arXiv subjects

Lingyang Liu

Publications and source records attributed to Lingyang Liu.

7 recordsLinked to original sources

arXiv2026

Controllability of quasilinear parabolic equations under multiplicative mobile controls

This paper addresses the controllability of a class of quasi-linear parabolic equations governed by multiplicative controls with mobile support. To prove the existence of such a control forcing the solution to rest at time $T>0$, we first establish the decay property of solutions for the uncontrolled system. Unlike the case of the linear heat equation, the nonlinearity in the principal part of the operator introduces significant challenges. These difficulties necessitate a novel approach, ultimately leading us to solve the controllability problem within the framework of classical solutions. Through a carefully constructed smooth transition, we demonstrate that there exists a multiplicative control driving the state exactly to rest at time $t=T$.

math.OC↗

arXiv2025

Stability of parabolic equations in non-cylindrical domains

This paper addresses the stability of a class of parabolic equations in non-cylindrical domains. We investigate the $L^\infty$-stability of systems for both nondegenerate and degenerate cases. Unlike in cylindrical domains, solutions to such problems may not exhibit exponential decay. An interesting phenomenon observed is that degeneracy has a positive impact on $L^\infty$-norm estimates for solutions to the system.

math.AP↗

arXiv2025

Local controllability of a free-boundary problem for a class of one-dimensional degenerate parabolic equations

This paper is devoted to a study of the controllability of a free-boundary problem for a class of one-dimensional degenerate parabolic equations with distributed controls, locally supported in space. We prove that for any $T>0$, if the initial state is sufficiently small, there exists a control that drives the state exactly to rest at time $t = T$. The proof is based on Schauder's fixed point theorem, combined with appropriate estimates for solutions to degenerate parabolic equations and for control functions.

math.OC↗

arXiv2021

p-Laplacian wave equations in non-cylindrical domains

This paper is devoted to studying the stability of p-Laplacian wave equations with strong damping in non-cylindrical domains. The method of proof based on some estimates for time-varying coefficients rising from moving boundary and a modified Kormonik inequality. Meanwhile, by selecting appropriate auxiliary functions, finally we obtain the polynomial stability (p > 2) and exponential stability (p = 2) for such systems in some unbounded development domains.

math.AP↗

arXiv2021

The stabilization of wave equations with moving boundary

In this paper, we consider the stabilization of wave equations with moving boundary. First, we show the solution behaviour of wave equation with Neumann boundary conditions, that is, the energy of wave equation with mixed boundary conditions may decrease, increase or conserve depending on the different range of parameter. Second, we prove the wellposedness and stabilization for the wave equation with time delay and moving boundary.

math.AP↗

arXiv2021

A note on damped wave equations with a nonlinear dissipation in non-cylindrical domains

In this paper, we study the large time behavior of a class of wave equation with a nonlinear dissipation in non-cylindrical domains. The result we obtained here relaxes the conditions for the nonlinear term coefficients (in precise, that is $β(t)|u|^ρu$) in \cite{alb} and \cite{ha} (which require $β(t)$ to be a constant or $β(t)$ to be decreasing with time $t$) and has less restriction for the defined regions.

math.AP↗

arXiv2021

Logarithmic wave equations in non-cylindrical domains

This paper is devoted to studying a type of logarithmic wave equation in non-cylindrical domains. Firstly, by the penalty method, we prove the existence of weak solutions to such kind of equations. Secondly, different from the dissipative wave equation, the energy defined in this problem is not always positive. Thus, some suitable initial data are selected to let the energy be positive. Finally, by a difference inequality, we derive a exponential decay estimate for the positive energy.

math.AP↗