Global Existence for a Fractionally Damped Nonlinear Jordan–Moore–Gibson–Thompson Equation

Abstract

In nonlinear acoustics, higher-order-in-time equations arise when taking into account a class of thermal relaxation laws in the modeling of sound wave propagation. In the literature, these families of equations came to be known as Jordan–Moore–Gibson–Thompson (JMGT) models. In this work, we show the global existence of solutions relying only on minimal assumptions on the nonlocal damping kernel. In particular, our result covers the until-now open question of global existence of solutions for the fractionally damped JMGT model with quadratic gradient nonlinearity. The factional damping setting forces us to work with non-integrable kernels, which require a tailored approach in the analysis to control. This approach relies on exploiting the specific nonlinearity structure combined with a weak damping provided by the nonlocality kernel.

1. Introduction

We consider the fractionally damped Jordan–Moore–Gibson–Thompson (JMGT) equation:

$$\begin{array}{r}\{\begin{array}{rl}& \tau {\psi }_{ttt}+{\psi }_{tt}-c^2\mathrm{\Delta }\psi -\tau c^2\mathrm{\Delta }{\psi }_t-\delta {\text{D}}_t^{1-\alpha }\mathrm{\Delta }(\tau {\psi }_t+\psi )=2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t+f(x,t),\\ & \psi (t=0)={\psi }_0,{\psi }_t(t=0)={\psi }_1,{\psi }_{tt}(t=0)={\psi }_2\\ & \psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0.\end{array}\end{array}$$

(1.1)

Here $x\in \mathrm{\Omega }$ and $t>0$ , where $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with $d\in \{1,2,3\}$ , is a bounded $C^{2,1}-$ regular domain. The parameters $\tau ,\delta$ are positive ( $>0$ ), while $\sigma \in \mathbb{R}$ . The fractional derivative parameter $\alpha$ belongs to $(0,1)$ . Here ${\text{D}}_t^{\gamma }$ denotes the Caputo–Djrbashian (Caputo, 1967; Dzhrbashyan, 1966) derivative of order $\gamma \in (0,1)$ , that is

$$\begin{array}{r}{\text{D}}_t^{\gamma }v=\frac{1}{\mathrm{\Gamma }(1-\gamma )}{\int }_0^t(t-s{)}^{-\gamma }{v}_t\text{d}s\text{for }v\in W^{1,1}(0,T),t\in (0,T).\end{array}$$

The function $f$ is a source term depending on the space and time variables $(x,t)$ but not on the solution $\psi$ .

Equation (1.1) is a special case of the models of nonlinear acoustics derived in Kaltenbacher and Nikolić (2022b) (see also; Jordan, 2014) based on incorporating the average of two generalized Cattaneo heat flux law proposed by Compte and Metzler (1997) in the set of governing equations of acoustic propagation. The first equation in (1.1) can be conveniently rewritten as an integro-differential equation with an appropriate kernel as follows

$$\begin{array}{r}\tau {\psi }_{ttt}+{\psi }_{tt}-c^2\mathrm{\Delta }\psi -\tau c^2\mathrm{\Delta }{\psi }_t-\delta \mathfrak{K}*\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)=2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t+f,\end{array}$$

(1.2)

where $*$ denotes the Laplace convolution, and is interpreted as

$$\begin{array}{r}(\mathfrak{K}*g)(t)={\int }_0^t\mathfrak{K}(s)g(t-s)\text{d}s,\text{for }\mathfrak{K},g\in L^1(0,t).\end{array}$$

In particular, to obtain the Djrbashian–Caputo fractional derivative of order $\alpha \in (0,1)$ , we can set:

$$\begin{array}{r}\mathfrak{K}:t\mapsto \frac{1}{\mathrm{\Gamma }(\alpha )}{t}^{\alpha -1}.\end{array}$$

(1.3)

Equation (1.2) is supplemented by zero-Dirchlet boundary conditions and appropriate initial data, discussed in Section 3.

1.1. Related Results

The study of the initial-boundary value problem of (1.1) (alternatively, (1.2)) with zero-Dirichlet data and sufficiently regular initial data has thus far focused on questions of local existence in a Hilbertian setting and asymptotic behavior (e.g., rate of convergence as $\alpha \to 0^{+}$ , and $\tau \to 0^{+}$ ) in order to connect different models of acoustics. We refer the reader to Kaltenbacher and Nikolić (2022b, 2024) and Meliani (2023) for such studies.

The general idea for the well-posedness proofs for the nonlinear problem in the absence of quadratic gradient nonlinearities ( $\sigma =0$ in (1.1) or (1.2)) is based on considering regular and small initial data

$$\begin{array}{r}({\psi }_0,{\psi }_1,{\psi }_2)\in (H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }))\times (H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega }),\end{array}$$

for which the existence of the solution can be established on a sufficiently small time interval $[0,T]$ ; see, for example, Kaltenbacher and Nikolić (2022b, 2024). In the presence of gradient nonlinearities, more regular initial data are needed. For instance, in Kaltenbacher and Nikolić (2024, Theorem 3.3) assumes initial data

$$\begin{array}{r}({\psi }_0,{\psi }_1,{\psi }_2)\in H^3(\mathrm{\Omega })\times H^3(\mathrm{\Omega })\times H^2(\mathrm{\Omega }).\end{array}$$

The authors of the above mentioned works did not address the global existence and long-time behavior of the solution, leaving these questions open for future investigation. The main challenge in investigating the question of global existence lies in the absence of uniform-in-time bounds within the energy estimates.

Interest in long-term behavior of solutions for the JMGT equation with fractional damping is more recent and was shown only for a linearized version of fractional JMGT equation and for a restricted family of kernels in Meliani and Said-Houari (2025a). Among the restrictions on the kernels found in Meliani and Said-Houari (2025a) is the need for exponentially decaying kernels, or at least $L^1({\mathbb{R}}^{+})$ kernels for a weaker result. Kernel (1.3) verifies neither condition, that is, it is not exponentially decaying kernels, nor is it in $L^1({\mathbb{R}}^{+})$ . In Meliani and Said-Houari (2025b), a global extensibility criterion was proposed for a nonlinear fractionally damped JMGT equation with the help of Brezis–Gallouët–Wainger inequalities (Brezis & Gallouet, 1980; Brezis & Wainger, 1980). In particular, we showed that as long as

$$\begin{array}{r}\Vert {\psi }_t{\Vert }_{H^{d/2}(\mathrm{\Omega })}+\Vert {\psi }_t{\Vert }_{H^{d/2-1}(\mathrm{\Omega })}<\mathrm{\infty },\end{array}$$

with $d$ being the spatial dimension, then a local solution to (1.2) belonging to a suitable solution class can be extended to be global in time. Our result relied on the assumption that the kernel is coercive (see Meliani & Said-Houari, 2025b, Assumption 1) and complemented the result of Nikolić and Winkler (2024) which showed that blow-up of the inviscid form of (1.2) (i.e., with $\delta =0$ ) is driven by an inflation of the $L^{\mathrm{\infty }}$ -norm of the solution $\Vert {\psi }_t{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}$ (as opposed to a gradient blow-up).

