Viscoelastic Wave Equation with Variable Exponents

• Viscoelastic wave equations with variable exponents are models that incorporate spatial and temporal variability in damping and source terms, enabling accurate representation of nonuniform dissipative behaviors.
• They leverage variable-exponent Lebesgue and Sobolev spaces along with convolution quadrature and sum-of-exponentials schemes to capture memory effects and fractional dynamics.
• The models yield explicit energy decay estimates and robust numerical algorithms, providing insights for applications in geophysics, materials science, and heterogeneous multiscale media.

Viscoelastic wave equations with variable exponents generalize classical viscoelastic wave models by allowing the nonlinearities governing both damping and source mechanisms, as well as memory kernels, to depend on spatial or temporal position through variable exponents. Such models arise in the study of complex dissipative materials, memory-rich continua, and heterogeneous multiscale media where material response is governed by position-dependent physical parameters, potentially reflecting evolving microstructure or fractal properties. The inclusion of variable exponents and fractional derivatives captures highly nonuniform attenuation, dispersion, and energy dissipation regimes, giving rise to rich mathematical and computational challenges.

1. Model Equations and Exponent Conditions

Viscoelastic wave equations with variable exponents typically take the form, posed on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$ $(n\geq3)$ with Dirichlet boundary conditions,

$$u_{tt}-\Delta u+\int_0^t g(t-s)\Delta u(s)\,ds + a|u_t|^{m(x)-2}u_t = b|u|^{p(x)-2}u$$

where $u: \Omega \times (0,\infty)\rightarrow \mathbb{R}$ is the displacement, $g$ is a nonincreasing relaxation kernel, $a>0$, $b>0$ are damping/source strength coefficients, and $p(x), m(x)$ are measurable functions mapping $\Omega$ into $[2,2n/(n-2)]$, satisfying

$$2 \leq p_1 := \text{ess inf}~p(x) \leq p(x) \leq p_2 := \text{ess sup}~p(x) < \frac{2n}{n-2},$$

with analogous bounds for $m(x)$. In addition, $p(x)$ and $m(x)$ are assumed to satisfy a log-Hölder continuity condition: $$|q(x)-q(y)| \leq \frac{A}{|\ln|x-y||} \quad \text{for}~|x-y|<\delta,~q=p~\text{or}~m.$$ Initial data is prescribed by $u(x,0)=u_0(x)\in H_0^1(\Omega)$, $u_t(x,0)=u_1(x)\in L^2(\Omega)$.

Generalizations incorporate source terms involving variable-exponent logarithmic nonlinearities and fractional time derivatives, as in

$$u_{tt} - \Delta u + \int_0^t g(t-s)\Delta u(s)\,ds + u_t = \alpha|u|^{p(x)-2}u\,\ln|u| [2601.01752].$$

2. Functional Settings: Variable-Exponent Spaces

Analysis is performed within the modular and norm structures of variable-exponent Lebesgue and Sobolev spaces:

• Variable-exponent Lebesgue space: $L^{q(x)}(\Omega)$, equipped with the modular $\rho(f)=\int_\Omega|f(x)|^{q(x)}dx$ and Luxemburg norm

$$\|f\|_{q(x)} = \inf \{ \lambda>0 : \rho(f/\lambda) \leq 1 \}.$$

• Variable-exponent Sobolev space: $W_0^{1,q(x)}(\Omega)$ defined as the closure of $C_0^\infty(\Omega)$ under

$\|u\|_{1,q(x)} := \|\nabla u\|_{q(x)}.$

Key embedding results for variable-exponent spaces include [Fan-Zhao]:

• $$L^{q_2(x)}(\Omega) \hookrightarrow L^{q_1(x)}(\Omega)$$ if $q_1(x)\leq q_2(x)$.
• $W_0^{1,2}(\Omega) \hookrightarrow L^{p(x)}(\Omega)$ for exponents in the subcritical range.

