Singularly Perturbed Time-Periodic Eigenvalue Problems With Spectral Fractional Laplace Operators

We study singularly perturbed time-periodic linear and nonlinear eigenvalue problems involving spectral fractional diffusion on smoothly bounded domains. These problems naturally arise as a mathematical means aiding the attempts to describe transport dynamics of complex systems, which are governed by anomalous diffusion and may rely upon time-periodic resources. In such systems, anomalous diffusion is often represented by fractional powers of linear second-order differential operators subject to three types of boundary conditions on a bounded domain. As a special case, we also investigate time-independent eigenvalue problems. The primary focuses are on the existence and uniqueness of principal eigenvalues and their dependence on the dispersal rate and fractional power, and especially, their asymptotic behavior as the dispersal rate tends to zero or infinity. Subsequently, we discuss the solvability and regularity of periodic solutions of time-dependent spectral fractional reaction-diffusion equations and their spatial profiles as the dispersal rate tends to zero or infinity. As an application, the established results are utilized to investigate the basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , a threshold that is crucial in studying population dynamics and infectious disease transmission. It is obtained in terms of the principal eigenvalue of an indefinite weight eigenvalue problem, which plays a key role in determining the existence and nonexistence of a time-periodic epidemic equilibrium as well as stabilities of the disease-free and endemic equilibria. The dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:math> on the dispersal rate is also examined.