Mobilisation kinetics of metals from microfibres in freshwater and under simulated digestive conditions.

This article investigates the uniform null controllability problem for a system of coupled parabolic equations with a periodic oscillating coefficient. Our approach combines spectral analysis and Carleman estimates. First, we analyze the spectral properties of an elliptic operator with an oscillating coefficient to control the low frequencies. Then the system is allowed to evolve freely to achieve the required decay. In the third step, we establish a Carleman estimate that leads to a suitable observability result. By combining the three steps, we prove the uniform null controllability of the system, which is then used to homogenize the associated coupled parabolic system.

In this paper, we deal with the problem of stabilisation for a class of retarded distributed bilinear systems of neutral type evolving in a Hilbert space. We first give a proof for the existence and uniqueness of the solution for the considered systems. Then, under a null controllability condition, we establish the stabilisation result and provide an explicit estimate of the energy decay. We also, examine the particular case where the semigroup associated with the linear part of the system is differentiable. Some illustrating applications to hyperbolic and parabolic equations are displayed.

ABSTRACT We in this manuscript restudy the long time decay, extinction and blow‐up for a singular p‐biharmonic parabolic equation with logarithmic nonlinearity, which appears in many branches of physics. In the framework of potential well theory and the existence of global solution, by a way of establishing a nonlinearly integral inequality without non‐increasing condition, we prove that ‐norm for the weak solutions is non‐increasing, and establish two decay and extinction theorems that incorporate two kinds of polynomial decay, two kinds of exponential decay and two kinds of finite time extinction. By a way of establishing an improved Hardy–Sobolev inequality and applying a non‐concavity method, we establish the four blow‐up theorems independent of the potential well depth with as the blow‐up criterion, where two of them are finite time blow‐up, one is at least exponential growth and blows up at least at infinity, the last one blows up at infinity, where is a nonlinear function of . These generalize previous research results from three aspects: long time decay, extinction and blow‐up.

This paper proposes a method for the design of spur gear tooth profiles using parabolic curve as its path line of action. A mathematical model, including the equation of the line of action and the characteristics of the gear profile designed by the proposed method are analyzed. A comparative study on the different root fillet radii with the involutes gear profile. The function of undercutting condition is derived from the model. The effects of the two design parameters, that present the size (or shape) of the parabolic curve relative to the gear size, are the shape of tooth profiles and the contact ratio are also studied through the design of spur gear drive. The strength, including the contact stresses and life of cycle, of the gear drive designed by using the proposed method and CAD model generated CATIA –V5 and is analyzed by an FEA simulation with ANSYS 14.0. A comparison of the above characteristics of the gear drive designed with the involutes gear drive is also carried out in this work. The results confirm that the proposed design method is more flexible to control the shape of the tooth profile by changing the parameters of the parabola curve.

Minimal time of the pointwise controllability for a degenerate/singular parabolic equation and related numerical results via B-splines

Preservation of geometry property of delayed parabolic equations

A relation between the Dirichlet and the Regularity problem for parabolic equations

Abstract We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of nonlinear parabolic equations describing the motion of curves belonging to a given two-dimensional manifold. Using the abstract theory of analytic semiflows, we prove the local existence, uniqueness of Hölder smooth solutions to the governing system of nonlinear parabolic equations for the position vector parametrization of evolving curves. We apply the method of flowing finite volumes in combination with the methods of lines for numerical approximation of the governing equations. Qualitative analytical results are illustrated by various numerical experiments.

