Wave Equation with Memory Effects

• Wave Equation with Memory is a class of evolution equations where past states influence current behavior through convolution integrals, modeling hereditary effects.
• It employs various memory kernels—exponential, power-law, or Mittag-Leffler—to capture complex damping, dispersion, and stabilization behaviors in viscoelastic and anomalous media.
• Recent advances highlight controllability, regularity, and efficient numerical inversion methods, linking theoretical insights to practical applications in engineering and physics.

A wave equation with memory is a class of evolution equations in which the present state of the system depends not only on current but also on past states via convolution integrals, typically modeling hereditary or aftereffect phenomena in materials such as viscoelastic solids, thermoviscoelastic fluids, or electromagnetic and gravitational fields exhibiting “memory” effects. This structure manifests mathematically as integro-differential operators, convolution with a memory kernel, or via fractional derivatives. These systems exhibit rich dynamics including altered dispersion, controllability properties, stabilization phenomena, and non-classical regularity, and are central to both modern mathematical physics and control theory.

1. Mathematical Structures and Canonical Models

The canonical form of the wave equation with memory on a domain $\Omega$ is

$$u_{tt}(x,t) - \Delta u(x,t) - \int_0^{+\infty} p(s) \Delta u(x,t-s)\, ds = f(x,t),$$

where $p(s)$ is the memory kernel. Strongly related forms include additional internal or boundary viscous damping, time delays, or modifications in the principal operator or the source term. Prominent examples include:

• Viscoelastic wave equation: Memory is modeled through convolution with a kernel (often exponentially decaying or a linear combination of exponentials), as in $K(t) = \sum_j C_j e^{-Y_j t}$ (Romanov et al., 2015).
• Fractional wave equations: The kernel takes a power-law or Mittag-Leffler form, representing anomalous diffusion-wave behavior (Sandev et al., 2019).
• Delay or feedback-modified equations: Additional delayed damping $\propto u_t(x, t-T)$ is incorporated, enabling explicit quantification of stabilization/destabilization mechanisms (Alabau-Boussouira et al., 2014).
• Nonlocal variants: Memory is embedded in both principal and lower-order parts, possibly with fractional Laplacian spatial operators (Biccari et al., 2019).

Augmented system reformulations are frequently employed (e.g., with history variables $n^t(x,s)$ or delay-localizing variables $z(x,p,t)$ (Alabau-Boussouira et al., 2014)) to cast the memory problem as an abstract evolution equation on a suitable Hilbert or Banach space.

2. Memory Kernels and Physical Meaning

The nature and properties of the memory kernel $p(\cdot)$ (or $K(\cdot), \eta(\cdot)$, etc.) crucially determine the system's qualitative behavior.

• Exponential-type kernels ($p(t) = p_0 e^{-a t}$): Lead to exponential energy decay and permit reduction to finite-dimensional ODE chains (e.g., generalized Maxwell bodies), common in viscoelasticity (Romanov et al., 2015, Guo et al., 2023).
• Sum-of-exponentials, Prabhakar, or Mittag-Leffler kernels: These support accurate modeling of hereditary effects spanning several time scales and enable efficient SOE-based numerical schemes (Guo et al., 2023).
• Power-law kernels: Arise in anomalous diffusion, yielding fractional equations with subdiffusive or superdiffusive scaling for the mean squared displacement (Sandev et al., 2019).
• Kernels defined via Bessel functions: As in certain Bessel-like media, the Laplace transform $$\widetilde \Phi(s) = (2/\sqrt{s\tau}) [I_1(\sqrt{s\tau}) / I_0(\sqrt{s\tau})]$$ encapsulates complex, frequency-dependent dispersion (Giusti et al., 13 Oct 2025).

Selection and analysis of the kernel shape (decay, positivity, complete monotonicity, or structural properties) is central: for example, decay rate restrictions such as $\int_0^{\infty} p(t)\, dt < 1$, $p'(t) \leq -a p(t)$ ensure dissipativity and exclude resonance phenomena (Alabau-Boussouira et al., 2014).

3. Stability and Energy Decay Mechanisms

Memory terms can have both stabilizing and destabilizing effects depending on their interplay with other feedback mechanisms.

• Viscoelastic Damping Stability: With exponentially decaying kernels and $\int_0^{\infty} p(t)\, dt < 1$, memory induces exponential energy decay, robust against small destabilizing time-delayed damping (Alabau-Boussouira et al., 2014). The core result is

$F(t) \leq C e^{-\omega t} F(0)$

provided the delay amplitude $|k|$ is below a threshold explicitly determined by the kernel and domain.

