psi = g_{b-a} *\Delta \psi _t. \end{aligned}$$

Note also that in case $a=b$, the limiting equation is strongly damped.

Proposition 3.5

Let $\tau _{\theta }>0$ and $\delta >0$ be fixed, and let the assumptions of Theorems 3.2 and 3.3 hold. Consider the family of solutions $\{\psi ^\varepsilon \}_{\varepsilon \in (0,\bar{\varepsilon })}$ of (3.1) with the kernel given by

$$\begin{aligned} \mathfrak {K}_\varepsilon (t)=\,\delta \left( \frac{\tau _{\theta }}{\varepsilon }\right) ^{a-b}\frac{1}{\varepsilon ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\varepsilon }\right) ^a\right) \quad \text {where } \, 0 < a \le b\le 1. \end{aligned}$$

Then the family $\{\psi ^\varepsilon \}_{\varepsilon \in (0,\bar{\varepsilon })}$ converges to the solution $\psi$ of

$$\begin{aligned} \left\{ \begin{aligned}&(1+2{k} \psi _t)\psi _{tt}-c^2 \Delta \psi - \mathfrak {K}_0*\Delta \psi _t + 2\ell \,\nabla \psi \cdot \nabla \psi _t = 0\quad{} & {} \text {in } \Omega \times (0,T), \\&\psi =0 \quad{} & {} \text {on } \partial \Omega \times (0,T),\\&(\psi , \psi _t)=(\psi _0, \psi _1), \quad{} & {} \text {in } \Omega \times \{0\}, \end{aligned} \right. \end{aligned}$$

with the kernel $\mathfrak {K}_0= \delta \tau _{\theta }^{a-b} g_{b-a}$ at the following rate:

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E }\lesssim \Vert (\mathfrak {K}_\varepsilon -\mathfrak {K}_0)*1\Vert _{L^1(0,T)} \sim \varepsilon ^a \quad \text{ as } \ \varepsilon \searrow 0. \end{aligned}$$

Proof

By Theorem 3.3 ($\mathfrak {K}_0$ satisfies (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$)), we have

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E }\le C \Vert (\mathfrak {K}_{\varepsilon }-\mathfrak {K}_{0})*1\Vert _{L^1(0,T)} \end{aligned}$$

To further establish the asymptotic behavior of the right-hand side as $\varepsilon \searrow 0$, we rely on the following asymptotic behavior established in [3, Proposition 4.1]:

$$\begin{aligned} \begin{aligned} \Vert (\mathfrak {K}_\varepsilon -\mathfrak {K}_0)*1\Vert _{L^1(0,T)}=&\, \delta T^{1+b-a}E_{a,2+b-a}\left( -\left( \frac{T}{\varepsilon }\right) ^a\right) \\ \sim&\, \delta \frac{T^{1+b-a}}{\Gamma (2+b-2a)} \left( \frac{T}{\varepsilon }\right) ^{-a} \qquad \text { as }\ \frac{T}{\varepsilon }\rightarrow \infty , \end{aligned} \end{aligned}$$

which yields the claimed rate of convergence when $0 < a \le b\le 1$. $\square$

4 Differences in the Analysis of the Nonlocal Blackstock Equation

We now turn our attention to the limiting behavior of the Blackstock equation and consider the following initial-boundary value problem:

$$\begin{aligned} \left\{ \begin{aligned}&\psi _{tt}^\varepsilon -c^2 (1-2k\psi _{t}^\varepsilon )\Delta \psi ^\varepsilon - \mathfrak {K}_\varepsilon * \Delta \psi _{t}^\varepsilon + 2 \ell \,\nabla \psi ^\varepsilon \cdot \nabla \psi _{t}^\varepsilon = 0 \quad{} & {} \text {in } \Omega \times (0,T), \\&\psi ^\varepsilon =0 \quad{} & {} \text {on } \partial \Omega \times (0,T),\\&(\psi ^\varepsilon , \psi _{t}^\varepsilon )=(\psi _0, \psi _1), \quad{} & {} \text {in } \Omega \times \{0\}. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(4.1)

We focus in this section on pointing out the differences in treating the Blackstock nonlinearity compared Kuznetsov’s. First off, the coercivity assumption (${\textbf{A}}^{\textbf {K}}_{3}$) can be weakened.

4.1 Blackstock-Specific Assumption on the Memory Kernel

In the absence of a nonlinearity involving $\psi ^\varepsilon _{tt}$, a weaker coercivity assumption on the kernel compared to what we had in (${\textbf{A}}^{\textbf {K}}_{3}$) suffices.

Since (${\textbf{A}}^{{\textbf {B}}}_{3}$) assumption is weaker than (${\textbf{A}}^{\textbf {K}}_{3}$), we refer to Sect. 3.1 and [3, Section 5.2] for discussions on the verification of this assumption. Assumption (${\textbf{A}}^{{\textbf {B}}}_{3}$) is in particular verified for all the kernels considered in this work.

4.2 Uniform Well-Posedness of the Nonlocal Blackstock Equation

The main result to prove in order to conduct the limiting behavior analysis is to show uniform well-posedness of (4.1) with respect to $\varepsilon$. The strategy will rely on studying a linearized problem combined with a Banach fixed-point argument, as before. However, here a different linearization is needed to tackle the Blackstock nonlinearity, namely

$$\begin{aligned} \begin{aligned} \psi ^\varepsilon _{tt}-c^2 \mathfrak {n}\Delta \psi ^\varepsilon - \mathfrak {K}_\varepsilon * \Delta \psi ^\varepsilon _{t}+ \nabla \mathfrak {l}\ \cdot \nabla \psi ^\varepsilon _t=0 \ \text {in }\Omega \times (0,T), \end{aligned} \end{aligned}$$

(4.2a)

with variable coefficients

$$\begin{aligned} \mathfrak {n}=1-2k\phi _t, \qquad \mathfrak {l}= 2 \ell \phi , \qquad k, \ell \in \mathbb {R}, \end{aligned}$$

supplemented by the initial and boundary conditions:

$$\begin{aligned} (\psi ^\varepsilon , \psi ^\varepsilon _t) \vert _{t=0}=\,(\psi _0, \psi _1), \qquad \psi ^\varepsilon \vert _{\partial \Omega }=0. \end{aligned}$$

(4.2b)

4.2.1 Uniform Well-Posedness of a Linearized Blackstock Equation with Variable Coefficients

Compared to Proposition 3.1, we need less regularity-in-time from our solution. This can be explained by the fact that the term $\psi _{tt}^\varepsilon$ appears linearly in the Blackstock equation. This leads us to consider a different space of solutions and bootstrap argument.

