Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator II: Small diffusion
Shuang Liu, Yuan Lou, Rui Peng, Maolin Zhou
Abstract
We investigate the effect of small diffusion on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions in one dimensional space. The asymptotic behaviors of the principal eigenvalues, as the diffusion coefficients tend to zero, are established for non-degenerate and degenerate spatial-temporally varying environments. A new finding is the dependence of these asymptotic behaviors on the periodic solutions of a specific ordinary differential equation induced by the drift. The proofs are based upon delicate constructions of super/sub-solutions and the applications of comparison principles.
Topics & Concepts
MathematicsEigenvalues and eigenvectorsDegenerate energy levelsMathematical analysisDiffusionOperator (biology)Parabolic partial differential equationOrdinary differential equationDifferential operatorZero (linguistics)Elliptic operatorPeriodic boundary conditionsSpace (punctuation)Partial differential equationBoundary value problemDifferential equationPhysicsChemistryTranscription factorRepressorBiochemistryThermodynamicsLinguisticsPhilosophyQuantum mechanicsGeneAdvanced Mathematical Modeling in EngineeringDifferential Equations and Numerical MethodsStability and Controllability of Differential Equations