Simultaneous stable determination of quasilinear terms for parabolic equations

In this work, we develop a disturbance suppression-oriented fuzzy sliding mode secured sampled-data controller for third-order parabolic partial differential equations that ought to cope with nonlinearities, hybrid cyber attacks, and modeled disturbances. This endeavor is mainly driven by formulating an observer model with a T–S fuzzy mode of execution that retrieves the latent state variables of the perceived system. Progressing onward, the disturbance observers are formulated to estimate the modeled disturbances emerging from the exogenous systems. In due course, the information received from the system and disturbance estimators, coupled with the sliding surface, is compiled to fabricate the developed controller. Furthermore, in the realm of security, hybrid cyber attacks are scrutinized through the use of stochastic variables that abide by the Bernoulli distributed white sequence, which combat their unpredictability. Proceeding further in this framework, a set of linear matrix inequality conditions is established that relies on the Lyapunov stability theory. Precisely, the refined looped Lyapunov–Krasovskii functional paradigm, which reflects in the sampling period that is intricately split into non-uniform intervals by leveraging a fractional-order parameter, is deployed. In line with this pursuit, a strictly (Φ1,Φ2,Φ3)−ϱ dissipative framework is crafted with the intent to curb norm-bounded disturbances. A simulation-backed numerical example is unveiled in the closing segment to underscore the potency and efficacy of the developed control design technique.

ABSTRACT This paper presents a comprehensive numerical framework for solving nonlinear time‐fractional parabolic equations with distributed delays and variable coefficients. The proposed method combines the high‐order Alikhanov ‐ temporal discretization on graded meshes with a novel predictor‐corrector quasi‐linearization technique for handling the coupled nonlinearity and delay terms. Spatial discretization is achieved through the standard Galerkin finite element method. We establish the unconditional stability of the scheme and derive optimal error estimates of order in the ‐norm and in the ‐norm. Furthermore, we prove a superconvergence result, demonstrating that through appropriate post‐processing, one can achieve an enhanced convergence rate of in the ‐norm for sufficiently smooth solutions. The theoretical analysis employs a discrete fractional Grönwall inequality and rigorous error estimates that account for the weak singularity at the initial time. In contrast to existing methods such as Peng et al. (2024), which use first‐order L1 temporal discretization, our scheme achieves second‐order temporal accuracy and systematically handles variable coefficients and general distributed delay kernels. Numerical experiments validate the theoretical results and demonstrate the method's effectiveness for biologically relevant models, including a fractional Nicholson's blowflies equation with spatial heterogeneity. Additional experiments with higher‐order finite elements confirm the higher‐order spatial convergence, and condition number analyses demonstrate the mesh‐independent performance of the predictor‐corrector linearization. The robustness of the algorithm is confirmed through long‐time simulations with strongly variable coefficients and distributed delay kernels.

We derive rigorous error bounds for exponential Runge–Kutta discretizations of parabolic equations with nonsmooth initial data. Our analysis is carried out in the framework of abstract semilinear evolution equations, allowing for operators with non-dense domains. The results provide a foundation for establishing error estimates for nonsmooth data in prototypical problems such as the Allen-Cahn and Burgers equations. Furthermore, we apply these estimates to the analysis of split exponential integrators, yielding convergence results expressed explicitly in terms of the prescribed data.

Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates

In this paper we investigate a class of nonlinear degenerate parabolic equations involving heterogeneous (p,q)-Laplacian operators and subject to Dirichlet boundary conditions. These equations model complex diffusion phenomena with mixed-phase behavior in heterogeneous media. Our aim is to establish existence and uniqueness results for weak solutions under minimal regularity assumptions on the source term f, without requiring any control at infinity. The main difficulties stem from the degeneracy of the operator, the non-standard (p,q)-growth conditions, and the discontinuity of material phases. To overcome these challenges, we develop a variational framework based on Orlicz–Sobolev space theory and employ a generalized version of the Minty–Browder theorem to ensure the surjectivity of the nonlinear operator. Our approach yields new energy estimates, compactness results in non-reflexive settings, and stability under L∞-perturbations of the data. This work provides a rigorous mathematical foundation for analyzing nonlinear diffusion problems in complex and irregular environments.

We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of p–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent p, which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.