When the convolution kernel $\mathfrak{K}$ is allowed to be the Dirac delta pulse ${\delta }_0$ , problem (1.2) reduces to the initial-boundary value problem for the nonlinear JMGT equation:

$$\begin{array}{r}\tau {\psi }_{ttt}+{\psi }_{tt}-c^2\mathrm{\Delta }\psi -(\delta +\tau c^2)\mathrm{\Delta }{\psi }_t-2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t=0,\end{array}$$

(1.4)

which can be seen as a special case of the thermally relaxed Kuznetsov equation (Jordan, 2014, Equation (72)), where the coefficient of nonlinearity is set to 1 (Jordan, 2014, p. 2197). Alternatively, equation (1.4) can be obtained as an approximation of the classical JMGT equation (1.6) under the assumption that ${\psi }_t^2\sim c^2|\mathrm{\nabla }\psi {|}^2$ valid when the acoustic field is close to a plane wave (Jordan, 2016).

The related relaxed Westervelt equation:

$$\begin{array}{r}\tau {\psi }_{ttt}+(1+2k{\psi }_t){\psi }_{tt}-c^2\mathrm{\Delta }\psi -(\delta +\tau c^2)\mathrm{\Delta }{\psi }_t=0,\end{array}$$

(1.5)

has received comparatively higher attention in recent years, as the absence of quadratic gradient nonlinearities makes it easier to study. Global existence and exponential decay for equation (1.5) has been established in Kaltenbacher et al. (2012) for $\delta >0$ . A similar study was carried out for its linearized counterpart ( $k=0$ , $\delta >0$ ) in Kaltenbacher et al. (2011) where the authors used the energy method together with a bootstrap argument to show, under a smallness assumption on the initial data, a global existence result and decay estimates for the solution. See also the decay results for the linearized problem in Pellicer and Said-Houari (2019) and Chen and Ikehata (2021). The interested reader is referred to Conejero et al. (2015), Kaltenbacher (2015), Kaltenbacher and Nikolić (2019), Marchand et al. (2012), Pellicer and Solà-Morales (2019), and Said-Houari (2022) for various studies related to (1.5).

As for equation (1.4), whose prominent feature is the quadratic gradient nonlinearity: $2\sigma \mathrm{\nabla }{\psi }_t\cdot \mathrm{\nabla }\psi$ , there exist far fewer results. A notable result is the global existence result of solution to the Cauchy problem for the JMGT equation with Kuznetsov nonlinearity:

$$\begin{array}{r}\tau {\psi }_{ttt}+(1+2k{\psi }_t){\psi }_{tt}-c^2\mathrm{\Delta }\psi -(\delta +\tau c^2)\mathrm{\Delta }{\psi }_t-2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t=0,\end{array}$$

(1.6)

is due to Racke and Said-Houari (2021). Naturally, their result also covers (1.5) by setting $\sigma =0$ . We also mention here the work of Chen et al. (2022) who proved finite-time blow-up of the solution for the inviscid (i.e., $\delta =0$ ) JMGT equation (1.6) in ${\mathbb{R}}^n$ for appropriately chosen initial data.

Our goal in this work is to show that the fractional damping is strong enough to prevent the formation of this blow-up and that we can show global boundedness of the solution for small enough data.

In this work, we use a similar bootstrapping idea as Racke and Said-Houari (2021) to prove global existence of solutions for small enough data.

1.2. Novelty

Our work answers the fundamental question of whether fractional damping is strong enough to produce sufficient smoothness for a solution to exist globally. Indeed, in equation (1.2), no damping can be extracted from the term $-\tau c^2\mathrm{\Delta }{\psi }_t$ as we are in the so-called critical case, thus our arguments will need to make primarily use of the term $-\delta \mathfrak{K}*\mathrm{\Delta }{\psi }_{tt}$ to prevent blow up of the solution. Working with nonlocal damping has clear drawbacks from the point of view of analysis. Indeed, a quick inspection of the convolution kernel $\mathfrak{K}$ , shows that it is not even $L^1({\mathbb{R}}^{+})$ making it difficult to control the nonlinear and nonlocal terms without first extracting some control on the norm of the solution by exploiting the kernel term; see Lemma 2.2 below. Thus, our main ingredient will be to exploit the concept of strongly positive kernels introduced by Nohel and Shea (1976, Corollary 2.2); see the characterization of strongly positive kernels given in Lemma 2.1. Furthermore, we need to exploit the structure of the nonlinearity, which can be written as a derivative in time: $2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t=\sigma {\mathrm{\partial }}_t(|\mathrm{\nabla }\psi {|}^2)$ . This will allow us to integrate the equation in time and obtain the necessary estimates to prove the global existence; see Section 4.1 for more details.

1.3. Organization of the Paper

The rest of the paper is organized as follows: in Section 2.2, we discuss the main assumptions made on the nonlocal damping kernel and relevant properties of convolutions. The main result of the paper (Theorem 3.1) is presented in Section 3. Section 4 is then devoted to the proof of Theorem 3.1 through energy arguments. Appendix A provides a local well-posedness result for the initial-boundary value problem related to (1.2) with a source term $f$ .

2. Preliminaries and Assumptions

In this section, we introduce a few notations, and list some necessary assumptions on the kernel $\mathfrak{K}$ . In addition, we collect some helpful embedding results and inequalities that we will repeatedly use in the proof of the main results.

2.1. Notations

Throughout the paper, the letter $C$ denotes a generic positive constant that does not depend on time, and can have different values on different occasions. We often write $f\lesssim g$ where there exists a constant $C>0$ , independent of parameters of interest such that $f\le Cg$ . We often omit the spatial and temporal domain when writing norms; for example, $\Vert \cdot {\Vert }_{L^p{L}^q}$ denotes the norm in $L^p(0,T;L^q(\mathrm{\Omega }))$ .