Proposition 4.1

Let $\varepsilon \in (0, \bar{\varepsilon })$ and k, $\ell \in \mathbb {R}$. Given $T>0$, let $\phi \in X_\phi$ and assume that there exist $\overline{\mathfrak {n}}$ and $\underline{\mathfrak {n}}$, independent of $\varepsilon$, such that

$$\begin{aligned} 0<\underline{\mathfrak {n}} \le \mathfrak {n}(\phi )=1-2k\phi _t(x,t)\le \overline{\mathfrak {n}} \quad \text {a.e. in } \ \Omega \times (0,T). \end{aligned}$$

Let assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{{\textbf {B}}}_{3}$) on the kernel hold. Furthermore, assume that the initial conditions satisfy

$$\begin{aligned} (\psi _0, \psi _1) \in {\left\{ \begin{array}{ll} {H_\diamondsuit ^4(\Omega )}\times {H_\diamondsuit ^3(\Omega )}\ \text {if } \mathfrak {K}_\varepsilon \equiv \varepsilon \delta _0 \ \text {or } \mathfrak {K}_\varepsilon \equiv 0,\\ {H_\diamondsuit ^4(\Omega )}\times {H_\diamondsuit ^4(\Omega )}\ \text {if } \mathfrak {K}_\varepsilon \not \equiv 0 \in L^1(0,T). \end{array}\right. } \end{aligned}$$

Then there exists $\Xi ^{\textbf {B}} >0$, independent of $\varepsilon$, such that if

$$\begin{aligned} \Vert \phi \Vert _{L^1(H^4(\Omega ))} + \Vert \phi _{t}\Vert _{L^1(H^3(\Omega ))} + \Vert \phi _{tt}\Vert _{L^2(H^2(\Omega ))} \le \Xi ^{\textbf {B}} , \end{aligned}$$

and if the final time $T=T(\Vert \phi \Vert _{X_\phi })$ is small enough, then there is a unique solution $\psi ^\varepsilon$ of (4.2a), (4.2b) in

$$\begin{aligned} X_{\psi }^{\textbf {B}} = \, W^{2,\infty }(0,T;{H_\diamondsuit ^2(\Omega )})\cap W^{1,\infty }(0,T;{H_\diamondsuit ^3(\Omega )}) \cap L^2(0,T;{H_\diamondsuit ^4(\Omega )}). \end{aligned}$$

(4.3)

This solution satisfies the estimate

$$\begin{aligned} \Vert \psi ^\varepsilon \Vert ^2_{X_\psi ^{\textbf {B}}} \le C_{lin }^{\textbf {B}}(T)\Bigl (\Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )}+ \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)} ^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )} {\Bigr )}.\qquad \end{aligned}$$

(4.4)

where $\eta = 0$ if $\mathfrak {K}_\varepsilon \in \{ 0, \varepsilon \delta _0 \}$ and $\eta =1$ otherwise.

Proof

Similarly to the approach in the proof of Proposition 3.1, we use a Galerkin procedure in a suitable Hilbert space and consider a time-differentiated semi-discrete problem. We set $p=\psi ^\varepsilon _t$, where, again, the dependence of p on $\varepsilon$ is omitted to simplify the notation. The time-differentiated semi-discrete nonlocal Blackstock equation is then given by

$$\begin{aligned} p_{tt}-c^2 \mathfrak {n}\Delta p - \mathfrak {K}_\varepsilon * \Delta p_{t}= F^{\textbf {B}}\end{aligned}$$

(4.5)

with the right-hand side

$$\begin{aligned} F^{\textbf {B}}(t)= -\nabla \mathfrak {l}_t \cdot \nabla p-\nabla \mathfrak {l}\cdot \nabla p_t +c^2 \mathfrak {n}_t\Delta \psi ^\varepsilon + \eta \mathfrak {K}_\varepsilon (t) \Delta p(0). \end{aligned}$$

(4.6)

Similarly to the proof of Proposition 3.1, we test this equation with $(-\Delta )^2p_t$ and integrate over space and (0, t), where now we can rely on assumption (${\textbf{A}}^{{\textbf {B}}}_{3}$) to conclude that

$$\begin{aligned} \int _0^t\bigl ((\mathfrak {K}_\varepsilon *\nabla \Delta p_t)(s), \nabla \Delta p_t(s)\bigr )_{L^2(\Omega )}\, d s \ge 0 . \end{aligned}$$

(4.7)

Most of the estimates remain the same, we therefore simply point out how to deal with the $\mathfrak {n}$ terms arising after integration by parts in (3.11) and (3.21). We first note that, when $\mathfrak {n}\ne 1$, (3.10) generalizes to

$$\begin{aligned}{} & {} -c^2(\mathfrak {n}\Delta p, (-\Delta )^2 p_{t})_{L^2}\\{} & {} \quad =\, \frac{c^2}{2}\frac{d }{d t}(\mathfrak {n}\nabla \Delta p, \nabla \Delta p)_{L^2} -\frac{c^2}{2} (\mathfrak {n}_t \nabla \Delta p, \nabla \Delta p)_{L^2}\\{} & {} \qquad -c^2(\nabla \Delta p \cdot \nabla \mathfrak {n}+\Delta p \Delta \mathfrak {n}, \Delta p_t)_{L^2}. \end{aligned}$$

