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The longtime behavior of a kind of fully magnetic effected nonlinear multi-dimensional piezoelectric beam with viscoelastic infinite memory is considered. The well-posedness of this nonlinear coupled PDEs’ system is showed by means of the semigroup theories and Banach fixed-point theorem. Based on frequency-domain analysis, it is proved that the corresponding coupled linear system can be indirectly stabilized exponentially by only one viscoelastic infinite memory term, which is located on one equation of these strongly coupled PDEs. Then, the exponential decay of the solution to the nonlinear coupled PDEs’ system is established by the energy estimation method under certain conditions.

We consider a nonlinear coupled PDE model for a single piezoelectric beam retaining the electromagnetic effects and a long-range strain memory. Nonlinear source terms in both mechanical and electromagnetic equations and a viscous magnetic damping term in the electromagnetic equation are considered in the model. The mathematical analysis of this model is particularly needed for certain class of fully dynamic piezoelectric materials demonstrating a viscoelastic memory or creep. With an injection of magnetic damping, the structure of the dynamical system associated with the solutions of this system allows using the quasi-stability theory in order to obtain the existence of global and exponential attractors.

A third-order in time nonlinear equation with memory term is considered. This particular model is motivated by high-frequency ultrasound technology which accounts for thermal and molecular relaxation. The resulting equations give rise to a quasilinear-like evolution with a potentially degenerate damping (Kaltenbacher in Evol Eqs Control Theory 4(4):447–491, 2015 ). Local and global (in time) existence of smooth solutions is studied. The main result of the paper states that with appropriate calibration of the memory kernel, solutions exist globally for sufficiently small and regular initial data. With exponentially decaying relaxation kernel said solutions exhibit exponential decay rates. The proof relies on the “barrier” method applied to a string of higher energy estimates, along with an abstract representation and the theory of viscoleastic evolutions developed in Pruss (Arch Math 92:158–173, 2009 , Evolutionary integral equations and applications. Birkhauser, 2012 ).

Cell blebbing is a fundamental morphodynamic process governed by the interplay of cytoplasmic pressure, cortical contractility, and membrane tension. Classical geometric flow models capture instantaneous mechanical effects but fail to represent hereditary and viscoelastic memory inherent to living cells. In this work, we propose a (q,τ) \\left{(}{q}{,}τ \\right) ‐fractional geometric flow framework for bleb morphogenesis, where the parameters q q and τ τ quantify deformation memory and stress‐relaxation tempering, respectively. Fluorescence microscopy frames from the WRAP dataset and synthetic simulations are segmented to extract time series of bleb height, effective radius, and fractional mean curvature. These observables are fitted using a predictor‐corrector numerical scheme for a (q,τ) \\left{(}{q}{,}τ \\right) ‐fractional evolution equation subject to an energy dissipation law. The numerical solver and segmentation pipeline are validated on synthetic and experimental data. The proposed model accurately reproduces both the rapid expansion and slow relaxation phases of bleb evolution, with residual errors below 10−3 10⁻³ and fitted parameters in biologically plausible ranges. Moreover, the total fractional energy exhibits monotonic decay, consistent with thermodynamic dissipation. The results demonstrate that (q,τ) \\left{(}{q}{,}τ \\right) ‐fractional geometric flows provide a unified and physically interpretable framework for coupling image‐based quantification with nonlocal curvature‐driven dynamics in cellular morphogenesis. We propose a (q,τ) \\left{(}{q}{,}τ \\right) ‐fractional geometric flow model for cell blebbing that incorporates hereditary memory and viscoelastic effects in curvature‐driven membrane dynamics. Image‐based measurements of bleb geometry are coupled with fractional evolution equations and validated numerically. The model accurately reproduces rapid expansion and slow relaxation phases of bleb morphogenesis, providing a quantitative link between fractional calculus and cellular mechanobiology.

We study existence of attractors for weak solutions of the regularized model for viscoelastic medium motion with memory in non-autonomous case. We apply the theory of trajectory attractors for non-invariant trajectory spaces and prove the existence of trajectory attractor, global attractor, uniform trajectory attractor, and uniform global attractor for this system.