We assume throughout that $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , where $n\in \{1,2,3\}$ , is a bounded $C^{2,1}$ -regular domain. This will allow us to use desired elliptic regularity results; see estimate (4.17). We also introduce, similarly to the work (Kaltenbacher et al., 2024a) where a similar quadratic nonlinearity was treated, the following functional Sobolev spaces which are of interest in the analysis:

$$\begin{array}{r}{H}_{\mathrm{♢}}^2(\mathrm{\Omega }):=H_0^1(\mathrm{\Omega })\cap H^2(\mathrm{\Omega }),H_{\mathrm{♢}}^3(\mathrm{\Omega }):=\{u\in H^3(\mathrm{\Omega }):u{|}_{\mathrm{\partial }\mathrm{\Omega }}=0,\mathrm{\Delta }u{|}_{\mathrm{\partial }\mathrm{\Omega }}=0\}.\end{array}$$

2.2. Assumption on the Kernel $\mathfrak{K}$

In order to state and prove our main result, we make the following assumptions on the kernel $\mathfrak{K}$ .

Assumption 1

We assume that $\mathfrak{K}$ is a locally integrable function on $[0,\mathrm{\infty })$ . Furthermore, we assume that it is twice differentiable with ${\mathfrak{K}}_t\ne 0$ satisfying

$$\begin{array}{r}(-1{)}^n{\mathfrak{K}}^{(n)}(t)\ge 0\mathrm{\forall }t>0,n=0,1,2\end{array}$$

We intend here to show that a kernel $\mathfrak{K}$ verifying Assumption 1 can be used to provide some damping; see Lemma 2.2 below. To this end let us recall a definition and a characterization of strongly positive definite kernels.

Definition 1 (Strongly positive kernel (Nohel & Shea, 1976))

A real function $h\in L_{\text{loc}}^1(0,\mathrm{\infty })$ is said to be strongly positive with constant $\eta$ if there exists $\eta >0$ such that the function $g:t\mapsto h(t)-\eta e^{-t}$ is of positive type. That is if

$$\begin{array}{r}{\int }_0^t(g*y)(s)y(s)\text{d}s\ge 0,\end{array}$$

or equivalently, if

$$\begin{array}{r}{\int }_0^t(h*y)(s)y(s)\text{d}s\ge \eta {\int }_0^t(e^{-t}*y)(s)y(s)\text{d}s.\end{array}$$

Note that $t\mapsto e^{-t}$ is a positive kernel (Nohel & Shea, 1976, pp. 279–280) and thus a strongly positive kernel is naturally also a positive kernel, that is,

$$\begin{array}{r}{\int }_0^t(h*y)(s)y(s)\text{d}s\ge 0.\end{array}$$

(2.1)

We recall the result from Nohel and Shea (1976, Corollary 2.2), which is fundamental in establishing the strong positivity of many kernel classes discussed in the literature.

Lemma 2.1 Characterization of strongly positive kernels

A twice differentiable function $h$ with $h_t\ne 0$ satisfying

$$\begin{array}{r}(-1{)}^n{h}^{(n)}(t)\ge 0\mathrm{\forall }t>0,n=0,1,2\end{array}$$

is a strongly positive kernel.

Lemma 2.2 Lemma 2.4 in Kawashima (1993) and Lemma 2.9 in Cannarsa and Sforza (2011)

Suppose that $y\in L_{\text{loc}}^2(0,\mathrm{\infty })$ and $y_t\in L_{\text{loc}}^2(0,\mathrm{\infty })$ . Let $\mathfrak{K}$ be a strongly positive kernels with constant $\eta$ . Then the following holds:

$$\begin{array}{r}{\int }_0^t|y(s){|}^2\text{d}s\le |y(0){|}^2+2{\eta }^{-1}{\int }_0^t(\mathfrak{K}*y)(s)y(s)\text{d}s+2{\eta }^{-1}{\int }_0^t(\mathfrak{K}*y_t)(s)y_t(s)\text{d}s.\end{array}$$

(2.2)

2.3. Examples of Relevant Kernels

Although the Abel kernel (1.3) is our primary example of interest, Assumption 1 is sufficiently general to encompass a broad class of kernels arising in the modeling of acoustic and visco-elastic phenomena. We mention here some of these kernels which verify Assumption 1:

The exponential kernel

$$\begin{array}{r}\mathfrak{K}(t)=e^{-\beta t},\beta >0,\end{array}$$

.

The exponentially regularized Abel kernel (along the lines of that found in Messaoudi et al. (2007))

$$\begin{array}{r}\mathfrak{K}=\frac{t^{\alpha -1}{e}^{-\beta t}}{\mathrm{\Gamma }(\alpha )},\end{array}$$

with $0<\alpha <1,$ $\beta >0$ .

The fractional Mittag–Leffler kernels encountered in the study of fractional second order wave equations in complex media in Kaltenbacher et al. (2024b)

$$\begin{array}{r}\mathfrak{K}=\frac{t^{\beta -1}}{\mathrm{\Gamma }(1-\alpha )}{E}_{\alpha ,\beta }(-t^{\alpha }),\end{array}$$

with $0<\alpha \le \beta \le 1$ , where $E_{\alpha ,\beta }$ is the two-parametric Mittag–Leffler function.

The polynomially decaying kernel

$$\begin{array}{r}\mathfrak{K}(t)=\frac{1}{(1+t{)}^p},p>1\end{array}$$

arising in viscoelasticity, see, for example, Muñoz Rivera and Lapa (1996).

2.4. Elliptic Regularity Estimates

We recall here a useful elliptic regularity result to be used in establishing the necessary energy estimates; see, for example, (4.17) below.

Lemma 2.3

Let $\mathrm{\Omega }$ be a bounded $C^{2,1}$ -regular domain. Let $\psi$ be a function such that $\mathrm{\Delta }\psi \in L^2(\mathrm{\Omega })$ and $\psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ , then

$$\begin{array}{r}\Vert \psi {\Vert }_{H^2(\mathrm{\Omega })}\lesssim \Vert \mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$

(2.3)

If instead $\mathrm{\nabla }\mathrm{\Delta }\psi \in L^2(\mathrm{\Omega })$ and $\psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=\mathrm{\Delta }\psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ , then

$$\begin{array}{r}\Vert \psi {\Vert }_{H^3(\mathrm{\Omega })}\lesssim \Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$

(2.4)

Proof.

Consider the following elliptic PDE

$$\begin{array}{r}\{\begin{array}{cc}-\mathrm{\Delta }\psi =g,& \text{on }\mathrm{\Omega },\\ \psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0& \end{array}\end{array}$$

with $g\in H_0^1(\mathrm{\Omega })$ (since $\mathrm{\Delta }\psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ ). Note that $\Vert g{\Vert }_{H_0^1(\mathrm{\Omega })}=\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega }}$ . Thus, from classic elliptic regularity results (Grisvard, 2011, Theorem 2.5.1.1), we have the estimate:

$$\begin{array}{r}\Vert \psi {\Vert }_{H^3(\mathrm{\Omega })}\lesssim \Vert g{\Vert }_{H^1(\mathrm{\Omega })}\sim \Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$

Estimate (2.3) is obtained in a similar manner.

