| Abstract: |
| Abstract: In this paper, we investigate constant steady states of the system \\\\begin{equation}\\\\tag{KS} \\\\begin{cases} \\\\begin{array}{llll} \\\\displaystyle u_t=\\\\Delta u-\\\\chi\\\\nabla\\\\cdot\\\\big(u\\\\nabla v\\\\big), &x\\\\in\\\\Omega,\\\\\\\\ \\\\displaystyle v_t=\\\\Delta v-a v+b u, &x\\\\in\\\\Omega \\\\end{array} \\\\end{cases} \\\\end{equation} subject to homogeneous Neumann boundary conditions, where $\\\\Omega\\\\subset\\\\Bbb{R}^N$ is a bounded domain and $a,b,\\\\chi>0$. The main contribution is identifying a critical threshold for constant steady states that governs the exchange of stability, as reflected in the dynamics of the solutions. Due to the lack of quantitative information on the threshold in the existing theory, this work provides a universal constant $\\\\sigma_0>0$ with an explicit value to determine the stability behavior of solutions. Whenever $\\\\sigma\\\\sigma_0$, we show that $\\\\big(\\\\sigma,\\\\frac{b\\\\sigma}{a}\\\\big)$ is Lyapunov unstable. Previous results on Lyapunov instability were limited to the two- or three-dimensional torus. We overcome this limitation in our instability analysis by introducing a new method based on a special paring of different frequencies. We emphasize that this method can lead to an establishment of Lyapunov instability whenever the linearized operator associated with (KS) possesses a positive eigenvalue. Thus, it enables us to consider general domains in arbitrary dimensions $N\\\\ge 2,$ and is also adaptable to studying unstable behaviors for a wide class of related problems. |
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