关于平流非局部算子：主本征对的多重性
On advective nonlocal operators: multiplicity of principal eigenpairs
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中文总结 AI 辅助
本文研究带平流的周期异质非局部扩散模型的主本征对多重性，因算子非预解紧无法用经典Krein-Rutman理论，分平流系数常号与非常号情形分析，还讨论其在非线性KPP型方程中的应用。
中文摘要 AI 辅助
我们研究带有平流的周期异质非局部扩散模型的主特征值与特征函数的存在性及多重性。所考虑的算子是预解正的，但非预解紧的，因此无法直接应用经典的Krein-Rutman理论。当平流系数为常号时，我们证明主特征值及对应归一化特征函数的存在性与唯一性。与之形成鲜明对比的是，当平流非为常号时，问题更为复杂并产生出人意料的结果：根据方程系数的不同，主本征问题要么存在唯一归一化解，要么存在连续统的解，在其边界处存在带有奇异测度分量的主特征向量；在后一情形中，所有构造的特征值均嵌入于算子的连续谱中。我们完全刻画了与正特征向量相关的特征值，即便该特征向量为Radon测度。最后，我们讨论其在带有非局部扩散的非线性KPP型方程中的应用，该方程具有非平凡平稳解的连续统，这与带有局部扩散的经典KPP方程的行为不同。
英文摘要
We study the existence and multiplicity of principal eigenvalues and eigenfunctions for a periodically heterogeneous nonlocal dispersal model with advection. The operator we consider is resolvent-positive but not resolvent-compact; therefore, the classical Krein-Rutman theory cannot be applied directly. When the advection coefficient has a constant sign, we prove the existence and uniqueness of the principal eigenvalue and the corresponding normalized eigenfunction. In sharp contrast, when the advection does not have a constant sign, the problem is more involved and leads to surprising results. Depending on the coefficients of the equation, the principal eigenproblem can either have a unique normalized solution or a continuum of solutions, at the boundary of which there exists a principal eigenvector with a singular measure component. In the latter situation, all the constructed eigenvalues are embedded in the continuous spectrum of our operator. We completely characterize the eigenvalues associated with positive eigenvectors, even when the eigenvector is a Radon measure. Finally, we discuss an application to a nonlinear KPP-type equation with nonlocal dispersal, which possesses a continuum of nontrivial stationary solutions, a different behavior from the classical KPP equation with local diffusion.