2.5. A Crucial Functional Inequality

We conclude the preliminaries section by introducing an important lemma that we will use to establish a uniform bound of the total energy of the solution; see Proposition 4.1. This lemma was proved in Strauss (1968, Lemma 3.7).

Lemma 2.4

Let $M=M(t)$ be a non-negative continuous function satisfying the inequality

$$\begin{array}{r}M(t)\le c_1+c_{2}M{(t)}^{\kappa },\end{array}$$

in some interval containing $0$ , where $c_1$ and $c_2$ are positive constants and $\kappa >1$ . If $M(0)\le c_1$ and

$$\begin{array}{r}{c}_1{c}_2^{1/(\kappa -1)}<(1-1/\kappa ){\kappa }^{-1/(\kappa -1)},\end{array}$$

then in the same interval

$$\begin{array}{r}M(t)<\frac{c_1}{1-1/\kappa }.\end{array}$$

3. Main Results

In this section, we present the main results of this paper and explain the strategy of the proof. For the convenience of the reader, we recall here the initial-boundary value problem under consideration:

$$\begin{array}{r}\{\begin{array}{rl}& \tau {\psi }_{ttt}+{\psi }_{tt}-c^2\mathrm{\Delta }\psi -\tau c^2\mathrm{\Delta }{\psi }_t-\delta \mathfrak{K}*\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)=2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t+f,\\ & \psi (t=0)={\psi }_0,{\psi }_t(t=0)={\psi }_1,{\psi }_{tt}(t=0)={\psi }_2\\ & \psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0.\end{array}\end{array}$$

(3.1)

Let us introduce the space of solutions used in the statement of Theorem 3.1:

$$\begin{array}{rl}\mathcal{U}(0,T)=& \{\psi \in W^{3,1}(0,T;H_0^1(\mathrm{\Omega }))\cap W^{2,\mathrm{\infty }}(0,T;H_{\mathrm{♢}}^2(\mathrm{\Omega }))\cap W^{1,\mathrm{\infty }}(0,T;H_{\mathrm{♢}}^3(\mathrm{\Omega }))|\\ & \sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t\in L^{\mathrm{\infty }}(0,T;H_0^1(\mathrm{\Omega }))\}.\end{array}$$

(3.2)

Theorem 3.1 Global existence for small data

Let $\mathrm{\Omega }$ be a $C^{2,1}$ domain. Let Assumption 1 on the kernel hold. On the problem data, assume the source term has the form

$$\begin{array}{r}f=\frac{\text{d}}{\text{d}t}\tilde{f}\in L^1({\mathbb{R}}^{+};H_{\mathrm{♢}}^2(\mathrm{\Omega }))\end{array}$$

with $\tilde{f}\in W^{1,1}({\mathbb{R}}^{+};H_{\mathrm{♢}}^2(\mathrm{\Omega }))$ , and let the initial data

$$\begin{array}{r}\psi (t=0)={\psi }_0\in H_{\mathrm{♢}}^3(\mathrm{\Omega }),{\psi }_t(t=0)={\psi }_1=-\frac{1}{\tau }{\psi }_0,{\psi }_{tt}(t=0)={\psi }_2\in H_{\mathrm{♢}}^2(\mathrm{\Omega }).\end{array}$$

(3.3)

We furthermore impose the zero-Dirichlet boundary condition:

$$\begin{array}{r}\psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0.\end{array}$$

Then, there exists a small $m>0$ such that if the initial data and source term verify

$$\begin{array}{r}\Vert {\psi }_0{\Vert }_{H^3(\mathrm{\Omega })}+\Vert {\psi }_2{\Vert }_{H^2(\mathrm{\Omega })}+\Vert \tilde{f}{\Vert }_{W^{1,1}({\mathbb{R}}^{+};H^2(\mathrm{\Omega }))}\le m,\end{array}$$

then the local solution $\psi \in \mathcal{U}$ to (3.1) exists globally in time.

Remark 1 (On the requirements on the data)

The seemingly peculiar form of the requirement on the source term $f$ has to do with our intention to integrate equation (3.1) in time in Section 4.1. We point out also that requiring special form of the initial data is not uncommon when studying fractional/nonlocal-in-time wave equations; for example, Kaltenbacher and Nikolić (2022b) and Meliani (2023), or even local-in-time Moore–Gibson–Thompson equations; for example, Chen and Ikehata (2021) where some of the initial data are set to $0$ . In our work, the initial data restriction leads to

$$\begin{array}{r}(\tau {\psi }_t+\psi )(t=0)=0\end{array}$$

which is used when integrating (3.1) in time.

3.1. Discussion of the Main Result

Before moving onto the proof, we briefly discuss the statements made above in Theorem 3.1.

Theorem 3.1 proves the existence of global-in-time solution to (3.1) for small enough data. The main difficulty in showing a global existence result lies in proving that the nonlocal term $\delta \mathfrak{K}*\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)$ provides sufficient control to ensure that the solution does not blow-up. We recall a notable existing result in the direction of global existence for the local JMGT equation with $\delta =0$ in the case of Westervelt-type nonlinearity (see (1.5)) is a conditional regularity result due to Nikolić and Winkler (2024). For the nonlocal JMGT equation with a general convolution kernels $\mathfrak{K}$ and Westervelt nonlinearity, a conditional regularity result was established recently in Meliani and Said-Houari (2025b). On the other hand, long-term behavior of nonlocal equations is, in general, poorly understood (Fritz et al., 2022). The combination of these facts, makes the study of (3.1) challenging and interesting.

Showing the large-time asymptotic behavior solutions is an interesting problem which strongly depends on the type of the kernel $\mathfrak{K}$ . For exponentially decaying kernel, it is possible to prove an exponential decay of the solution in particular cases which do not cover (1.2); see, for example, Meliani and Said-Houari (2025a) and Messaoudi et al. (2007). However, proving decay rates for more general classes of kernels is more difficult. This is the case only in particular instances of lower-order equation with a simpler structure compared to (1.2); see, for example, Fritz et al. (2022) and Vergara and Zacher (2008, 2015). Studying the asymptotic behavior of nonlinear equations with higher-order derivatives is a challenging task which remains open for the fractional JMGT equation at hand.

The proof of Theorem 3.1 is given in Section 4.

