Homoclinic solutions of discrete $p$-Laplacian equations containing both advance and retardation

1.   Introduction

We consider the following Cauchy problem for the JMGT-thermoviscoelastic plate with Cattaneo-type heat conduction:

with initial data

where $\tau, \rho, k^*, k, m, \tau_0, \kappa$ are positive constants, and the critical parameter is given by $K: = k-\tau k^*$.

The linear Jordan-Moore-Gibson-Thompson equation (JMGT) is expressed as follows:

where $A$ is a strictly positive operator in a Hilbert space, and $\tau, \delta, \beta, \gamma$ are positive constants.

Equation (1.3) originally arises as a model for wave propagation in viscous, thermally relaxing fluids (cf. ^[5,6]). A similar form of the equation appears in the standard linear solid model (cf. ^[4]) and in the formulation of a relaxation parameter within the Green–Naghdi type Ⅲ theory (cf. ^[3,10]), particularly when $A = -\Delta$. Furthermore, Eq (1.3) serves as a potential model for vertical displacements in viscoelastic plates (cf. ^[7]) when $A = \Delta^2$.

In recent years, there has been growing interest in the study of problem (1.3). In ^[2], the authors investigated the MGT-viscoelastic plate coupled with the Fourier law and type Ⅲ heat conduction, proving that the corresponding semigroups are analytic in the subcritical case $K > 0$. Subsequently, ^[1] focused on the MGT-viscoelastic plate with Cattaneo heat conduction and established the following decay result in the subcritical case $K > 0$:

● the subcritical case $K > 0$:

where $U = (u_t+\tau u_{tt}, \Delta u_t, \Delta(u+\tau u_t), \theta, q)^T$ and $U_0 = U(x, 0)$.

In this work, we improve upon the results in ^[1] for the subcritical case $K > 0$ and establish the decay result for the critical case $K = 0$. Additionally, we analyze the eigenvalues to demonstrate the optimality of the decay results in both cases. The specific decay rates are as follows:

● the subcritical case $K > 0$:

where $W = (u_t+\tau u_{tt}, \Delta u_t, \Delta(u+\tau u_t), \theta, q)^T$ and $W_0 = W(x, 0)$.

● the critical case $K = 0$:

where $Z = (u_t+\tau u_{tt}, \Delta(u+\tau u_t), \theta, q)^T$ and $Z_0 = Z(x, 0)$.

The paper is organized as follows. In Section 2, we introduce some notations and present our main results. Section 3 is devoted to proving the decay estimates for the JMGT-thermoviscoelastic plate with Cattaneo heat conduction. Finally, in Section 4, we establish the optimality of the decay rates obtained.

Before closing this section, we give some notations to be used below. Let the Fourier transform of a function $f = f(x)$ be denoted by $\hat{f} = \hat{f}(\xi)$, defined as

and let the complex conjugate of $\hat{u}$ be denoted by $\bar{\hat{u}}$.

2.   Preliminaries and main results

In this section, we state our main results.

Taking the Fourier transform of system (1.1)-(1.2), we have

with initial data

where $\xi\in\mathbb{R}^n$. By the new variables

we obtain

Then, we state the following pointwise estimates and decay results.

Theorem 2.1. Let

where $(\hat{u}(\xi, t), \hat{\theta}(\xi, t), \hat{q}(\xi, t))$ is the Fourier image of the solution $(u(x, t), \theta(x, t), q(x, t))$. Assume that $K > 0$. Then, $\hat{W}$ satisfies the following pointwise estimate

for any $t\ge 0$, where $\rho_1(\xi): = \frac{|\xi|^2}{1+|\xi|^2}$, and where $C, c > 0$ are independent of $t, \xi$, and the initial data.