Similarly to before, by the time-differentiated semi-discrete PDE (4.5), we have $F^{\textbf {B}}=0$ on the boundary and thus $(F^{\textbf {B}}, (-\Delta )^2p_t)_{L^2}=(\Delta F^{\textbf {B}}, \Delta p_t)_{L^2}$. Testing (4.5) with $(-\Delta )^2 p_t$ and utilizing (4.7) therefore yields

$$\begin{aligned}{} & {} \frac{1}{2} \Vert \Delta p_t\Vert _{L^2(\Omega )}^2 \big \vert _0^t +\frac{c^2}{2} \Vert \sqrt{\mathfrak {n}}\nabla \Delta p\Vert _{L^2(\Omega )}^2\big \vert _0^t \nonumber \\{} & {} \quad \le \, \int _0^t\Bigl (\frac{c^2}{2}(\mathfrak {n}_t\nabla \Delta p,\nabla \Delta p)_{L^2} +(c^2\nabla \Delta p\cdot \nabla \mathfrak {n}+ c^2\Delta p\, \Delta \mathfrak {n}+\Delta F^{\textbf {B}},\Delta p_t)_{L^2} \Bigr )\, d s .\nonumber \\ \end{aligned}$$

(4.8)

The terms on the right-hand side of (4.8) can be estimated as follows. We first have

$$\begin{aligned}\int _0^t \frac{c^2}{2}(\mathfrak {n}_t\nabla \Delta p,\nabla \Delta p)_{L^2} \, d s \lesssim \, \Xi ^{\textbf {B}} \Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))}^2. \end{aligned}$$

Additionally,

$$\begin{aligned}{} & {} \int _0^t (c^2\nabla \Delta p\cdot \nabla \mathfrak {n}+ c^2\Delta p\, \Delta \mathfrak {n},\Delta p_t)_{L^2} \, d s \\{} & {} \quad \le c^2 \Vert \nabla \Delta p\cdot \nabla \mathfrak {n}+\Delta p\, \Delta \mathfrak {n}\Vert _{L^1(L^2(\Omega ))} \Vert \Delta p_t\Vert _{L^{\infty }_t(L^2(\Omega ))}. \end{aligned}$$

The first term on the right in the above bound can be further estimated as follows:

$$\begin{aligned} \begin{aligned}&\Vert \nabla \Delta p\cdot \nabla \mathfrak {n}+\Delta p\, \Delta \mathfrak {n}\Vert _{L^1(L^2(\Omega ))}\\&\quad \lesssim \, \Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))}\Vert \nabla \mathfrak {n}\Vert _{L^1(L^\infty (\Omega ))} +\Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))}\Vert \Delta \mathfrak {n}\Vert _{L^1(L^3(\Omega ))}\\&\quad \lesssim \, \Xi ^{\textbf {B}}(\Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))} +\Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))}) \end{aligned} \end{aligned}$$

since $\Delta p=0$ on $\partial \Omega$. It thus remains to estimate the term $\int _0^t (\Delta F^{\textbf {B}},\Delta p_t)_{L^2} \, d s .$ Recalling how $F^{\textbf {B}}$ is defined in (4.6), we have the following identity:

$$\begin{aligned} \Delta F^{\textbf {B}}= & {} \, c^2 \Delta \mathfrak {n}_t\Delta \psi ^\varepsilon + 2c^2 \nabla \mathfrak {n}_t\cdot \nabla \Delta \psi ^\varepsilon +c^2 \mathfrak {n}_t\Delta ^2 \psi ^\varepsilon \\{} & {} +\Delta (-\nabla \mathfrak {l}_t \cdot \nabla p-\nabla \mathfrak {l}\cdot \nabla p_{t} )+ \eta \mathfrak {K}_\varepsilon (t) \Delta ^2 p(0), \end{aligned}$$

We use the same splitting approach as in the proof of Proposition 3.1

$$\begin{aligned} \int _0^t (\Delta F^{\textbf {B}},\Delta p_t)_{L^2} \, d s \le \, \Vert \Delta F^{\textbf {B}}+g\Vert _{L^1(L^2(\Omega ))} \Vert \Delta p_t\Vert _{L^{\infty }_t(L^2(\Omega ))} +\left| \int _0^t (g,\Delta p_t)_{L^2} \, d s \right| , \end{aligned}$$

with $g=\nabla \mathfrak {l}\cdot \nabla \Delta p_t$. Then, since $\Delta p_t=0$ on $\partial \Omega$, we can estimate (see inequality (3.16))

$$\begin{aligned} \left| \int _0^t (g,\Delta p_t)_{L^2} \, d s \right| \lesssim \;\Xi ^{\textbf {B}} \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2. \end{aligned}$$

We further have

$$\begin{aligned}{} & {} \Vert \Delta F^{\textbf {B}}+g\Vert _{L^1(L^2(\Omega ))}\nonumber \\{} & {} \quad \lesssim \, \Vert \Delta \mathfrak {n}_t\Vert _{L^1(L^2(\Omega ))} {\Vert \Delta \psi ^\varepsilon \Vert _{L^{\infty }_t(L^\infty (\Omega ))}} +\Vert \nabla \mathfrak {n}_t\Vert _{L^1(L^3(\Omega ))} {\Vert \nabla \Delta \psi ^\varepsilon \Vert _{L^{\infty }_t(L^6(\Omega ))}} \nonumber \\{} & {} \qquad +\Vert \mathfrak {n}_t\Vert _{L^2(L^\infty (\Omega ))} \Vert \Delta ^2 \psi ^\varepsilon \Vert _{L^{2}(L^2(\Omega ))} +\Vert \nabla \Delta \mathfrak {l}_t \cdot \nabla p \Vert _{L^1(L^2(\Omega ))}\nonumber \\{} & {} \qquad +\Vert \nabla \mathfrak {l}_t \cdot \nabla \Delta p \Vert _{L^1_t(L^2(\Omega ))}+\Vert D^2 \mathfrak {l}_t: D^2 p \Vert _{L^1_t(L^2(\Omega ))}\nonumber \\{} & {} \qquad +\Vert \nabla \Delta \mathfrak {l}\cdot \nabla p_t \Vert _{L^1_t(L^2(\Omega ))} +\Vert D^2 \mathfrak {l}: D^2 p_t \Vert _{L^1_t(L^2(\Omega ))}\nonumber \\{} & {} \qquad + \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,t)} \Vert \Delta ^2 \psi _1\Vert _{L^2(\Omega )}. \end{aligned}$$

(4.9)