4. Proof of Theorem 3.1

The proof of Theorem 3.1 will be done though a bootstrap argument. We define first the energy associated to (3.1) as

$$\begin{array}{r}\mathbf{E}(t):=\underset{\nu \in (0,t)}{sup}{E}_1(\nu )+E_2(\nu )\end{array}$$

(4.1)

where

$$\begin{array}{r}{E}_1(t):=\frac{1}{2}{\int }_{\mathrm{\Omega }}(|\mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt}){|}^2+c^2|\mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t){|}^2)\text{d}x\end{array}$$

and

$$\begin{array}{r}{E}_2(t):=\frac{1}{2}{\int }_{\mathrm{\Omega }}(|\mathrm{\Delta }(\psi +\tau {\psi }_t){|}^2+c^2|\mathrm{\nabla }\mathrm{\Delta }(\xi +\tau \psi ){|}^2)\text{d}x,\end{array}$$

where $\xi ={\int }_0^t\psi \text{d}s+{\xi }_0$ with ${\xi }_0$ defined through the elliptic problem (4.6) below.

We define the associated dissipation rate

$$\begin{array}{r}\mathbf{D}(t):=\frac{\delta }{\eta }{\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s.\end{array}$$

(4.2)

We introduce

$$\begin{array}{r}\mathbf{Y}(t):=\mathbf{E}(t)+\mathbf{D}(t).\end{array}$$

Our main goal is to prove by a continuity argument that $\mathbf{Y}(t)$ is uniformly bounded for all time provided that the initial energy $\mathbf{E}(0)=\mathbf{Y}(0)$ and source term $\Vert \tilde{f}{\Vert }_{W^{1,1}({\mathbb{R}}^{+},L^2(\mathrm{\Omega }))}$ are sufficiently small. Such a functional will be shown to verify the inequality

$$\begin{array}{r}\mathbf{Y}(t)\lesssim \mathbf{Y}(0)+\Vert \tilde{f}{\Vert }_{W^{1,1}({\mathbb{R}}^{+};H^2(\mathrm{\Omega }))}+{\mathbf{Y}}^2(t).\end{array}$$

(4.3)

where the hidden constant in (4.3) is independent of $t$ . Equipped with the above inequality, we can show by using Lemma 2.4 that the energy of the solution is uniformly bounded as time goes to infinity. Hence, our primary goal is to prove (4.3). To do so, it is natural to compute the time evolution of $\mathbf{E}(t)$ .

Our starting point in the analysis is the following energy estimate:

Lemma 4.1

We have for all $t\ge 0$ , the identity

$$\begin{array}{r}{E}_1(t)+\delta {\int }_0^t{\int }_{\mathrm{\Omega }}\mathfrak{K}*\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)\cdot \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)\text{d}x\text{d}s\le E_1(0)+{\mathrm{R}}_1,\end{array}$$

(4.4)

where

$$\begin{array}{r}{\mathrm{R}}_1:={\int }_0^t{\int }_{\mathrm{\Omega }}2\sigma \mathrm{\Delta }(\mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t)\cdot \mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt})\text{d}x\text{d}s+{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Delta }f\cdot \mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt})\text{d}x\text{d}s.\end{array}$$

Proof.

We intend to test the first equation of system (3.1) by ${\mathrm{\Delta }}^2({\psi }_t+\tau {\psi }_{tt})$ which lacks the sufficient regularity to be a valid test function. Therefore, similarly to the idea of, for example, Conti et al. (2023, Theorem 8.3) and Meliani and Said-Houari (2025a, Lemma 5.2), we will carry out the testing in a semi-discrete setting (Galerkin approximation), such that the differential inequality (4.4) will make sense at first for the Galerkin approximations of the equation. Taking the limit in the Galerkin procedure will then ensure that the inequality also holds for the solution of the infinite dimensional problem.

Thus, multiplying the first equation of (3.1) by ${\mathrm{\Delta }}^2({\psi }_t+\tau {\psi }_{tt})$ and integrating by parts over $\mathrm{\Omega }$ , we obtain

$$\begin{array}{rl}& \frac{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{\mathrm{\Omega }}|\mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt}){|}^2\text{d}x+\frac{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{\mathrm{\Omega }}{c}^2|\mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t){|}^2\text{d}x\\ & +\delta {\int }_{\mathrm{\Omega }}\mathfrak{K}*\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)\cdot \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t)\text{d}x\\ & ={\int }_{\mathrm{\Omega }}2\sigma (\mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t){\mathrm{\Delta }}^2({\psi }_t+\tau {\psi }_{tt})\text{d}x+{\int }_{\mathrm{\Omega }}f{\mathrm{\Delta }}^2({\psi }_t+\tau {\psi }_{tt})\text{d}x.\end{array}$$

We next integrate by parts the right-hand side using the fact that $f\in H_{\mathrm{♢}}^2(\mathrm{\Omega })$ and thus $f{|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ and recalling also through the PDE (3.1) and the function space $\mathcal{U}$ (see (3.2)), that $2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t{|}_{\mathrm{\partial }\mathrm{\Omega }}=\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t){|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ , which yields after integrating over $(0,t)$ the desired estimate.

4.1. Control of $\Vert \mathrm{\nabla }{\psi }_t{\Vert }_{L^2(0,T;L^2(\mathrm{\Omega }))}$

One of the main challenges in closing the estimates for the nonlinear equation is controlling the norm $\Vert \mathrm{\nabla }{\psi }_t{\Vert }_{L^2(0,T;L^2(\mathrm{\Omega }))}$ ; see proof of Proposition 4.1. Since, we are in the critical case (in terms of the coefficient (Kaltenbacher et al., 2012)), we need to use crucially the properties of the kernel. In particular, we need to rely on Lemma 2.2 together with an approach inspired by Lions and Strauss (1965, Section 1.9). The idea is to exploit the structure of the nonlinearity which can be written as a derivative in time: $2\sigma \mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t=\sigma (|\mathrm{\nabla }\psi {|}^2{)}_t$ . Then, by averaging in time, we can use (2.2) to derive (4.12) through appropriate energy estimates.

Define

$$\begin{array}{r}\xi ={\xi }_0+{\int }_0^t\psi (s)\text{d}s,\end{array}$$

(4.5)

with ${\xi }_0$ being the solution to the following elliptic problem on $\mathrm{\Omega }$ :

$$\begin{array}{r}\{\begin{array}{rlc}-c^2\mathrm{\Delta }{\xi }_0& =\sigma |\mathrm{\nabla }{\psi }_0{|}^2-\tau {\psi }_2-{\psi }_1+\tau c^2\mathrm{\Delta }{\psi }_0+\tilde{f}(0),& \text{on }\mathrm{\Omega },\\ {\xi }_0& =0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$$

(4.6)

Note that due to the regularity of the source term and initial data (3.3), the right-hand side is in $H^1(\mathrm{\Omega })$ . Thus, there exists a unique ${\xi }_0\in H^3(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ which solves (4.6); see, for example, Grisvard (2011, Theorem 2.5.1.1).