Solutions exhibiting weak initial singularities arise in various equations, including diffusion and subdiffusion equations. When employing the well-known L1 scheme to solve subdiffusion equations with weak singularities, numerical simulations reveal that this scheme exhibits varying convergence rates for different choices of model parameters (i.e., domain size, final time $T$, and reaction coefficient $\kappa$). This elusive phenomenon is not unique to the L1 scheme but is also observed in other numerical methods for reaction-diffusion equations such as the backward Euler (IE) scheme, Crank-Nicolson (C-N) scheme, and two-step backward differentiation formula (BDF2) scheme. The existing literature lacks an explanation for the existence of two different convergence regimes, which has puzzled us for a long while and motivated us to study this inconsistency between the standard convergence theory and numerical experiences. In this paper, we provide a general methodology to systematically obtain error estimates that incorporate the exponential decaying feature of the solution. We term this novel error estimate the ‘decay-preserving error estimate’ and apply it to the aforementioned IE, C-N, and BDF2 schemes. Our decay-preserving error estimate consists of a low-order term with an exponential coefficient and a high-order term with an algebraic coefficient, both of which depend on the model parameters. Our estimates reveal that the varying convergence rates are caused by a trade-off between these two components in different model parameter regimes. By considering the model parameters, we capture different states of the convergence rate that traditional error estimates fail to explain. This approach retains more properties of the continuous solution. We validate our analysis with numerical results.

Abstract We are concerned with semilinear parabolic equations, with a time-dependent source term of the form $$h(t)u^q$$ h ( t ) u q with $$q>1$$ q > 1 , posed on an infinite graph. We assume that the bottom of the $$L^2$$ L 2 -spectrum of the Laplacian on the graph, denoted by $$\lambda _1(G)$$ λ 1 ( G ) , is positive. In dependence of q , h ( t ) and $$\lambda _1(G)$$ λ 1 ( G ) , we show global in time existence or finite time blow-up of solutions.

High-energy laser systems are fundamental to modern advanced manufacturing and defense engineering, where effective thermal management is the key bottleneck limiting beam quality and operational longevity. This paper presents a robust numerical framework for simulating transient temperature fields in laser gain media, governed by a nonlinear parabolic partial differential equation (PDE). We first establish a comprehensive thermophysical model incorporating conductive heat transfer, convective boundary cooling, and crucially, the temperature-dependent thermal conductivity of the solid-state crystal (e.g., YAG or Sapphire). This results in a nonlinear heat conduction equation. For its numerical solution, an implicit Finite Difference Method (FDM) scheme is implemented, ensuring unconditional stability for the stiff problem arising from intense internal heat generation. A predictor-corrector iteration is designed to handle the nonlinearity introduced by the variable conductivity. Applied to a simplified cylindrical laser rod geometry, the model successfully simulates the evolution of temperature and thermal stress under pulsed operation. The numerical solution reveals a significant thermal lensing effect, with a maximum temperature rise of 152.3 K and a corresponding focal power of 0.85 m⁻¹. Comparative analysis demonstrates that neglecting the nonlinearity (assuming constant conductivity) leads to a 12.7% underestimation of the peak temperature and a mischaracterization of the stress profile. This work provides a reliable and adaptable computational tool for the design and optimization of thermal management systems in high-power laser engineering, highlighting the critical role of applied mathematics in tackling complex multiphysics challenges.

Underwater sound propagation modeling is crucial for ocean environmental monitoring, underwater communication, and target localization. Traditional underwater acoustics models are limited by high computational costs and restricted adaptability, while data-driven machine learning methods lack physical constraints, leading to poor generalization and reliance on large datasets. Although Physics-Informed Neural Networks have recently emerged to integrate physical priors, they still face challenges in achieving accurate long-range extrapolation. To address this limitation, we propose U-PARANET, a physics-informed machine learning method that incorporates the parabolic equation as a hard constraint directly into its architecture. The model leverages the parabolic equation's recursive, range-stepping structure within a neural network framework, enhancing stability and mitigating error accumulation over long-range propagation. Validation on both simulated and experimental data shows that U-PARANET accurately predicts transmission loss and phase structures, with good agreement in spatial field patterns. Specifically, the mean absolute error for transmission loss prediction is 1.40 dB in an ideal shallow-water environment, 1.06 dB in a simulation using SWellEx-96 environmental parameters, and 2.87 dB on SWellEx-96 experimental data. In conclusion, the proposed method exhibits excellent long-range modeling capabilities, demonstrating robust extrapolation in challenging, realistic environments.