The $\mathfrak {l}$-terms are then estimated similarly to before (see (3.17), (3.18), and (3.19)) to obtain

$$\begin{aligned}{} & {} \int _0^t (\Delta F^{\textbf {B}},\Delta p_t)_{L^2} \, d s \nonumber \\{} & {} \quad \lesssim \, \Xi ^{\textbf {B}} \left( \Vert \Delta p_t\Vert _{L^\infty _t(L^2(\Omega ))}^2+\Vert \nabla \Delta p \Vert ^2_{L^\infty _t(L^2(\Omega ))} \right) \nonumber \\{} & {} \qquad +\Vert \mathfrak {n}_t\Vert _{L^2(H^2(\Omega ))} {\Vert \Delta ^2 \psi ^\varepsilon \Vert _{L^{2}(L^2(\Omega ))}} + \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,t)} \Vert \Delta ^2 \psi _1\Vert _{L^2(\Omega )}. \end{aligned}$$

(4.10)

Bootstrap argument for $\psi ^\varepsilon \in L^2(0,T;{H_\diamondsuit ^4(\Omega )})$. Similarly to Proposition 3.1, if $\ell \not =0$, but now also if $\mathfrak {n}_t\not =0$, we have to estimate the $\Delta ^2 \psi ^\varepsilon$ term in (4.10). To this end, we test

$$\begin{aligned} {[}\mathfrak {K}_\varepsilon *\partial _t+\mathfrak {n}c^2]\Delta \psi ^\varepsilon =r:= p_t+\nabla \mathfrak {l}\cdot \nabla p \end{aligned}$$

with $\Delta ^3\psi ^\varepsilon$ and integrating by parts, due to assumption (${\textbf{A}}_{2}$), we have

$$\begin{aligned} \begin{aligned}&c^2\Vert \sqrt{\mathfrak {n}}\Delta ^2\psi ^\varepsilon \Vert _{L^2_t(L^2(\Omega ))}^2\\&\quad \le \, \int _0^t\Bigl \{(\Delta r,\Delta ^2\psi ^\varepsilon )_{L^2}-c^2\Delta [\mathfrak {n}\Delta \psi ^\varepsilon ],\Delta ^2\psi ^\varepsilon )_{L^2}\Bigr \}\, d s + C_{{\textbf {A}}_2}\Vert \Delta ^2 \psi _0\Vert ^2_{L^2(\Omega )}\\&\quad \le \Vert \Delta ^2\psi ^\varepsilon \Vert _{L^2_t(L^2(\Omega ))} \left\| \Delta p_t-\Delta [\nabla \mathfrak {l}\cdot \nabla p]-c^2\Delta \mathfrak {n}\Delta \psi ^\varepsilon -c^2\nabla \mathfrak {n}\nabla \Delta \psi ^\varepsilon \right\| _{L^2_t(L^2(\Omega ))} \\&\qquad + C_{{\textbf {A}}_2}\Vert \psi _0\Vert ^2_{H^4(\Omega )}, \end{aligned} \end{aligned}$$

where we have used the fact that r and $\Delta ^2 \psi ^\varepsilon$ vanish on the boundary. Contrary to the proof of Proposition 3.1, here we do not have access to an intermediate estimate (cf. (3.20)), therefore we don’t know a priori if $\Delta p_t-\Delta [\nabla \mathfrak {l}\cdot \nabla p]-c^2\Delta \mathfrak {n}\Delta \psi ^\varepsilon -c^2\nabla \mathfrak {n}\nabla \Delta \psi ^\varepsilon \in L^2(0,T;L^2(\Omega ))$. Instead, we use the following inequalities to estimate the corresponding $L^2(0,T; L^2(\Omega ))$-norm:

$$\begin{aligned} \Vert \Delta [\nabla \mathfrak {l}\cdot \nabla p]\Vert _{L_t^2(L^2(\Omega ))} \lesssim&\; \Vert \mathfrak {l}\Vert _{L_t^2(H^3(\Omega ))} \Vert \nabla \Delta p\Vert _{L_t^\infty (L^2(\Omega ))}, \end{aligned}$$

together with

$$\begin{aligned} \Vert \Delta \mathfrak {n}\Delta \psi ^\varepsilon \Vert _{L_t^2(L^2(\Omega ))}\lesssim & {} \; \Vert \mathfrak {n}\Vert _{L^2_t(H^3(\Omega ))} \Vert \nabla \Delta \psi ^\varepsilon \Vert _{L^\infty _t(L^2(\Omega ))} \nonumber \\\lesssim & {} \; \Vert \mathfrak {n}\Vert _{L^2_t(H^3(\Omega ))} (\Vert \nabla \Delta p\Vert _{L^1_t(L^2(\Omega ))} + \Vert \nabla \Delta \psi _0\Vert _{H^3(\Omega )})\nonumber \\\lesssim & {} \; \Vert \mathfrak {n}\Vert _{L^2_t(H^3(\Omega ))} (T\Vert \nabla \Delta p\Vert _{L^\infty _t(L^2(\Omega ))} + \Vert \nabla \Delta \psi _0\Vert _{H^3(\Omega )}),\nonumber \\ \end{aligned}$$

(4.11)

and, since $\Delta \psi \vert _{\partial \Omega }=0$, by elliptic regularity

$$\begin{aligned} \Vert \nabla \mathfrak {n}\nabla \Delta \psi ^\varepsilon \Vert _{L_t^2(L^2(\Omega ))}\lesssim & {} \; \Vert \mathfrak {n}\Vert _{L_t^\infty (H^3(\Omega ))} \Vert \nabla \Delta \psi ^\varepsilon \Vert _{L_t^2(L^2(\Omega ))}\\\lesssim & {} \; \Vert \mathfrak {n}\Vert _{L_t^\infty (H^3(\Omega ))} \Vert \Delta ^2\psi ^\varepsilon \Vert _{L_t^2(L^2(\Omega ))}. \end{aligned}$$

We then rely on the smallness of $\Vert \mathfrak {n}_t\Vert _{L^2(L^\infty (\Omega ))} \le \Xi ^{\textbf {B}}$ (and eventually of final time T), which multiplies $\Vert \Delta ^2\psi ^\varepsilon \Vert _{L^2(L^2(\Omega ))}$ in (4.9), to absorb the arising terms and obtain (4.4). This yields the desired estimates. $\square$

Note that in Proposition 4.1, the need of small T arises from estimate (4.11). This requirement can be alleviated by establishing additional estimates by testing (4.2a) by $\Delta ^2\psi _{t}^\varepsilon$. To avoid increased technicality, and since our goal is ultimately to establish the well-posedness of the nonlinear equation (4.1), we do not pursue this refinement.

