Asymptotic profiles in diffusive logistic equations

This paper is concerned with the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of nonlinear diffusion functions. Both the classical reaction–diffusion equation and nonlocal dispersal equation are investigated. We prove the sharp change occurs in reaction–diffusion equation by regularity estimates and compact principle. Due to lack of compactness and regularities theory for nonlocal problems, we obtain the sharp changes by nonlocal estimates and comparison arguments. Our result reveals the nonlinear term plays quite different roles between evolution problem and stationary solution problem.

[1]  J. López-Gómez.  Metasolutions of Parabolic Equations in Population Dynamics , 2015 .

[2]  Noemi Wolanski,et al.  Boundary fluxes for nonlocal diffusion , 2007 .

[3]  Jian-Wen Sun,et al.  Limiting Solutions of Nonlocal Dispersal Problem in Inhomogeneous Media , 2021, Journal of Dynamics and Differential Equations.

[4]  Peter Hess,et al.  On some linear and nonlinear eigenvalue problems with an indefinite weight function , 1980 .

[5]  J. López-Gómez.  Linear Second Order Elliptic Operators , 2013 .

[6]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[7]  Emmanuel Chasseigne,et al.  Asymptotic behavior for nonlocal diffusion equations , 2006 .

[8]  Julián López-Gómez,et al.  Sharp patterns of positive solutions for some weighted semilinear elliptic problems , 2021, Calculus of Variations and Partial Differential Equations.

[9]  Henri Berestycki,et al.  Le nombre de solutions de certains problèmes semi-linéaires elliptiques , 1981 .

[10]  J. López-Gómez.  The Maximum Principle and the Existence of Principal Eigenvalues for Some Linear Weighted Boundary Value Problems , 1996 .

[11]  Yuan Lou,et al.  Random dispersal vs. non-local dispersal , 2009 .

[12]  Song-Sun Lin,et al.  On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function , 1980 .

[13]  Julio D. Rossi,et al.  A logistic equation with refuge and nonlocal diffusion , 2009 .

[14]  D. Daners,et al.  Global Dynamics of Generalized Logistic Equations , 2018 .

[15]  Wan-Tong Li,et al.  Asymptotic behavior for nonlocal dispersal equations , 2010 .

[16]  P. Rabinowitz,et al.  The effects of spatial heterogeneities on some multiplicity results , 2015 .

[17]  J. López-Gómez,et al.  Semiclassical analysis of general second order elliptic operators on bounded domains , 2000 .

[18]  Jian-Wen Sun.  Sharp profiles for periodic logistic equation with nonlocal dispersal , 2020, Calculus of Variations and Partial Differential Equations.

[19]  Wan-Tong Li,et al.  A nonlocal dispersal logistic equation with spatial degeneracy , 2015 .

[20]  V. Hutson,et al.  The evolution of dispersal , 2003, Journal of mathematical biology.

[21]  P. Bates,et al.  Traveling Waves in a Convolution Model for Phase Transitions , 1997 .

[22]  J. López-Gómez.  Approaching metasolutions by solutions , 2001, Differential and Integral Equations.

[23]  Jian-Wen Sun.  Effects of dispersal and spatial heterogeneity on nonlocal logistic equations , 2021, Nonlinearity.

[24]  Chufen Wu,et al.  Forced waves and gap formations for a Lotka–Volterra competition model with nonlocal dispersal and shifting habitats , 2021 .

[25]  José M. Mazón,et al.  Nonlocal Diffusion Problems , 2010 .

[26]  Yan Li,et al.  Bistable traveling waves for a lattice competitive-cooperative system with delay , 2021 .

[27]  Paul C. Fife,et al.  Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions , 2003 .

[28]  Haim Brezis,et al.  Remarks on sublinear elliptic equations , 1986 .

[29]  SPATIAL PATTERNS FOR POPULATION MODELS IN A HETEROGENEOUS ENVIRONMENT , 2004 .