Furthermore, let $W = (u_t+\tau u_{tt}, \Delta u_t, \Delta(u+\tau u_t), \theta, q)^T$, where $(u(x, t), \theta(x, t), q(x, t))$ is the solution of problem (1.1), (1.2), and $W_0 = W(x, 0)\in H^{s}(\mathbb{R}^n)\cap L^{1}(\mathbb{R}^n)$, where $s$ is nonnegative. Then, $W$ satisfies the decay estimate

for all $0\leq p \leq s$.

Remark 2.2. The decay result (2.5) does not exhibit the regularity-loss phenomenon. In consideration of ^[1], the decay estimate presented here aligns with the exponential stability of the MGT-viscoelastic plate with Cattaneo-type heat conduction in a bounded domain. At the same time, we improve the result in unbounded domain obtained in ^[1]. Noting the asymptotic expansion of the eigenvalues in Section 4, we find that the exponent in pointwise estimate (2.4) is optimal. Thus, the decay estimate (2.5) is optimal.

Remark 2.3. Note that the decay estimate (2.5) and the MGT-viscoelastic plate with the Gurtin-Pipkin thermal law in ^[11] exhibit the same decay rate when $K > 0$, despite the absence of a regularity-loss phenomenon in (2.5).

Theorem 2.4. Let

where $(\hat{u}(\xi, t), \hat{\theta}(\xi, t), \hat{q}(\xi, t))$ is the Fourier image of the solution $(u(x, t), \theta(x, t), q(x, t))$. Assume that $K = 0$. Then, $\hat{Z}$ has the following pointwise estimate

for any $t\ge 0$, where $\rho_2(\xi): = \frac{|\xi|^2}{(1+|\xi|^2)^2}$. Furthermore, let $Z = (u_t+\tau u_{tt}, \Delta(u+\tau u_t), \theta, q)^T$, where $(u(x, t), \theta(x, t), q(x, t))$ is the solution of problem (1.1)-(1.2), and $Z_0 = Z(x, 0)\in H^{s}(\mathbb{R}^n)\cap L^{1}(\mathbb{R}^n)$, where $s$ is nonnegative. Then, $Z$ satisfies the following decay estimate

for all $0\leq p+l\leq s$.

Remark 2.5. According to ^[11], we find that the MGT-viscoelastic plate with Gurtin-Pipkin thermal law has the same decay result as (2.7) under the condition $K = 0$, including both the decay rate and regularity-loss phenomenon.

3.   Decay estimates

In this section, we consider the decay estimates of the norm related to (1.1)-(1.2).

3.1.   Decay estimates–the case $K > 0$

In this subsection, we define the energy functional of system (2.3) as

which is equivalent to $|\hat{W}(\xi, t)|^2$. To derive our main result, we begin by stating and proving several lemmas.

Lemma 3.1. Let $(\hat{u}, \hat{\varphi}, \hat{w}, \hat{\theta}, \hat{q})$ be the solution of (2.3). Assume that $K > 0$. Then, $\hat{E}(\xi, t)$ satisfies

Lemma 3.2. The following inequality holds true:

for any $\varepsilon_1 > 0$, where

Proof. We can easily obtain

By virtue of Young's inequality, for any $\varepsilon_1 > 0$, we have

Combining (3.3)–(3.5), we obtain the desired result (3.2). □

Lemma 3.3. The functional

satisfies

for any $\varepsilon_2, \varepsilon_2' > 0$.

Proof. It is easy to obtain

Applying Young's inequality with $\varepsilon_2, \varepsilon_2' > 0$, we get

Thanks to (3.7)–(3.10), we deduce (3.6). □

Lemma 3.4. Define the functional

Then,

for any $\varepsilon_3 > 0$.

