The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions.

研究了具有非局部边界条件的非线性标量抛物方程的唯一稳态的全局渐近稳定性。该方程式描述了受反馈控制回路影响的温度曲线的变化。可以将其视为基本恒温器的模型，其中参数根据边界点与参考温度的偏差幅度来控制热流强度。已知当参数超过临界值时，系统会发生Hopf分叉。通过分析密切相关的非线性Volterra积分方程，可以获得参考稳态为全局渐近稳定的最大参数范围的表征结果。它的内核是从线性热方程的基本解的轨迹中得出的。将Popov准则的一种版本改编并应用于Volterra积分方程，以获得其解的渐近衰减的充分条件。