A one-dimensional model is derived in order to study how the elasticity (internal elastic energy) of viscoelastic and elastoplastic materials, such as biopolymers (muscles and grain flour dough) or metals, changes due to the action of external forces. For such materials, the model takes the form of an initial-boundary value problem, corresponding to Newton's second law, which is coupled to an auxiliary (stress-strain) state equation which characterizes the nature of the interaction between the material and the external forces. In the oscillatory loading of muscles and the mixing of grain flour, as well as of the fatiguing of metals, the state equation must model how the stress depends on the earlier history of the strain as well as describe how the material gains or loses elastic energy due to the action of the loading. One is thereby led to model the auxiliary stress-strain relationship as a constitutive relationship involving a Duhem-Madelung hysteresis operator. As well as discussing the formulation of such models along with the properties of Duhem-Madelung hysteresis operators, this paper examines the existence and uniqueness for the solutions of such coupled systems. In addition, some global estimates are derived for these solutions, and their asymptotic behavior, as the time increases, is studied under the assumption that a part of the internal (elastic) energy dissipates during the interaction and, hence, the associated Duhem-Madelung hysteron has negative spin.

Existence, uniqueness and regularity of weak solutions of semilinear second order Volterra integro-differential equations in Hubert space are established by the variational method. As an application we give a well-posedness result for the semilinear viscoelastic equations with long memory.

This paper is concerned with the identification problems of unknown parameters in viscoelastic materials with long nonlinear memory. The unknown parameters are diffusion constants and kernels in nonlinear memory terms, and the identification of such parameters is studied by means of quadratic optimal control theory due to Lions [10]. The existence of optimal parameters is proved, and the necessary condition is established for distributive and terminal values observation by using the transposition method.

A colloidal particle is a prominent example of a stochastic system, and, if suspended in a simple viscous liquid, very closely resembles the case of an ideal random walker. A variety of new phenomena have been observed when such colloid is suspended in a viscoelastic fluid instead, for example pronounced nonlinear responses when the viscoelastic bath is driven out of equilibrium. Here, using a micron-sized particle in a micellar solution, we investigate in detail, how these nonlinear bath properties leave their fingerprints already in equilibrium measurements, for the cases where the particle is unconfined or trapped in a harmonic potential. We find that the coefficients in an effective linear (generalized) Langevin equation show intriguing inter-dependencies, which can be shown to arise only in nonlinear baths: for example, the friction memory can depend on the external potential that acts only on the colloidal particle (as recently noted in simulations of molecular tracers in water in (2017 Phys. Rev. X 7 041065)), it can depend on the mass of the colloid, or, in an overdamped setting, on its bare diffusivity. These inter-dependencies, caused by so-called fluctuation renormalizations, are seen in an exact small time expansion of the friction memory based on microscopic starting points. Using linear response theory, they can be interpreted in terms of microrheological modes of force-controlled or velocity-controlled driving. The mentioned nonlinear markers are observed in our experiments, which are astonishingly well reproduced by a stochastic Prandtl-Tomlinson model mimicking the nonlinear viscoelastic bath. The pronounced nonlinearities seen in our experiments together with the good understanding in a simple theoretical model make this system a promising candidate for exploration of colloidal motion in nonlinear stochastic environments.

In this work, a model for the quasistatic frictionless contact between a viscoelastic body with long memory and a foundation is studied. The material constitutive relation is assumed to be nonlinear and the contact is modelled with the normal compliance condition, i.e., the obstacle is assumed to be deformable. The evolution of the mechanical damage of the material, caused by the opening and growth of microcracks under excessive stress or strain, is modelled by a nonlinear partial differential equation. We discuss the existence and uniqueness of weak solution for the mechanical problem and we introduce a fully discrete scheme based on the finite element method and finite differences to approximate this solution. The existence and uniqueness of the approximate solutions are presented and some estimates of the numerical error are provided. Finally, we show some results obtained in the simulation of a two-dimensional test problem.