Abstract
The dynamics of nonlinear PDEs has been studied thoroughly during past decades, and a typical result may be the existence, structure, and dimensions of attractors. As is known so far, most work focus on PDEs with distributed nonlinearities, but little is known about those with boundary nonlinearities. On the other hand, chaos is found in the one-dimensional (1D) wave equation with nonlinear boundary conditions of van der Pol type due to imbalance of energy flows. So in this paper we consider the linear heat equations with nonlinear Neumann boundary conditions of van der Pol type, and explore their dynamics. We prove the existence and structure of the attractor by classical theories of parabolic PDEs and infinite-dimensional dynamical systems. Moreover, the existence and attraction of the equilibrium when parameters enter some regimes is proven as well. Finally we conclude that there is no chaos in the 1D heat equation with van der Pol boundary energy flows.
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