Similarly to the Kuznetsov case, we can establish a uniform lower-order estimate for (4.2). Under the assumption of Proposition 4.1, testing (4.2) with $\psi _{t}$ and integrating over space and time yields, after usual manipulations,

$$\begin{aligned}{} & {} \frac{1}{2} \left\{ \Vert \psi _{t}^\varepsilon (t)\Vert ^2_{L^2}+c^2\Vert \sqrt{\mathfrak {n}} \nabla \psi ^\varepsilon (t)\Vert ^2_{L^2} \right\} \Big \vert _0^t \le \, -c^2 \int _0^t(\psi ^\varepsilon \nabla \mathfrak {n}, \nabla \psi _{t}^\varepsilon )_{L^2}\, d s \nonumber \\{} & {} \quad +\frac{1}{2} c^2\Vert \mathfrak {n}_t\Vert _{L^1(L^\infty (\Omega ))}\Vert \nabla \psi ^\varepsilon \Vert ^2_{L^\infty (0,t;L^2(\Omega ))} - \int _0^t( \nabla \mathfrak {l}\cdot \nabla \psi _{t}^\varepsilon , \psi _{t}^\varepsilon )\, d s . \end{aligned}$$

(4.12)

To bound the first term on the right, we use integration by parts in time and Hölder’s inequality to obtain

$$\begin{aligned} -c^2 \int _0^t(\psi ^\varepsilon \nabla \mathfrak {n}, \nabla \psi _{t}^\varepsilon )_{L^2}\, d s\lesssim & {} \, \Vert \psi ^\varepsilon \Vert _{L^\infty _t(L^4(\Omega ))}\Vert \nabla \mathfrak {n}\Vert _{L^\infty (L^4(\Omega ))} \Vert \nabla \psi ^\varepsilon \Vert _{L^\infty _t(L^2(\Omega ))}\\{} & {} + \Vert \psi _{t}^\varepsilon \Vert _{L^\infty _t(L^2(\Omega ))}\Vert \nabla \mathfrak {n}\Vert _{L^1(L^\infty (\Omega ))} \Vert \nabla \psi ^\varepsilon \Vert _{L^\infty _t(L^2(\Omega ))}\\{} & {} + \Vert \psi ^\varepsilon \Vert _{L^\infty _t(L^6(\Omega ))}\Vert \nabla \mathfrak {n}_t\Vert _{L^1(L^3(\Omega ))} \Vert \nabla \psi ^\varepsilon \Vert _{L^\infty _t(L^2(\Omega ))}. \end{aligned}$$

The last term is treated using estimate (3.22). Note that, due to the embedding $W^{1,1}(0,T)\hookrightarrow L^\infty (0,T)$, we have

$$\begin{aligned} \Vert \nabla \mathfrak {n}\Vert _{L^\infty (L^4(\Omega ))} \lesssim \Vert \phi _{t}\Vert _{L^1(H^2(\Omega ))} + \Vert \phi _{tt}\Vert _{L^2(H^2(\Omega ))} \lesssim \Xi ^{\textbf {B}}, \end{aligned}$$

and

$$\begin{aligned}&\Vert \nabla \mathfrak {n}\Vert _{L^1(L^\infty (\Omega ))} \lesssim \Vert \phi _t\Vert _{L^1(H^3(\Omega ))} \lesssim \Xi ^{\textbf {B}}, \\&\Vert \nabla \mathfrak {n}_t\Vert _{L^1(L^3(\Omega ))} \lesssim \Vert \phi _{tt}\Vert _{L^2(H^2(\Omega ))} \lesssim \Xi ^{\textbf {B}}. \end{aligned}$$

Thus, utilizing the above estimates and the smallness of $\Xi ^{\textbf {B}}$ (and eventually final time T), leads to the uniform bound in $\varepsilon$:

$$\begin{aligned} \Vert \psi _{t}^\varepsilon (t)\Vert ^2_{L^2(\Omega )} + \Vert \nabla \psi ^\varepsilon (t)\Vert ^2_{L^2(\Omega )} \lesssim \Vert \psi _1\Vert ^2_{L^2(\Omega )} + \Vert \nabla \psi _0\Vert ^2_{L^2(\Omega )} \end{aligned}$$

a.e. in time. The next theorems and propositions are extensions of the results of Sect. 3 to the Blackstock equation setting.

4.3 Well-Posedness and Limiting Behavior of the Nonlocal Blackstock Equation

The well-posedness of the nonlinear initial-boundary value problem 4.1 relies again on setting up a fixed-point mapping $\mathcal {T}^{\textbf {B}}:\phi \mapsto \psi ^\varepsilon$, where $\phi$ will belong to a ball in a suitable Bochner space and $\psi ^\varepsilon$ solves the associated linearized problem (4.2). The proof is similar to that of Theorem 3.2 so we omit it here.

Theorem 4.2

Let $\varepsilon \in (0, \bar{\varepsilon })$ and k, $\ell \in \mathbb {R}$. Furthermore, let $(\psi _0, \psi _1)$ be as in Proposition 4.1 and such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )} \le r^2, \qquad \text {if } \mathfrak {K}_\varepsilon = \varepsilon \delta _0 \ \text {or } \mathfrak {K}_\varepsilon \equiv 0, \\ \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^4(\Omega )} \le r^2, \qquad \text {if } \mathfrak {K}_\varepsilon \not \equiv 0 \in L^1(0,T). \end{array}\right. } \end{aligned}$$

where r does not depend on $\varepsilon$. Let assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{{\textbf {B}}}_{3}$) on the kernel hold. Then, there exist a data size $r_0=r_0(r)>0$, and final time $T=T(r)>0$, both independent of $\varepsilon$, such that if

$$\begin{aligned} \Vert \psi _0\Vert ^2_{H^1(\Omega )}+\Vert \psi _1\Vert ^2_{L^2(\Omega )} \le r_0^2, \end{aligned}$$