• Time-Delayed Damping: Isolated, such delay can destabilize the system (e.g., for negative $k$ the system can exhibit anti-damping and exponential growth). Memory counteracts this, provided the dissipativity induced by $p(\cdot)$ dominates (Alabau-Boussouira et al., 2014).
• General Damped Models: The simultaneous presence of local/internal (possibly fractional) damping and hereditary (memory) damping can produce either global existence with exponential/polynomial decay or finite-time blow-up, primarily governed by the sign and magnitude of initial energy and detailed Lyapunov functionals (Boulaaras et al., 2020).
• Dispersive Memory: In Bessel-like media, dispersive memory alters phase and group velocities (with $v_p(\omega) < v_g(\omega) < c$ for all $\omega$), leading to anomalous dispersion—a fundamentally different asymptotic regime than in memoryless wave propagation (Giusti et al., 13 Oct 2025).

4. Controllability and Moment-Based Methods

Controllability of wave equations with memory is a nuanced topic involving lifting properties from the memoryless problem and careful spectral-moment analysis:

• Lifting via Cosine Operators: If the memoryless equation is controllable in time $2T$, then—under natural conditions—the memory-affected system is controllable in time $T+\varepsilon$ ($\forall\,\varepsilon>0$), where the memory acts as a singular perturbation that slightly increases the minimal control time (Pandolfi, 2014).
• Exact Controllability with Distributed Control: For kernels that are linear combinations of exponentials, exact controllability to rest (i.e., driving both displacement and its time derivative to zero, with state remaining at rest after control deactivation) is achievable with distributed control, with precisely constructed bounded control functions via moment equations (Romanov et al., 2015).
• Memory-Type Null Controllability: Stronger notions require vanishing not only the present state but also the memory component at a final time. This is possible with a moving control mechanism, typically necessary due to the instantaneous propagation of the memory (ODE) component (Lu et al., 2015, Biccari et al., 2019, Biccari et al., 2018). The Moving Geometric Control Condition (MGCC) ensures that both wave-like and memory-like characteristics are intercepted by the control region.
• Spectral and Moment Problems: Many controllability results reduce to Aronszajn-type moment problems for the eigenfunction expansions of the solution, with careful handling of biorthogonal families (e.g., via entire functions of sine type and Riesz basis theory) required for nonharmonic spectra induced by the memory operator (Biccari et al., 2018).

5. Regularity, Well-Posedness, and Singular Limits

Memory terms significantly influence the regularity of solutions and allow for unifying frameworks between different equations:

• Regularity Transfer via Volterra Theory: Through MacCamy’s trick (integration by parts in the memory integral), regularity of the memoryless wave equation can be “lifted” to the memory setting. For the Moore–Gibson–Thompson (MGT) equation, an extra unit of spatial regularity is gained for the time derivative compared to standard hyperbolic models (Bucci et al., 2017).
• Boundary Data and Trace Regularity: Square-integrable boundary data propagate into the domain analogously to the memoryless situation, with sharp trace estimates available via operator theory and Volterra equations (Bucci et al., 2017).
• Singular Limit Behavior: The viscoelastic damped wave equation with memory converges (in suitable norms) to the MGT equation as the “thermal relaxation” parameter tends to zero, with the convergence rate influenced by the regularity and moment conditions on initial data (Chen, 2020). This connects hyperbolic, viscoelastic, and thermoacoustic models.

6. Inverse Problems and Numerical Methods

Wave equations with memory pose substantial analytic and computational challenges due to their nonlocality in time. Developments in this direction include:

• Inverse Memory Kernel Identification: By supplementing the direct problem with integral overdetermination conditions (e.g., average velocity or boundary measurement), global existence and uniqueness of the memory kernel can be achieved via contraction mapping in Sobolev spaces after reformulation to homogeneous boundary conditions and suitable energy estimates (Totieva et al., 12 May 2025).
• Efficient Numerical Schemes: For practical simulation of fractional (constant-Q) viscoelastic wave equations, sum-of-exponentials (SOE) approximations to the memory kernel, constructed via generalized Gaussian quadrature, efficiently compress the history variable requirement without degrading accuracy—enabling high-dimensional simulation and connection to generalized Maxwell body models (Guo et al., 2023, Xiong et al., 2021).
• Laplace Inversion and Transient Response: Direct numerical inversion of the Laplace transform (notably via Talbot’s method) supports accurate computation of transient wave responses in media with highly dispersive, memory-induced dynamics, such as Bessel-like wave propagation (Giusti et al., 13 Oct 2025).

7. Applications and Broader Implications

Wave equations with memory underlie modeling and control in viscoelasticity (both bulk and boundary dissipation), composite and hereditary fluids, seismic wave propagation with frequency-independent Q, gravitational and electromagnetic wave memory effects, and non-Fickian anomalous diffusion. Advances in well-posedness, stabilization, controllability, and computational implementation directly translate into improved design protocols for vibration damping, seismic imaging, optimal feedback control, and inverse material characterization.

This body of research demonstrates that memory can both enrich and complicate classical PDE dynamics—introducing nontrivial spectral, control, and regularity phenomena, while necessitating sophisticated analytic and computational tools to unravel, stabilize, and exploit such systems across engineering, physics, and applied mathematics.