Proof. Multiplying (2.3)₄ and (2.3)₅ by $i\tau_0 \xi\bar{\hat{q}}$ and $(-i\tau_0\xi\bar{\hat{\theta}})$, respectively, adding the resulting equations, and taking the real part, we have

To eliminate ${\rm{Re}}(i\xi^3\hat{w}\bar{\hat{q}})$, we multiply (2.3)₂ and (2.3)₅ by $im\tau\tau_0 \xi^3\bar{\hat{q}}$ and $-im\tau\tau_0 \xi^3\bar{\hat{\varphi}}$, respectively. Then, combining the resulting equations and taking real parts, we have

Summing up (3.12) and (3.13), we arrive at

Young's inequality yields, for any $\varepsilon_3 > 0$,

Hence, plugging (3.15)–(3.17) into (3.14), we arrive at (3.11). □

We now proceed to prove our main result.

Proof of Theorem 2.1. We define the Lyapunov functional as follows:

where $N, N_2,$ and $N_3$ are positive constants to be determined later. By utilizing the previously established lemmas, we obtain

At this stage, we aim to determine the constants in Eq (3.18). We begin by selecting

Next, we fix $N_2 > \frac{1}{m-\varepsilon_2}$ and choose $\varepsilon_2' < \frac{k^*}{\rho N_2}-\frac{2\varepsilon_1}{N_2}$. Then, we select $N_3$ such that

Finally, we choose $N$ sufficiently large to satisfy

Consequently, we obtain, with a positive constant $C_1$,

where

From the definitions of $\hat{E}(\xi, t)$ and $L_1(\xi, t)$, we know that there exist two positive constants $C_2$ and $C_3$ such that the following relation holds

Thus, Eq (3.19) transforms into

Finally, the estimate in (3.20) leads to the desired result (2.4), allowing us to derive the decay estimate (2.5). The proof of (2.5) is the same as the one of Theorem 3.6 in ^[8], so we omit it here. □

3.2.   Decay estimates–the case $K=0$

Based on Lemmas 3.1–3.3 and the condition $K = 0$, we have the following conclusion.

Lemma 3.5. Under the condition $K = 0$, the energy functional (3.1) becomes

and then $\hat{\mathcal{E}}(\xi, t)$ satisfies

and the following inequality holds true:

for any $\epsilon_1 > 0$ and $\epsilon_2, \epsilon_2' > 0$.

Proof. It is straightforward to obtain Eqs (3.22) and (3.23). It follows from (3.7) that

Using Young's inequality, we get

where $\epsilon_2, \epsilon_2' > 0$. Collecting the above estimates, we obtain (3.24). □

Lemma 3.6. The functional

satisfies

for any $\epsilon_3, \epsilon_3' > 0$.

Proof. Taking (3.12) into account, we arrive at

Taking advantage of Young's inequality, we obtain (3.25). The proof is complete. □

Proof of Theorem 2.4. We define the new Lyapunov functional $L_2(\xi, t)$ associated to the case $K = 0$ as follows:

Taking the derivative of (3.27) with respect to $t$ and making use of the above lemmas, we derive

At this point, we choose our constants carefully like before. First, we pick

Next, we choose

Then, we fix $\epsilon_3'$ satisfying

Finally, we choose $N$ large enough such that

Thus, we arrive at

where

From the definition of $\hat{\mathcal{E}}(\xi, t)$ and (3.27), it is obviously that $L_2(\xi, t)\sim (1+|\xi|^2)^2\hat{\mathcal{E}}(\xi, t)$. Then,

Thus, we achieve the desired pointwise estimate (2.6), which leads to the conclusion (2.7). The proof process of (2.7) is similar to the proof of Theorem 3.1 in ^[9], so we omit it here. □

4.   Asymptotic expansion of eigenvalues

From now on, we study the asymptotic expansion of the eigenvalues for $|\xi|\rightarrow 0$ and $|\xi| \rightarrow \infty$ to show the optimality.

