arXiv subjects

Yikan Liu

Publications and source records attributed to Yikan Liu.

At least 19 recordsLinked to original sources

arXiv2026

Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential

This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.

math.NA↗

arXiv2026

Strict positivity property of inhomogeneous subdiffusion equations and its application to coupled subdiffusion systems

The positivity of solutions to subdiffusion equations has been widely studied, mainly in the context of homogeneous problems for single equations. In this article, we fill the missing strict positivity for inhomogeneous subdiffusion equations with nonnegative and nontrivial source terms by connecting Green's functions for fractional and classical diffusion equations via special functions. As a direct application, we further investigate the strict positivity property of coupled subdiffusion systems with nonnegative and partially nontrivial initial values or sources. Under suitable cooperativeness and connectivity conditions, the strict positivity turns out to propagate not only in time but also across different components of the system, reflecting the intrinsic interactions induced by the coupling structure. These results provide a unified framework for understanding positivity properties of both scalar and coupled subdiffusion equations, offering new insights beyond the classical maximum principle approach.

math.AP↗

arXiv2026

Simultaneous recovery of multiple parameters in nonlocal diffusion equations from internal measurements

This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain a stable approximate solution of the inverse problem. Numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate our theoretical findings.

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arXiv2026

Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems

This article is concerned with the inverse problem on determining the temporal component of the source term in a coupled time-fractional diffusion system by a single point observation. Under a non-degeneracy condition on the known spatial component, we establish the Lipschitz stability by observing all solution components by a series representation of the mild solution. To reduce the observation data, we prove the strict positivity of some fractional integral of the solution to the homogeneous problem by a modified Picard iteration. This, together with a coupled Duhamel's principle, lead us to the uniqueness of the inverse problem by observing any single solution component under a specific structural constraint on the unknown. Numerically, we propose an iterative regularizing ensemble Kalman method (IREKM) for the simultaneous recovery of the temporal sources. Through extensive numerical tests, we demonstrate its accuracy, robustness against noise, and scalability with respect to the number of components. Our findings highlight the essential roles of the non-degeneracy condition, measurement configuration, and fractional structural constraints in ensuring reliable reconstructions. The proposed framework provides both rigorous theoretical guarantees and a practical algorithmic approach for multi-component source identification in fractional diffusion systems.

math.AP↗

arXiv2026

Identifying the source term in a viscoelastic membrane with a Riemann-Liouville time derivative by the partial interior observation

This paper studies an inverse source problem for a viscoelastic membrane, where the material's memory effect is characterized by the Riemann-Liouville fractional derivative. The problem is to recover the unknown source term from the limited interior observation data. We propose an optimal control framework to address this ill-posed inverse problem. The first-order optimality condition leads to a coupled system of forward and backward fractional partial differential equations. A numerical algorithm combining the finite element method and a conjugate gradient iterative scheme is then developed for the reconstruction of the source term. Several numerical examples are provided to demonstrate the effectiveness and robustness of the proposed method.

math.NA↗

arXiv2026

Energy decay of a viscoelastic wave equation with variable exponent logarithmic nonlinearity and weak damping

In this paper, we investigate the energy decay of the solution to a viscoelastic wave equation with variable exponents logarithmic nonlinearity and weak damping in a bounded domain. We establish an explicit general decay result under mild conditions on the relaxation function $g$. Furthermore, under the general assumption $g'(t)\leq-\zeta(t)G(g(t))$ with some suitably given $\zeta$ and $G$, we derive a refined decay estimate improving existing results. In particular, uniform exponential and polynomial decay rates are obtained under a further special situation $g'(t)\leq-\xi(t)g^q(t)$ with $1\leq q<2$, extending earlier studies that were restricted to the case $1\leq q<\frac{3}{2}$.

math.AP↗

arXiv2025

Threshold dynamics in time-delay systems: polynomial $β$-control in a pressing process and connections to blow-up

This paper addresses a press control problem in straightening machines with small time delays due to system communication. To handle this, we propose a generalized $β$-control method, which replaces conventional linear velocity control with a polynomial of degree $β\ge 1$. The resulting model is a delay differential equation (DDE), for which we derive basic properties through nondimensionalization and analysis. Numerical experiments suggest the existence of a threshold initial velocity separating overshoot and non-overshoot dynamics, which we formulate as a conjecture. Based on this, we design a control algorithm under velocity constraints and confirm its effectiveness. We also highlight a connection between threshold behavior and finite-time blow-up in DDEs. This study provides a practical control strategy and contributes new insights into threshold dynamics and blow-up phenomena in delay systems.

math.OC↗

arXiv2025

Inverse source problem with a posteriori interior measurements for space-time fractional diffusion equations

This paper investigates an inverse source problem for space-time fractional diffusion equations from a posteriori interior measurements. The uniqueness result is established by the memory effect of fractional derivatives and the unique continuation property. For the numerical reconstruction, the inverse problem is reformulated as an optimization problem with the Tikhonov regularization. We use the Levenberg-Marquardt method to identity the unknown source from noisy measurements. Finally, we give some numerical examples to illustrate the efficiency and accuracy of the proposed algorithm.

math.NA↗

arXiv2025

Sharp decay estimates and numerical analysis for weakly coupled systems of two subdiffusion equations

This paper investigates the initial-boundary value problem for weakly coupled systems of time-fractional subdiffusion equations with spatially and temporally varying coupling coefficients. By combining the energy method with the coercivity of fractional derivatives, we convert the original partial differential equations into a coupled ordinary differential system. Through Laplace transform and maximum principle arguments, we reveal a dichotomy in decay behavior: When the highest fractional order is less than one, solutions exhibit sublinear decay, whereas systems with the highest order equal to one demonstrate a distinct superlinear decay pattern. This phenomenon fundamentally distinguishes coupled systems from single fractional diffusion equations, where such accelerated superlinear decay never occurs. Numerical experiments employing finite difference methods and implicit discretization schemes validate the theoretical findings.