Detailed treatment of variable-exponent memory convolution operators involving non-positive type Abel kernels, as in (Li et al., 1 May 2025, Zheng et al., 2024), requires careful splitting into positive-definite and perturbative components.

3. Relaxation Kernels, Memory Effects, and Fractional Operators

The relaxation kernel $g(t)$ encodes the material memory effect, subject to:

• $g\in C^1(\mathbb{R}^+)$, nonincreasing, $g(0)>0$, $1-\int_0^\infty g(s)\,ds >0$.
• The general decay condition $g'(t)\leq -\xi(t)g(t)$, with $\xi(\cdot)$ non-increasing, $\xi(0)>0$, and $\int_0^\infty\xi(s)\,ds=+\infty$.

Special cases:

• Exponential decay: $\xi(t)\equiv \text{const}>0$.
• Polynomial decay: $g'(t)+C g^\alpha(t)\leq 0$ for $\alpha\in(1,2)$.

Variable-exponent fractional operators arise via kernels of form $k(t)=t^{-\alpha(t)}/\Gamma(1-\alpha(t))$ (Abel type), or more generally, employing Caputo or Riemann–Liouville time-fractional derivatives with order $\alpha(x)$ or $\alpha(t)$. Memory kernels may lose positive-definiteness and monotonicity when exponents vary, necessitating reformulations based on convolution splittings and product-integration schemes (Zheng et al., 2024, Li et al., 1 May 2025, Jia et al., 8 Nov 2025, Guo et al., 2023).

4. Well-Posedness, Regularity, and Singularity Properties

Global existence, uniqueness, and regularity of weak solutions can be established under small initial energy assumptions and appropriate regularity of data and exponents: $0<E(0)<E_*$ with $E(t)$ the total energy functional: $$E(t)=\frac12\|u_t(t)\|_{L^2}^2 +\frac12\left(1-\int_0^t g(s)\,ds\right)\|\nabla u(t)\|_{L^2}^2 +\frac12\int_0^t g(t-s)\|\nabla u(t)-\nabla u(s)\|_{L^2}^2ds -\int_\Omega\frac{b}{p(x)}|u(t)|^{p(x)}dx.$$

Well-posedness in the context of non-positive type variable-exponent memory kernels is achieved via perturbation methods:

• Splitting $k(t)=k_{\alpha_0}(t)+r(t)$, with fixed reference order $\alpha_0$ and $r(t)$ representing the variable-exponent perturbation.
• Establishing solution regularity in weighted Sobolev-Bochner spaces, e.g., $u\in H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^2(\Omega))$ (Li et al., 1 May 2025).

Initial singularity profiles in high-order norms are governed by the reference exponent $\alpha_0=\alpha(0)$, yielding $u(t)=O(t^{-\alpha_0/2})$ in $H^2$ as $t\to0$.

5. Energy Dissipation and Asymptotic Decay Rates

Rigorous decay estimates for the energy functional follow from differential inequalities: $$E'(t)\leq -a\int_\Omega|u_t|^{m(x)} dx -\frac12 g(t)\|\nabla u\|_2^2 +\frac12 \int_0^t g'(t-s)\|\nabla u(t)-\nabla u(s)\|_2^2 ds$$ ensuring nonincreasing energy. Under the general kernel condition $g'(t)\leq -\xi(t)g(t)$, explicit decay rates are available:

• For $m_2>2$:

$$E(t)\leq E(0)\left[1+K(m_2-2)\int_0^t\xi(s)ds\right]^{-1/(m_2-2)}$$

• For $m_2=2$ (linear damping):

$E(t)\leq E(0)\exp(-K\int_0^t\xi(s)\,ds)$

• For polynomially decaying kernels $g'(t)+C g^\alpha(t)\leq 0$ $(\alpha\in(1,2))$, polynomial and exponential decay rates are similarly established (Liao et al., 2020, Peng et al., 5 Jan 2026).

