arXiv subjects

Ruixiang Xing

Publications and source records attributed to Ruixiang Xing.

7 recordsLinked to original sources

arXiv2026

Linear stability of the first bifurcation in a tumor growth free boundary problem via local bifurcation structure

In this paper, we consider a 3-dimensional free boundary problem modeling tumor growth with the Robin boundary condition. The system involves a positive parameter $\mu$ which reflects the intensity of tumor aggressiveness. Huang, Zhang and Hu [Nonlinear Anal. Real World Appl. 2017(35), 483-502] have shown that for each $\mu_n$ ($n$ even) in a strictly increasing sequence $\{ \mu_n \}(n\geq 2)$, there exists a stationary bifurcation solution $(\sigma_n(\varepsilon),p_n(\varepsilon),r_n(\varepsilon))$ with $\mu = \mu_n(\varepsilon)$ bifurcating from $\mu_n$. We first derive that the bifurcation curve $(r_2(\varepsilon),\mu_2(\varepsilon))$ exhibits a transcritical bifurcation with $\mu_2'(0)<0$. Moreover, we show that the stationary bifurcation solution $(\sigma_2(\varepsilon),p_2(\varepsilon),r_2(\varepsilon))$ is linearly unstable for small $|\varepsilon|$ under non-radially symmetric perturbations. In contrast to the linear stability of the radially symmetric stationary solution, the lack of explicit expressions for bifurcation solutions adds great difficulty in analyzing their linear stability. The novelty of this paper lies in the use of the bifurcation curve's structure to overcome the above difficulties. Moreover, this linear stability result is not established using the standard method, due to an eight-dimensional generalized kernel at eigenvalue 0 for the linearized operator.

math.AP↗

arXiv2025

Symmetry-breaking bifurcation of periodic solutions for a free-boundary tumor model

In this paper, we consider a free boundary multi-layer tumor model that incorporates a $T-$periodic provision of external nutrients $\Phi(t)$. The simplified model contains three parameters: the mean of periodic external nutrients $\Phi(t)$, the threshold concentration $\widetilde{\sigma}$ for proliferation and the cell to cell adhesiveness coefficient $\gamma$. We first study the flat solution and give a complete classification about $\frac{1}{T} \int_0^T \Phi(t) d t$ and $\widetilde{\sigma}$ according to global stability of zero equilibrium solution or global stability of the positive periodic solution. Precisely, (i) a zero flat solution is globally stable under the flat perturbations if and only if $\widetilde{\sigma} \geqslant \frac{1}{T} \int_0^T \Phi(t) d t$; (ii) If $\widetilde{\sigma}<\frac{1}{T} \int_0^T \Phi(t) d t$, then there exists a unique positive flat solution $\left(\sigma_*(y, t), p_*(y, t), { \rho_*(t)}\right)$ with period $T$ and it is a global attractor of all positive flat solutions for all $\gamma>0$. We further investigate periodic solutions bifurcating from the flat periodic solution $\left(\sigma_*(y, t), p_*(y, t), { \rho_*(t)}\right)$. By periodicity and symmetry, we not only give symmetry-breaking periodic solutions for all positive parameter $\gamma_j$, but also show the existence of a plethora of periodic bifurcations. For the free boundary tumor problem, this is the first result of the existence of periodic bifurcations.

math.AP↗

arXiv2022

A Degenerate Hopf Bifurcation Theorem in Infinite Dimensions

A Hopf bifurcation theorem is established for the abstract evolution equation $\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,λ)$ in infinite dimensions under the degeneracy condition $Re μ^{\prime}(λ_0)= 0$ and suitable assumptions. The stability properties of bifurcating periodic solutions are also derived. Interestingly, it is shown that a transcritical Hopf bifurcation still can occur at $λ_0$ although the stability property of the trivial solutions does not change near $λ_0$. Our results do not require the analyticity of $F$. The main tools are the Lyapunov--Schmidt reduction and a Morse lemma. Applications to a multi-parameter diffusive predator--prey system discover new branches of periodic solutions.

math.FA↗

arXiv2021

The existence of periodic solution and asymptotic behavior of solutions for a multi-layer tumor model with a periodic provision of external nutrients

In this paper, we consider a multi-layer tumor model with a periodic provision of external nutrients. The domain occupied by tumor has a different shape (flat shape) than spherical shape which has been studied widely. The important parameters are periodic external nutrients $Φ(t)$ and threshold concentration for proliferation $\widetildeσ$. In this paper, we give a complete classification about $Φ(t)$ and $\widetildeσ$ according to global stability of zero equilibrium solution or global stability of the positive periodic solution. Precisely, if $\frac{1}{T} \int_{0}^{T} Φ(t)d t\leqslant\widetildeσ$, then the zero equilibrium solution is globally stable while if $\frac{1}{T} \int_{0}^{T} Φ(t)d t>\widetildeσ$, then there exists a unique positive T-periodic solution and it is globally stable.

math.AP↗

arXiv2021

The linear stability for a free boundary problem modeling multi-layer tumor growth with time delay

We study a free boundary problem modeling multi-layer tumor growth with a small time delay $τ$, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper we establish the well-posedness of the problem, namely, first we prove that there exists a unique flat stationary solution $(σ_*, p_*, ρ_*, ξ_* )$ for all $μ>0$. The stability of this stationary solution should depend on the tumor aggressiveness constant $μ$. It is also unrealistic to expect the perturbation to be flat. We show that, under non-flat perturbations, there exists a threshold $μ_*>0$ such that $(σ_*, p_*, ρ_*, ξ_*)$ is linearly stable if $μ<μ_*$ and linearly unstable if $μ>μ_*$. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.

math.AP↗

arXiv2020

The existence and linear stability of periodic solution for a free boundary problem modeling tumor growth with a periodic supply of external nutrients

We study a free boundary problem modeling tumor growth with a T-periodic supply $Φ(t)$ of external nutrients. The model contains two parameters $μ$ and $\widetildeσ$. We first show that (i) zero radially symmetric solution is globally stable if and only if $\widetildeσ\ge \frac{1}{T} \int_{0}^{T} Φ(t) d t$; (ii) If $\widetildeσ<\frac{1}{T} \int_{0}^{T} Φ(t) d t$, then there exists a unique radially symmetric positive solution $\left(σ_{*}(r, t), p_{*}(r, t), R_{*}(t)\right)$ with period $T$ and it is a global attractor of all positive radially symmetric solutions for all $μ>0$. These results are a perfect answer to open problems in Bai and Xu [Pac. J. Appl. Math. 2013(5), 217-223]. Then, considering non-radially symmetric perturbations, we prove that there exists a constant $μ_{\ast}>0$ such that $\left(σ_{*}(r, t), p_{*}(r, t), R_{*}(t)\right)$ is linearly stable for $μ<μ_{\ast}$ and linearly unstable for $μ>μ_{\ast}$.

math.AP↗

arXiv2015

On the Existence of Positive Solutions for Some Nonlinear Boundary Value Problems II

We study a class of boundary value problems with $φ$-Laplacian (e.g., the prescribed mean curvature equation, in which $φ(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(φ(u')\right)'=λf(u)\; \text{ on }(-L, L),\quad u(-L)=u(L)=0,$ \end{center} where $λ$ and $L$ are positive parameters. For convex $f$ with $f(0)=0$, we establish various results on the exact number of positive solutions as well as global bifurcation diagrams. Some new bifurcation patterns are shown. This paper is a continuation of Pan and Xing [13], where the case $f(0)>0$ has been investigated.

math.CA↗