then there is a unique solution $\psi$ in $X_\psi ^{\textbf {B}}$ (defined in (4.3)) of equation (4.1), which satisfies the following estimate:

$$\begin{aligned} \Vert \psi ^\varepsilon \Vert ^2_{X_\psi ^{\textbf {B}}} \le \, C_{nonlin }^{\textbf {B}}\left( \, \Vert \psi _0\Vert ^2_{H^4(\Omega )}+\Vert \psi _1\Vert ^2_{H^3(\Omega )}+ \eta \Vert \mathfrak {K}_\varepsilon \Vert _{L^1(0,T)} ^2\Vert \Delta ^2 \psi _1\Vert ^2_{L^2(\Omega )}\right) . \end{aligned}$$

where $\eta = 0$ if $\mathfrak {K}_\varepsilon \in \{ 0, \varepsilon \delta _0 \}$ and $\eta =1$ otherwise. Here, $C_{nonlin }^{\textbf {B}}=C_{nonlin }^{\textbf {B}}(\Omega ,\overline{\phi }, T)$ does not depend on the parameter $\varepsilon$.

To determine the limiting behavior of solutions to (4.1) as $\varepsilon \searrow 0$, we state an analogous result to Theorem 3.3 on the continuity of the solution with respect to the memory kernel.

Theorem 4.3

Let $\varepsilon _1, \varepsilon _2 \in (0, \bar{\varepsilon })$. Under the assumptions of Theorem 4.2, for sufficiently small T, the following estimate holds:

$$\begin{aligned} \Vert \psi ^{\varepsilon _1}-\psi ^{\varepsilon _2}\Vert _{E }\lesssim \Vert (\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})*1\Vert _{L^1(0,T)}, \end{aligned}$$

where $\Vert \cdot \Vert _{E }$ is defined in (1.4).

Proof

The difference $\overline{\psi }=\psi ^{\varepsilon _1}-\psi ^{\varepsilon _2}$ solves

$$\begin{aligned}{} & {} \overline{\psi }_{tt}-c^2 (1 - 2 k\psi _t^{\varepsilon _1} ) \Delta \overline{\psi }- \mathfrak {K}_{\varepsilon _1}* \Delta \overline{\psi }_{t}\nonumber \\{} & {} \quad =\, - 2k\overline{\psi }_t \Delta \psi ^{\varepsilon _2}- 2\ell \left( \nabla \psi ^{\varepsilon _1} \cdot \nabla \overline{\psi }_t +\nabla \overline{\psi }\cdot \nabla \psi ^{\varepsilon _2}_t \right) +(\mathfrak {K}_{\varepsilon _1}-\mathfrak {K}_{\varepsilon _2})* \Delta \psi _t^{\varepsilon _2}\nonumber \\ \end{aligned}$$

(4.13)

with zero boundary and initial conditions. As in the proof of Theorem 3.3, we test (4.13) by $\overline{\psi }_{t}$ and find ourselves in a similar setting to that of the lower-order estimate (4.12). We can then proceed similarly to the proof of Theorem 3.3 to treat the last two right-hand-side terms. The remaining term can be estimated as follows:

$$\begin{aligned} \left| -2k\int _0^t \int _{\Omega }\overline{\psi }_t^2 \Delta \psi ^{\varepsilon _2} \, d xd s\right| \lesssim \, \Vert \Delta \psi ^{\varepsilon _2}\Vert _{L^1(L^\infty (\Omega ))}\Vert \overline{\psi }_t\Vert _{L^\infty _t(L^2(\Omega ))}^2. \end{aligned}$$

Note that thanks to Theorem 4.2, we have a uniform bound on

$$\begin{aligned} \Vert \Delta \psi ^{\varepsilon _2}\Vert _{L^1(L^\infty (\Omega ))} \lesssim \Vert \Delta \psi ^{\varepsilon _2}\Vert _{L^1(H^2(\Omega ))}. \end{aligned}$$

We can thus proceed similarly to the proof of Theorem 3.3 to arrive at the desired statement statement. $\square$

With this continuity result for the nonlocal Blackstock equation in hand, we can state the following counterparts of Corollary 3.4 and Proposition 3.5.

Corollary 4.4

Under the assumptions of Theorems 4.2 and 4.3 with the kernel

$$\begin{aligned} \mathfrak {K}_\varepsilon = \varepsilon \mathfrak {K}, \quad \varepsilon \in (0, \bar{\varepsilon }) \end{aligned}$$

satisfying assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{{\textbf {B}}}_{3}$), the family of solutions $\{\psi ^\varepsilon \}_{\varepsilon \in (0, \bar{\varepsilon })}$ of (4.1) converges in the energy norm to the solution $\psi$ of the initial-boundary value problem for the inviscid Blackstock equation:

$$\begin{aligned} \left\{ \begin{aligned}&\psi _{tt}-c^2 (1-2k\psi _t)\Delta \psi + 2\ell \, \nabla \psi \cdot \nabla \psi _t = 0 \quad{} & {} \text {in } \Omega \times (0,T), \\&\psi =0 \quad{} & {} \text {on } \partial \Omega \times (0,T),\\&(\psi , \psi _t)=(\psi _0, \psi _1), \quad{} & {} \text {in } \Omega \times \{0\}, \end{aligned} \right. \end{aligned}$$

at a linear rate

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E }\lesssim \varepsilon \quad \text{ as } \ \varepsilon \searrow 0. \end{aligned}$$

Proof

Follows analogously to Corollary 3.4. The details are omitted. $\square$

The thermal relaxation time limit result for specific families of Mittag–Leffler kernels is given by the following result.