math.AP↗

arXiv2025

Energy decay and blow-up of viscoelastic wave equations with polynomial nonlinearity and damping

This paper is concerned with the energy decay and the finite time blow-up of the solution to a viscoelastic wave equation with polynomial nonlinearity and weak damping. We establish explicit and general decay results for the solutions by imposing polynomial conditions on the relaxation function, provided that the initial energy is sufficiently small. Furthermore, we derive an upper bound for the blow-up time when the initial energy is less than the depth of the potential well by utilizing Levine's convexity method. Additionally, we provide a lower bound for the blow-up time if the solution blows up.

math.AP↗

arXiv2025

Numerical reconstructions of a source term in a mobile-immobile diffusion model from the partial interior observation

We consider an inverse source problem in the two-time-scale mobile-immobile fractional diffusion model from partial interior observation. Theoretically, we combine the fractional Duhamel's principle with the weak vanishing property to establish the uniqueness of this inverse problem. Numerically, we adopt an optimal control approach for determining the source term. A coupled forward-backward system of equations is derived using the first-order optimality condition. Finally, we construct a finite element conjugate gradient algorithm for the numerical inversion of the source term. Several experiments are presented to show the utility of the method.

math.NA↗

arXiv2025

Energy decay of nonlocal viscoelastic equations with nonlinear damping and polynomial nonlinearity

This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial energy decay for the solution under relatively weak assumptions regarding the kernel of the nonlocal term. More specifically, we improve the decay rate of the energy by additionally imposing a certain convexity assumption on the kernel. Several examples are provided to confirm such improvements to faster polynomial or even exponential decays.

math.AP↗

arXiv2025

Forward and backward problems for coupled subdiffusion systems

In this article, we investigate both forward and backward problems for coupled systems of time-fractional diffusion equations, encompassing scenarios of strong coupling. For the forward problem, we establish the well-posedness of the system, leveraging the eigensystem of the corresponding elliptic system as the foundation. When considering the backward problem, specifically the determination of initial values through final time observations, we demonstrate a Lipschitz stability estimate, which is consistent with the stability observed in the case of a single equation. To numerically address this backward problem, we refer to the explicit formulation of Tikhonov regularization to devise a multi-channel neural network architecture. This innovative architecture offers a versatile approach, exhibiting its efficacy in multidimensional settings through numerical examples and its robustness in handling initial values that have not been trained.

math.AP↗

arXiv2025

Numerical reconstruction of orders in coupled systems of subdiffusion equations

In this paper, we develop a numerical algorithm for an inverse problem on determining fractional orders of time derivatives simultaneously in a coupled subdiffusion system. Following the theoretical uniqueness, we reformulate the order inverse problem as a discrete minimization problem, so that we derive a concise Gauss-Newton iterative method. Abundant numerical tests demonstrate the efficiency and accuracy of the proposed algorithm.

math.NA↗

arXiv2025

Exponential stability for an infinite memory wave equation with frictional damping and logarithmic nonlinear terms

This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in $\mathbb R^d$ ($d\ge3$) with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density $\rho(x)$ under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if $\rho(x)$ takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the system.

math.AP↗

arXiv2024

Blowup in $L^1(Ω)$-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms

This article is concerned with semilinear time-fractional diffusion equations with polynomial nonlinearity $u^p$ in a bounded domain $Ω$ with the homogeneous Neumann boundary condition and positive initial values. In the case of $p>1$, we prove the blowup of solutions $u(x,t)$ in the sense that $\|u(\,\cdot\,,t)\|_{L^1(Ω)}$ tends to $\infty$ as $t$ approaches some value, by using a comparison principle for the corresponding ordinary differential equations and constructing special lower solutions. Moreover, we provide an upper bound for the blowup time. In the case of $0<p<1$, we establish the global existence of solutions in time based on the Schauder fixed-point theorem.

math.AP↗

arXiv2023

Blow-up for time-fractional diffusion equations with superlinear convex semilinear terms

This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $Ω$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and not identically vanishing initial value. The order of the fractional derivative in time is between $1$ and $0$, and the elliptic part is with time-independent coefficients. We prove (i) The solution with any initial value blow-up if the eigenvalue $λ_1$ of the elliptic operator with the minimum real part is non-positive. (ii) Otherwise, the solution blows up if a weighted $L^1$-norm of initial value is greater than some critical value give by $λ_1$. We provide upper estimates of the blow-up times. The key is a comparison principle for time-fractional ordinary differential equations.

math.AP↗

arXiv2023

Recovery of Multiple Parameters in Subdiffusion from One Lateral Boundary Measurement

This work is concerned with numerically recovering multiple parameters simultaneously in the subdiffusion model from one single lateral measurement on a part of the boundary, while in an incompletely known medium. We prove that the boundary measurement corresponding to a fairly general boundary excitation uniquely determines the order of the fractional derivative and the polygonal support of the diffusion coefficient, without knowing either the initial condition or the source. The uniqueness analysis further inspires the development of a robust numerical algorithm for recovering the fractional order and diffusion coefficient. The proposed algorithm combines small-time asymptotic expansion, analytic continuation of the solution and the level set method. We present extensive numerical experiments to illustrate the feasibility of the simultaneous recovery. In addition, we discuss the uniqueness of recovering general diffusion and potential coefficients from one single partial boundary measurement, when the boundary excitation is more specialized.

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