Then $\xi$ verifies the following equation

$$\begin{array}{r}\tau {\xi }_{ttt}+{\xi }_{tt}-c^2\mathrm{\Delta }\xi -\tau c^2\mathrm{\Delta }{\xi }_t-\delta \mathfrak{K}*\mathrm{\Delta }(\tau {\xi }_{tt}+{\xi }_t)=\sigma \mathrm{\nabla }|{\xi }_t{|}^2+\tilde{f}.\end{array}$$

(4.7)

This equation is obtained by integrating equation (1.2) in time on $(0,t)$ , keeping in mind (4.5) and using the following relations:

$$\begin{array}{rl}& {\xi }_t=\psi ={\int }_0^t{\psi }_t\text{d}s+{\psi }_0,{\xi }_{tt}={\psi }_t={\int }_0^t{\psi }_{tt}\text{d}s+{\psi }_1,\\ & {\xi }_{ttt}={\psi }_{tt}={\int }_0^t{\psi }_{ttt}\text{d}s+{\psi }_2,\mathfrak{K}*(\tau {\xi }_{tt}+{\xi }_t)=\mathfrak{K}*(\tau {\psi }_t+\psi )={\int }_0^t\mathfrak{K}*(\tau {\psi }_{tt}+{\psi }_t)\text{d}s,\\ & \sigma |\mathrm{\nabla }{\xi }_t{|}^2=\sigma |\mathrm{\nabla }\psi {|}^2={\int }_0^{t}2\sigma \mathrm{\nabla }{\psi }_t\cdot \mathrm{\nabla }\psi \text{d}s+\sigma |\mathrm{\nabla }{\psi }_0{|}^2.\end{array}$$

We also used the initial condition on ${\xi }_0$ given in (4.6) to eliminate the time-independent residual terms that results from integrating (1.2) with respect to time.

To simplify the study of the equation of $\xi$ , we reformulate (4.7) into the following system of equations:

$$\begin{array}{r}\{\begin{array}{ll}{\xi }_t=\psi ,& \\ z=\tau {\xi }_t+\xi ,& \\ z_{tt}-c^2\mathrm{\Delta }z-\delta \mathfrak{K}*\mathrm{\Delta }{z}_t=\sigma |\mathrm{\nabla }\psi {|}^2+\tilde{f},& \end{array}\end{array}$$

(4.8)

Exploiting the similarity in structure between (3.1) and (4.8), we give a first estimate for the functional $E_2$ , which we recall is defined as

$$\begin{array}{r}{E}_2(t)=\frac{1}{2}{\int }_{\mathrm{\Omega }}(|\mathrm{\Delta }(\psi +\tau {\psi }_t){|}^2+c^2|\mathrm{\nabla }\mathrm{\Delta }(\xi +\tau \psi ){|}^2)\text{d}x,\end{array}$$

We then have the following estimate.

Lemma 4.2

We have for all $t\ge 0$ , the identity

$$\begin{array}{r}{E}_2(t)+\delta {\int }_0^t{\int }_{\mathrm{\Omega }}\mathfrak{K}*\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi )\cdot \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi )\text{d}x\text{d}s\le E_2(0)+{\mathrm{R}}_2,\end{array}$$

(4.9)

where

$$\begin{array}{r}{\mathrm{R}}_2:=-{\int }_0^t{\int }_{\mathrm{\Omega }}\sigma \mathrm{\nabla }|\mathrm{\nabla }\psi {|}^2\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\text{d}s+{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Delta }\tilde{f}\cdot \mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\text{d}s.\end{array}$$

Proof.

Applying the Laplace operator $\mathrm{\Delta }$ to the third equation of (4.8) and multiplying it by $\mathrm{\Delta }(\psi +\tau {\psi }_t)=\mathrm{\Delta }{z}_t$ , we integrate by parts over $\mathrm{\Omega }$ to obtain

$$\begin{array}{rl}& \frac{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{\mathrm{\Omega }}|\mathrm{\Delta }(\psi +\tau {\psi }_t){|}^2\text{d}x+c^2{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\mathrm{\Delta }(\xi +\tau \psi )\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\\ & +\delta {\int }_{\mathrm{\Omega }}\mathfrak{K}*\mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\\ & =-{\int }_{\mathrm{\Omega }}\sigma \mathrm{\nabla }|\mathrm{\nabla }\psi {|}^2\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x+{\int }_{\mathrm{\Omega }}\mathrm{\Delta }\tilde{f}\cdot \mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x,\end{array}$$

(4.10)

where we exploited the a priori information on the vanishing boundary value of $\psi ,$ $\mathrm{\Delta }\psi ,$ ${\psi }_t,$ and $\mathrm{\Delta }{\psi }_t$ ; see solution space (3.2). Due to the choice of data, we know that $\mathrm{\Delta }\tilde{f}{|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ .

Exploiting the first equation in (4.8) (adding $\tau {\psi }_t$ to both sides) and testing with $-c^2{\mathrm{\Delta }}^3({\psi }_t+\tau \psi )$ , we have

$$\begin{array}{r}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{\mathrm{\Omega }}{c}^2|\mathrm{\nabla }\mathrm{\Delta }(\xi +\tau \psi ){|}^2\mathrm{d}x=c^2{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\mathrm{\Delta }(\xi +\tau \psi )\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x.\end{array}$$

(4.11)

Summing up (4.10) and (4.11) and integrating with respect to $t$ , we get obtain the desired estimate.

Now, collecting the inequalities (4.4) and (4.9), we show that although we are in the critical regime, the convolution term helps to gain a dissipative term that is crucial in controlling the nonlinearity and closing the energy estimate. This result is contained in the following lemma.

Lemma 4.3

We have for all $t\ge 0$

$$\begin{array}{r}{E}_1(t)+E_2(t)+\frac{\delta }{\eta }{\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s\le E_1(0)+E_2(0)+{\mathrm{R}}_1+{\mathrm{R}}_2.\end{array}$$

(4.12)

Proof.

Taking advantage of Lemma 2.2, we obtain

$$\begin{array}{rl}& {\int }_0^t(\mathfrak{K}*\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t),\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_{tt}+{\psi }_t))+(\mathfrak{K}*\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ),\mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ))\text{d}s\\ & \ge \frac{1}{\eta }{\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s,\end{array}$$

for some $\eta >0$ . Hence, summing up (4.4) and (4.9) yields the desired control of the term $\Vert \tau \mathrm{\nabla }\mathrm{\Delta }{\psi }_t+\mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(0,t;L^2(\mathrm{\Omega }))}$ .