Proposition 4.5

Let $\tau _{\theta }>0$ and $\delta >0$ be fixed, and let the assumptions of Theorems 4.2 and 4.3 hold. Consider the family of solutions $\{\psi ^\varepsilon \}_{\varepsilon \in (0,\bar{\varepsilon })}$ of (3.1) with the kernel given by

$$\begin{aligned} \mathfrak {K}_\varepsilon (t)=\,\delta \left( \frac{\tau _{\theta }}{\varepsilon }\right) ^{a-b}\frac{1}{\varepsilon ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\varepsilon }\right) ^a\right) \quad \text {where } \, 0 < a \le b\le 1. \end{aligned}$$

Then, the family $\{\psi ^\varepsilon \}_{\varepsilon \in (0,\bar{\varepsilon })}$ converges to the solution $\psi$ of

$$\begin{aligned} \left\{ \begin{aligned}&\psi _{tt}-c^2 (1-2k\psi _t)\Delta \psi - \mathfrak {K}_0*\Delta \psi _t + 2\ell \,\nabla \psi \cdot \nabla \psi _t = 0\quad{} & {} \text {in } \Omega \times (0,T), \\&\psi =0 \quad{} & {} \text {on } \partial \Omega \times (0,T),\\&(\psi , \psi _t)=(\psi _0, \psi _1), \quad{} & {} \text {in } \Omega \times \{0\}, \end{aligned} \right. \end{aligned}$$

with the kernel $\mathfrak {K}_0= \delta \tau _{\theta }^{a-b} g_{b-a}$ at the following rate

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E }\lesssim \Vert (\mathfrak {K}_\varepsilon -\mathfrak {K}_0)*1\Vert _{L^1(0,T)} \sim \varepsilon ^a \quad \text{ as } \ \varepsilon \searrow 0. \end{aligned}$$

Proof

Follows analogously to Proposition 3.5. The details are omitted. $\square$

4.4 Comparison with the Westervelt Equation

In the context of nonlinear acoustics, the potential form Westervelt equation is obtained by setting $\ell =0$ and adjusting the value of the nonlinearity parameter $k$ in (1.1) (we denote the modified parameter by ):

(4.14)

In this setting, the results of Sect. 3 remain valid, but we can also obtain a bit more. Indeed, when $\ell =0$, we do not require (${\textbf{A}}_{2}$) as the bootstrap argument on page 14 is not needed. Then, under similar restrictions on final time, regularity and size of data to those of Theorem 3.2, we expect (4.14) to be well-posed in

$$\begin{aligned} W^{3,1}(0,T;H_0^1(\Omega )) \cap W^{2,1}(0,T;{H_\diamondsuit ^2(\Omega )}) \cap W^{1,\infty }(0,T;{H_\diamondsuit ^3(\Omega )}). \end{aligned}$$

It is possible to show, under the following additional assumption on $\tau _{\theta }$:

$$\begin{aligned} \tau _{\theta }= \tau _{\theta }(\varepsilon ) \quad \text {and } \left( \frac{\tau _{\theta }}{\varepsilon }\right) ^{a-b} = \rho ^{a-b} = \text {constant}, \end{aligned}$$

that the family of solutions $\{\psi ^\varepsilon \}_{\varepsilon \in (0,\bar{\varepsilon })}$ of (3.1) with the kernel given by

$$\begin{aligned} \mathfrak {K}_\varepsilon (t)=\,\delta \left( \frac{\tau _{\theta }}{\varepsilon }\right) ^{a-b}\frac{1}{\varepsilon ^b}t^{b-1}E_{a,b}\left( -\left( \frac{t}{\varepsilon }\right) ^a\right) , \qquad 0<b<a \le 1, \end{aligned}$$

converges in the energy norm ($\Vert \cdot \Vert _{E }$) to the solution $\psi$ of the inviscid problem:

at the following rate:

$$\begin{aligned} \Vert \psi ^\varepsilon -\psi \Vert _{E }\lesssim \Vert \mathfrak {K}_\varepsilon *1\Vert _{L^1(0,T)} \sim \varepsilon ^{a-b} \quad \text{ as } \ \varepsilon \searrow 0. \end{aligned}$$

This result is comparable to the one established in [3, Proposition 4.2] where the pressure form of the Westervelt equation (see (1.3)) was analyzed. We note, however, that the regularity requirements on initial data are higher here since the quasilinear coefficient contains a time derivative ($1+2k\psi _{t}^\varepsilon$ instead of $1+2\tilde{k}u^\varepsilon$), thus requiring higher-order energy arguments to control it.

5 Discussion

In this work, we have investigated the nonlinear Kuznetsov and Blackstock equations with a general nonlocal dissipation term which encompasses the case of fractional-in-time damping. We rigorously studied their $\varepsilon$-uniform local well-posedness and their limiting behavior with respect to the parameter $\varepsilon$. As we have seen, the limiting behavior and rate of convergence is influenced by the dependence of the memory kernel $\mathfrak {K}_\varepsilon$ on $\varepsilon$. In particular, we have established limiting results for equations involving a class of Abel and Mittag–Leffler kernels. The difference between the Kuznetsov and Blackstock equations stems from the position of the nonlinearity which influences the manipulation of the equations and thus which terms need to be controlled. In particular, because the term $\psi _{tt}^\varepsilon$ appears only linearly in the Blackstock equation, its solution spaces need not be as regular in time as Kuznetsov’s and the requirements on the kernel are weaker as well. However, the convergence rate to their respective limiting behavior is qualitatively similar for both equations.

The framework developed in this work and leading to Theorem 3.3 and Theorem 4.3 allows extending the limiting study to other parameters of interest, as long as the parameter-dependent kernels satisfy assumptions (${\textbf{A}}_{1}$), (${\textbf{A}}_{2}$), and (${\textbf{A}}^{\textbf {K}}_{3}$) (when considering Kuznetsov’s nonlinearities) or the weaker (${\textbf{A}}^{{\textbf {B}}}_{3}$) (when interested in Blackstock’s). Among others, the limit as the fractional order $\alpha \nearrow 1$ in the Abel kernels can be studied.