Let $\hat{V} = (\hat{u}, \hat{\varphi}, \hat{w}, \hat{\theta}, \hat{q})^{T}$ and $\hat{V}_0 = (\hat{u}_0, \hat{\varphi}_0, \hat{w}_0, \hat{\theta}_0, \hat{q}_0)^{T}$. Then, we can rewrite system (2.1)-(2.2) as

where

For (4.1), the solution is given by

where $e^{t\hat{\Phi}(i\xi)}$ denotes the matrix exponential with

Setting $\zeta = i\xi$, we get

Let $\lambda_j(\zeta)$ denote the eigenvalues of the matrix $\hat{\Phi}(\zeta).$ By the direct calculation, we find that the characteristic polynomial of $\hat{\Phi}(\zeta)$ is

Lemma 4.1. The real parts of the eigenvalues of (2.1)-(2.2) satisfy the following asymptotic expansion:

for $|\xi|\rightarrow 0$.

Proof. We consider $\lambda_j(\zeta)$ the following asymptotic expansion:

for $|\zeta|\rightarrow 0$. Straightforward computations yield

where $\phi_j$ are the roots of equation $\rho X^3-\rho\kappa^2 X^2+(m^2+k^*)X-\kappa^2 k^* = 0$. To demonstrate that ${\rm{Re}}(\phi_j) > 0$, we set

Since $f(0)f(\kappa^2) < 0$, we conclude that $f$ has at least one real root $X = \phi_1$ in the interval $(0, \kappa^2)$. We express Eq (4.5) in the form

with $d_1 = -\rho \kappa^2 +\phi_1\rho < 0$ and $d_0 = \frac{\kappa^2 k^*}{\phi_1} > 0$. For the remaining roots $\phi_2$ and $\phi_3$, we find that

This implies that if $\phi_2$ and $\phi_3$ are real, they are both positive; if $\phi_2$ and $\phi_3$ are complex conjugates, then

Thus, we have arrived at the desired result in (4.3). This completes the proof. □

When $|\zeta|\rightarrow \infty$, we rewrite $\hat{\Phi}(\zeta)$ as $\hat{\Phi}(\zeta) = \zeta^2\hat{\Psi}(\zeta^{-1})$, where $\hat{\Psi}(\zeta^{-1}) = B-\zeta^{-1}A-\zeta^{-2}L-\zeta^{2}D$, and consider the eigenvalues $\mu_j(\zeta^{-1})$, for $j = 1, 2, 3, 4, 5$ of the matrix $\hat{\Psi}(\zeta^{-1})$. Meanwhile, these eigenvalues $\mu_j(\zeta^{-1})$ are the solutions to the characteristic equation

Lemma 4.2. When $K > 0$, the real parts of the eigenvalues of (2.1)-(2.2) satisfy the asymptotic expansion

for $|\xi| \rightarrow \infty$.

When $K = 0$, the real parts of the eigenvalues of (2.1)-(2.2) satisfy the asymptotic expansion

for $|\xi| \rightarrow \infty$.

Proof. As $|\zeta|\rightarrow \infty$, similar calculation as before yields

Consequently, our conclusion holds. □

5.   Conclusions

In this work, we have investigated the Cauchy problem for the JMGT-viscoelastic plate system coupled with Cattaneo-type heat conduction, focusing on the optimal decay rates of solutions in both the subcritical and critical cases. Our main contributions can be summarized as follows:

(1) The subcritical case: We proved that the system exhibits exponential decay without regularity loss, improving upon previous results in the literature. This indicates that the dissipation mechanism in this regime preserves the initial regularity of solutions.

(2) The critical case: In contrast, we observed a regularity-loss phenomenon in the decay rates, demonstrating a fundamental difference in the long-time behavior compared to the subcritical case.

Author contributions

Danhua Wang performed the formal analysis and wrote the manuscript; Kewang Chen performed the validation and review. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they haven't used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The first author is supported by the Basic Science (Natural Science) Research project of higher education institutions in Jiangsu Province (Grant No. 24KJB110021) and Youth Special project of Nanjing Xiaozhuang University (No. 2022NXY38).

Conflict of interest

The authors declare that there is no conflict of interest.