We show through the following lemma that the $L^2$ - $L^2$ control provided by Lemma 4.3 for the quantity

$$\begin{array}{r}{\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s\end{array}$$

can be used to extract suitable control for the individual component $\mathrm{\nabla }\mathrm{\Delta }\psi$ and $\mathrm{\nabla }\mathrm{\Delta }{\psi }_t$ in the spirit of Lasiecka (2017, Lemma 3.5).

Lemma 4.4

Let $t\ge 0$ , then we have the following bounds

$$\begin{array}{r}\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^p(0,t;L^2(\mathrm{\Omega }))}^2\lesssim {\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^2,\end{array}$$

for all $p\in [2,\mathrm{\infty }]$ and

$$\begin{array}{r}\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_t{\Vert }_{L^2(0,t;L^2(\mathrm{\Omega }))}^2\lesssim {\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s+{\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s.\end{array}$$

The hidden constants are independent of time $t$ .

Proof.

For convenience of notation for this proof, let

$$\begin{array}{r}w:=\tau {\psi }_t+\psi .\end{array}$$

We start with the relation

$$\begin{array}{r}\tau \mathrm{\nabla }\mathrm{\Delta }{\psi }_t+\mathrm{\nabla }\mathrm{\Delta }\psi =\mathrm{\nabla }\mathrm{\Delta }w,\end{array}$$

(4.13)

then testing with $\mathrm{\nabla }\mathrm{\Delta }\psi$ and integration over $\mathrm{\Omega }$ yields that

$$\begin{array}{r}\frac{\tau }{2}\frac{\text{d}}{\text{d}t}\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}^2+\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}^2\le \Vert \mathrm{\nabla }\mathrm{\Delta }w{\Vert }_{L^2(\mathrm{\Omega })}\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}.\end{array}$$

Integrating over $(0,\nu )$ , and taking the supremum of $\nu \in (0,t)$ , we obtain

$$\begin{array}{r}\underset{\nu \in (0,t)}{sup}\Vert \mathrm{\nabla }\mathrm{\Delta }\psi (\nu ){\Vert }_{L^2(\mathrm{\Omega })}^2+{\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s\lesssim {\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }w{\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^2,\end{array}$$

where the hidden constant only depends on the parameters of the problem but not on time $t$ . By interpolation of $L^p$ spaces, we infer that

$$\begin{array}{r}\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^p(0,t;L^2(\mathrm{\Omega }))}^2\lesssim {\int }_0^t\Vert \mathrm{\nabla }\mathrm{\Delta }w{\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^2,\end{array}$$

for $p\in [2,\mathrm{\infty }]$ .

For the second bound, we rely on the triangle inequality

$$\begin{array}{r}\Vert \tau \mathrm{\nabla }\mathrm{\Delta }{\psi }_t{\Vert }_{L^2(0,t;L^2(\mathrm{\Omega }))}\le \Vert \mathrm{\nabla }\mathrm{\Delta }(\tau {\psi }_t+\psi ){\Vert }_{L^2(0,t;L^2(\mathrm{\Omega }))}+\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(0,t;L^2(\mathrm{\Omega }))}.\end{array}$$

Proposition 4.1

Let the Assumptions of Theorem 3.1 hold. Then, for all time $t>0$ , it holds that

$$\begin{array}{r}\mathbf{E}(t)+\mathbf{D}(t)\le C_{*}[\mathbf{E}(0)+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^4+\Vert \tilde{f}{\Vert }_{W^{1,1}({\mathbb{R}}^{+};H^2(\mathrm{\Omega }))}^2+{\mathbf{D}}^2(t)],\end{array}$$

(4.14)

where $\mathbf{E}(t)$ and $\mathbf{D}(t)$ are defined in (4.1) and (4.2) respectively. Here, the constant $C_{*}\ge 1$ is independent of time $t$ .

Proof.

It suffices to estimate the terms ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ in (4.12), which we do as follows: for an arbitrary $\epsilon >0$ , there exist $C(\epsilon )$ such that

$$\begin{array}{rl}{\mathrm{R}}_1& ={\int }_0^t{\int }_{\mathrm{\Omega }}2\sigma \mathrm{\Delta }(\mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t)\cdot \mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt})\text{d}x\text{d}s+{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Delta }f\cdot \mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt})\text{d}x\text{d}s\\ & \le C(\epsilon )\Vert \mathrm{\Delta }f{\Vert }_{L^1(0,t;L^2(\mathrm{\Omega }))}^2+2\epsilon \Vert \mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt}){\Vert }_{L^{\mathrm{\infty }}(0,t;L^2(\mathrm{\Omega }))}^2+C(ϵ){\int }_0^t\Vert \mathrm{\Delta }(\mathrm{\nabla }\psi \cdot \mathrm{\nabla }{\psi }_t){\Vert }_{L^2(\mathrm{\Omega })}^2\text{d}s.\end{array}$$

(4.15)

We next provide a bound on

$$\begin{array}{r}\begin{array}{rl}\Vert \mathrm{\Delta }(\mathrm{\nabla }\psi & \cdot \mathrm{\nabla }{\psi }_t){\Vert }_{L^2(\mathrm{\Omega })}\\ & \lesssim \Vert \mathrm{\nabla }\psi {\Vert }_{H^2(\mathrm{\Omega })}\Vert \mathrm{\nabla }{\psi }_t{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}+\Vert \mathrm{\nabla }\psi {\Vert }_{W^{1,4}(\mathrm{\Omega })}\Vert \mathrm{\nabla }{\psi }_t{\Vert }_{W^{1,4}(\mathrm{\Omega })}+\Vert \mathrm{\nabla }{\psi }_t{\Vert }_{H^2(\mathrm{\Omega })}\Vert \mathrm{\nabla }\psi {\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\\ & \lesssim \Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_t{\Vert }_{L^2(\mathrm{\Omega })}\Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })},\end{array}\end{array}$$

(4.16)

where in the last line we used the Sobelev embeddings $H^2(\mathrm{\Omega })\hookrightarrow L^{\mathrm{\infty }}(\mathrm{\Omega })\cap W^{1,4}(\mathrm{\Omega })$ as well as the elliptic estimate (2.4)

$$\begin{array}{r}\Vert \psi {\Vert }_{H^3(\mathrm{\Omega })}\lesssim \Vert \mathrm{\nabla }\mathrm{\Delta }\psi {\Vert }_{L^2(\mathrm{\Omega })},\end{array}$$

(4.17)

since $\mathrm{\Delta }\psi {|}_{\mathrm{\partial }\mathrm{\Omega }}=0$ due to the definition of the solution space in (3.2).