Solvability of the Semi-discrete Problem

We provide here details on the Galerkin procedure regarding the existence of a unique semi-discrete approximation of the nonlocal Kuznetsov equation used in Sect. 3. For ease of notation, we drop the superscript $\varepsilon$ below when denoting the approximate solution $\psi ^n$ of the problem. We approximate the exact solution by

$$\begin{aligned} \psi ^n(x,t) =\sum _{i=1}^n \xi ^n_i(t)v_i(x), \end{aligned}$$

where $\{v_i\}_{i=1}^\infty$ are smooth eigenfunctions of the Dirichlet-Laplacian operator. With $V_n=span \{v_1, \ldots , v_n\}$, the semi-discrete problem is given by

$$\begin{aligned} (\mathfrak {m}\psi _{tt}^n- c^2 \Delta \psi ^n- \mathfrak {K}_\varepsilon *\Delta \psi _t^n+\nabla \mathfrak {l}\cdot \nabla \psi _t^n, v_j)_{L^2}= 0, \end{aligned}$$

for all $j=1, \ldots , n$, with approximate initial conditions $(\psi ^n, \psi _t^n)\vert _{t=0}=(\psi _{0}^n, \psi _{1}^n)$ taken as $L^2$ projections of $(\psi _0, \psi _1)$ onto $V_n$. That is,

$$\begin{aligned} \psi _{0}^n= \sum _{i=1}^n \xi ^n_{0, i} v_i(x), \quad \psi _{1}^n= \sum _{i=1}^n \xi ^n_{1, i} v_i(x) \end{aligned}$$

with $\xi ^n_{0, i} =(\psi _0, v_i)_{L^2}$ and $\xi ^n_{1, i} =(\psi _1, v_i)_{L^2}$. With $\varvec{\xi }=[\xi ^n_1 \ \xi ^n_2 \ \ldots \ \xi ^n_n]^T$, the approximate problem can be rewritten in matrix form

$$\begin{aligned} \mathbb {M}_{\mathfrak {m}}(t)\varvec{\xi }_{tt}+ c^2 \mathbb {K} \varvec{\xi }+ \mathbb {K}\, \mathfrak {K}_\varepsilon *\varvec{\xi }_t+\mathbb {L}\varvec{\xi }_t= 0 \end{aligned}$$

(A1)

where the entries of matrices $\mathbb {M}_{\mathfrak {m}}(t)=[\mathbb {M}_{\mathfrak {m}, ij}]$, $\mathbb {K}=[\mathbb {K}_{ij}]$, and $\mathbb {L}_\mathfrak {l}(t)=[\mathbb {L}_{\mathfrak {l}, ij}]$ are given by

$$\begin{aligned} \mathbb {M}_{\mathfrak {m}, ij}(t)= (\mathfrak {m}v_i, v_j)_{L^2}, \quad \mathbb {K}_{ij}= ( \nabla v_i, \nabla v_j)_{L^2}, \mathbb {L}_{\mathfrak {l}, ij}(t)=(\nabla \mathfrak {l}\cdot \nabla v_i, v_j)_{L^2}; \end{aligned}$$

recall that the variable coefficients $\mathfrak {m}$ and $\mathfrak {l}$ are defined in (3.2b). If we introduce the vectors of coordinates of the approximate initial data in the basis:

$$\begin{aligned} \varvec{\xi }_0=[\xi ^n_{0,1} \ \xi ^n_{0,2} \ \ldots \ \xi ^n_{0,n}]^T, \quad \varvec{\xi }_1=[\xi ^n_{1,1} \ \xi ^n_{1,2} \ \ldots \ \xi ^n_{1,n}]^T, \end{aligned}$$

then by setting $\varvec{\mu }=\varvec{\xi }_{tt}$, we have

$$\begin{aligned} \varvec{\xi }_t(t)=1*\varvec{\mu }+\varvec{\xi }_1, \quad \varvec{\xi }(t)=\varvec{\xi }_0 +t \varvec{\xi }_1+1*1*\varvec{\mu }. \end{aligned}$$

Therefore, the semi-discrete problem (A1) can be rewritten as

$$\begin{aligned} \mathbb {M}_{\mathfrak {m}}(t)\varvec{\mu }+ c^2 \mathbb {K} \left( \varvec{\xi }_0 +t \varvec{\xi }_1+1*1*\varvec{\mu }\right) + \mathbb {K}\, \mathfrak {K}_\varepsilon *(1*\varvec{\mu }+\varvec{\xi }_1)+\mathbb {L}_\mathfrak {l}(t)(1*\varvec{\mu }+\varvec{\xi }_1) =0. \end{aligned}$$

Since $\mathbb {M}(t)$ is positive definite due to assumption (3.4), the semi-discrete problem can be seen as a system of Volterra integral equations in the following form:

$$\begin{aligned} \varvec{\mu }+\int _0^t\varvec{K} (t, s)\varvec{\mu }(s)\, d s = \varvec{{f}}(t) \end{aligned}$$

with

$$\begin{aligned} \varvec{K}(t,s)= \mathbb {M}_{\mathfrak {m}}(t)^{-1}\left\{ c^2 \mathbb {K} {(1*1)(t-s)} + \mathbb {K}\,(\mathfrak {K}_\varepsilon *1)(t-s)+\mathbb {L}_\mathfrak {l}(t)\right\} , \end{aligned}$$

and the right-hand side

$$\begin{aligned} \varvec{{f}}(t)=\mathbb {M}_{\mathfrak {m}}(t)^{-1}\left\{ -c^2 \mathbb {K} \left( \varvec{\xi }_0 +t \varvec{\xi }_1\right) - \mathbb {K}\,(\mathfrak {K}_\varepsilon *\varvec{\xi }_1)(t)-\mathbb {L}_\mathfrak {l}(t)\varvec{\xi }_1\right\} . \end{aligned}$$

Additionally, by (3.3), we have $\mathbb {M}_{\mathfrak {m}}$, $\mathbb {L}_\mathfrak {l}\in C[0,T]$ and thus

$$\begin{aligned} \varvec{K} \in C(D) \quad \text {with} \ D=\{(t,s):\, 0 \le s \le t \le T\},\ \text {and } \varvec{{f}} \in C[0,T]. \end{aligned}$$

By the existence theory for systems of Volterra integral equations of the second kind [27, Ch. 2, Theorem 4.2], there is a unique $\varvec{\mu }\in C[0,T]$. Combined with initial data, from $\varvec{\xi }_{tt}=\varvec{\mu }$, we can then conclude that there exists a unique $\varvec{\xi }\in C^2(0,T)$ and thus $\psi ^n \in C^{2}(0,T; V_n)$. Owing to the higher regularity of the coefficients $\mathfrak {m}$ and $\mathfrak {l}$ in time assumed in (3.3), we can employ an additional bootstrap argument in (A1) to show that $\psi ^n \in W^{3,1}(0,T; V_n)$.