For models incorporating logarithmic nonlinearity and weak damping, both general polynomial decay and refined uniform exponential/polynomial decay estimates are proven under suitable kernel hypotheses involving convex functions $G$, e.g.,

$$E(t)\leq k_2 G_1^{-1}\left(k_1\int_{t_1}^t\zeta(s)ds\right)$$

or

$$E(t)\leq E(0)\left[1+K'(q-1)\int_0^t\xi(s)ds\right]^{-1/(q-1)}$$

for $g'(t)\leq -\xi(t)g^q(t)$, $1\leq q<2$ (Peng et al., 5 Jan 2026).

6. Numerical Schemes and Fast Algorithms

Numerical analysis of viscoelastic wave equations with variable exponents focuses on schemes robust to the indefiniteness and non-monotonicity of memory kernels. High-order accurate methods include:

• Convolution quadrature (CQ) schemes based on second-order BDF or product-integration rules, yielding $\alpha_0$-order or uniformly second-order temporal accuracy, paired with spatial Ritz–Galerkin discretization (Zheng et al., 2024, Jia et al., 8 Nov 2025).
• Adaptive quadrature-based sum-of-exponentials (SOE) compressions of Caputo derivatives for time-fractional models (Guo et al., 2023). SOE representations enable mapping to generalized Maxwell-body rheologies, yielding memory-variable systems with nearly optimal storage efficiency.
• Fast divide-and-conquer algorithms exploiting the translational invariance of discrete convolution weights, reducing temporal complexity from $O(MN^2)$ to $O(MN\log^2 N)$ for $M$ spatial degrees of freedom and $N$ time steps (Jia et al., 8 Nov 2025).

Error analyses guarantee second-order convergence in space and up to second order (uniformly or $\alpha_0$-order) in time under regularity constraints on initial data and variable exponents.

7. Physical Interpretation, Modeling Implications, and Extensions

Variable-exponent viscoelastic wave equations model physical phenomena including anomalous dispersion, frequency-independent attenuation, and evolving material creep regimes. Spatial and temporal dependence of exponents can describe:

• “Fractal-dimension” driven memory effects and microstructure evolution under cyclic loads (Li et al., 1 May 2025).
• Diffusive wave propagation intermediate between classical diffusion and pure wave motion (Zheng et al., 2024).
• Frequency-independent $Q$ attenuation in geophysical settings (Guo et al., 2023).

A plausible implication is that flexible modeling of exponent functions $p(x),~m(x),~\alpha(x,t)$ enables simulation of multiscale materials with both strong nonlinearity and long-range memory. Computational advances such as SOE compression and fast algorithms support large-scale simulation and seismic inversion.

Mathematical innovations, including variable-exponent space theory, memory kernel decomposition, and advanced energy estimates, broaden the analytic and computational toolkit for tackling viscoelastic wave propagation in nonuniform and evolving media.

---

Paper ID	Focus	Key Contribution
(Liao et al., 2020)	Variable-exponent damping/source, energy decay	General explicit decay rates, first variable-exponent analysis
(Li et al., 1 May 2025)	Variable-exponent memory, well-posedness	Perturbation analysis, initial singularity profile
(Peng et al., 5 Jan 2026)	Logarithmic nonlinearity, decay rates	Refined polynomial/exponential decay under weak damping
(Jia et al., 8 Nov 2025)	Variable-order fractional wave, numerics	Fast FDAC algorithm, high-order error analysis
(Zheng et al., 2024)	Variable-exponent fractional diffusion-wave	Reformulation for indefinite kernels, high-order methods
(Guo et al., 2023)	SOE approximations, geophysical application	Maxwell-body equivalence, uniform error bounds

These results collectively delineate the state-of-the-art for analysis and simulation of viscoelastic wave equations with variable exponents, establishing well-posedness, explicit decay rates, robust high-order algorithms, and applications to material sciences and geophysics.