Thus, going back to (4.15), we obtain

$$\begin{array}{rl}{\mathrm{R}}_1\le & C(\epsilon )\Vert \mathrm{\Delta }f{\Vert }_{L^1(0,t;L^2(\mathrm{\Omega }))}^2+2\epsilon \Vert \mathrm{\Delta }({\psi }_t+\tau {\psi }_{tt}){\Vert }_{L^{\mathrm{\infty }}(0,t;L^2(\mathrm{\Omega }))}^2+C(\epsilon ){\mathbf{D}}^2(t)+C(\epsilon )\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^4,\end{array}$$

where we have used Lemma 4.3.

We now turn our attention to the term

$$\begin{array}{r}{\mathrm{R}}_2=-{\int }_0^t{\int }_{\mathrm{\Omega }}\sigma \mathrm{\nabla }|\mathrm{\nabla }\psi {|}^2\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\text{d}s+{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Delta }\tilde{f}\cdot \mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\text{d}s,\end{array}$$

whose terms can be estimated as follows:

$$\begin{array}{rl}|{\int }_0^t{\int }_{\mathrm{\Omega }}\sigma & \mathrm{\nabla }|\mathrm{\nabla }\psi {|}^2\cdot \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t)\text{d}x\text{d}s|\\ & \lesssim \epsilon \Vert \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t){\Vert }_{L^{\mathrm{\infty }}(0,t;L^2(\mathrm{\Omega }))}^2+C(\epsilon ){\Vert |\mathrm{\nabla }\psi {|}^2\Vert }_{L^1(0,t;H^1(\mathrm{\Omega }))}^2\\ & \lesssim \epsilon \Vert \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t){\Vert }_{L^{\mathrm{\infty }}(0,t;L^2(\mathrm{\Omega }))}^2+C(\epsilon ){\Vert |\mathrm{\nabla }\psi {|}^2\Vert }_{L^1(0,t;H^2(\mathrm{\Omega }))}^2\\ & \lesssim \epsilon \Vert \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t){\Vert }_{L^{\mathrm{\infty }}(0,t;L^2(\mathrm{\Omega }))}^2+C(\epsilon ){\Vert \mathrm{\nabla }\psi \Vert }_{L^2(0,t;H^2(\mathrm{\Omega }))}^4\\ & \lesssim \epsilon \Vert \mathrm{\nabla }\mathrm{\Delta }(\psi +\tau {\psi }_t){\Vert }_{L^{\mathrm{\infty }}(0,t;L^2(\mathrm{\Omega }))}^2+C(\epsilon ){\Vert \psi \Vert }_{L^2(0,t;H^3(\mathrm{\Omega }))}^4,\end{array}$$

where in the last line we used that $H^2(\mathrm{\Omega })$ is an algebra for $\mathrm{\Omega }\in {\mathbb{R}}^d$ with $d\le 3$ . Using Lemma 4.4, we furthermore estimate:

$$\begin{array}{r}{\Vert \psi \Vert }_{L^2(0,t;H^3(\mathrm{\Omega }))}^4\lesssim {\mathbf{D}}^2(t)+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^4\end{array}$$

The estimate for the source term $\tilde{f}$ is treated similarly to above, this yields the inequality:

$$\begin{array}{rl}{\mathrm{R}}_2\le & C(\epsilon )\Vert \mathrm{\Delta }\tilde{f}{\Vert }_{L^1(0,t;L^2(\mathrm{\Omega }))}^2+2\epsilon \mathbf{E}(t)+C(\epsilon ){\mathbf{D}}^2(t)+C(\epsilon )\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^4.\end{array}$$

Thus, by choosing $\epsilon$ small enough, we obtain (4.14).

Proof of Theorem 3.1. Proof of Theorem 3.1.

Let $t>0$ and define $\mathbf{Y}(t):=\mathbf{E}(t)+\mathbf{D}(t)$ . Hence, using (4.14), we obtain

$$\begin{array}{r}\mathbf{Y}(t)\lesssim \mathbf{Y}(0)+\Vert \tilde{f}{\Vert }_{W^{1,1}({\mathbb{R}}^{+};H^2(\mathrm{\Omega }))}^2+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^4+\mathbf{Y}{(t)}^2.\end{array}$$

Applying Lemma 2.4, we deduce that $\mathbf{Y}(t)$ is uniformly bounded provided that

$$\begin{array}{r}\mathbf{Y}(0)+\Vert \tilde{f}{\Vert }_{W^{1,1}({\mathbb{R}}^{+};H^2(\mathrm{\Omega }))}^2+\Vert \mathrm{\nabla }\mathrm{\Delta }{\psi }_0{\Vert }_{L^2(\mathrm{\Omega })}^4,\end{array}$$

where $\mathbf{Y}(0)=\Vert {\psi }_2{\Vert }_{L^2(\mathrm{\Omega })}^2+\Vert \mathrm{\nabla }\mathrm{\Delta }({\xi }_0+\tau {\psi }_0){\Vert }_{L^2(\mathrm{\Omega })}^2$ , is sufficiently small. This complete the proof of Theorem 3.1. We refer the reader to, for example, Racke and Said-Houari (2021, p. 8) and Ji et al. (2022, p. 15) for similar arguments, sometimes based on a bootstrap argument by Tao (2006, Proposition 1.21) comparable to Lemma 2.4.

5. Conclusion

In this work, and even though we are in the critical regime in terms of the coefficients, we have shown that fractional damping is sufficient to ensure global existence for the nonlinear JMGT equation with quadratic gradient nonlinearity when the problem data are small. We also ensure that the solution’s energy remains small as time goes to infinity. A natural future question will be to study the precise asymptotic behavior as time goes to infinity. In particular, we would like to know under which conditions on the problem data and on the memory kernel, the solution’s energy decays, and to determine the corresponding decay rate.

Future work will also be tasked with proving the global existence of the fractional JMGT equation with Westervelt nonlinearity (i.e., ${\psi }_{tt}{\psi }_t$ ). In the present framework, we do not obtain sufficient control ( $L^2$ -control specifically) on the quantity ${\psi }_{tt}$ to argue global existence for Westervelt-type with fractional damping. Notice that this differs from the strong damping case ( $\mathfrak{K}={\delta }_0$ , $\delta >0$ ), where the full energy norm of the linearized equation can be shown to decay; see, for example, Kaltenbacher et al. (2011). Thus, one likely needs additional assumptions to be made on the memory kernel to achieve this result.