Задача со свободной границей для квазилинейного уравнения реакции-диффузии, возникающих в экологии

3 Дихотомия распространения-исчезновения

Этот раздел посвящен дихотомии распространения-исчезновения видов в окружающей среде.

Позволять есть решение задачи (2)-(6). Тогда справедливо следующее.
Исчезновение: и
Распространение: и равномерно для любого ограниченного множества где является единственным положительным решением следующей задачи

Доказательство аналогично доказательству теоремы 3.5 в [27].
Литература

1. 
   Asrakulova D.S., Elmurodov A.N. A reaction-diffusion-advection competition model with a free boundary, Uzb. Math. J. 2021. No.3, 25-37.
   
2. 
   Cannon J.R. The One dimensional heat equation. Cambridge: Cambridge University Press, 1984. – 500 p.
   
3. 
   Cantrell R.S., Cosner C. Spatial ecology via reaction-diffusion equations. John Wiley and Sons Ltd., Chichester, UK, 2003. – 428 p.
   
4. 
   Crank J. Free and Moving Boundary Problem, Oxford, 1984. – 425 p.
   
5. 
   Chen X., Friedman A. A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal. 2000, Vol.32, pp. 778-800.
   
6. 
   Ciliberto C. Formule di maggiorazione e teoremi di esistenza per le soluzioni delle equazioni paraboliche in due variabili Ricerche di Matem. 1954, 82. pp. 40-75.
   
7. 
   Du Y.H., Lin Z.G. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 2010, Vol. 42, pp. 377-405.
   
8. 
   Du Y., Lin Z.G. The diffusive competition model with a free boundary: invasion of a superior or inferior competitor Discrete Contin. Dyn. Syst. Ser. B. 2014, 19. pp. 3105-3132.
   
9. 
   Friedman A. Partial Differential Equations of Parabolic Type. Courier Dover Publications, 2008.
   
10. 
    Fasano A. Mathematical Models of some diffusive processes with free boundaries, 2008, – 137 p.
    
11. 
    Gupta S. C. The Classical Stefan Problem: Basic concepts, modelling and analysis with quasi-analytical solutions and methods, Elsevier, 2017, – 717 p.
    
12. 
    Gu H., Lin Z. G., Lou B. D. Long time behavior for solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 2015, Vol. 269, pp. 1714-1768.
    
13. 
    Kaneko Y., Matsuzawa H. Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations // J. Math. Anal. Appl., 2015, Vol. 428, No. 1, pp. 43-76.
    
14. 
    Kruzhkov S.N. Nonlinear parabolic equations with two variables, Proc. Moscow Mat. Soc.1967, 16, 329-346.
    
15. 
    Ladyzenskaja O.A., Solonnikov V.A and Ural’ceva N.N. Linear and quasilinear equations of parabolic type, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1968.
    
16. 
    Lin Z.G. A free boundary problem for a predator-prey model, Nonlinearity, 2007, Vol.20, No.8, pp. 1883-1892.
    
17. 
    Leung A. W. Nonlinear systems of partial differential equations: applications to life and physical sciences, World Scientific, 2009, – 541 p.
    
18. 
    Meyrmanov A. M. Zadacha Stefana. Novosibirsk: Nauka, 1986, -240 p. (Russian).
    
19. 
    Murray J.D. Mathematical Biology II: Spatial Models and Biomedical Applications., Springer, Berlin 2003, – 551 p.
    
20. 
    Okubo A., Levin S.A. Diffusion and Ecological Problems. Springer, 2002, – 470 p.
    
21. 
    Pao C.V. Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, 1992, – 777 p.
    
22. 
    Takhirov J.O., Rasulov M.S. Problem with free boundary for systems of equations of reaction-diffusion type. Ukr. Math. J. 2018, Vol.69. pp.1968-1980.
    
23. 
    Takhirov J.O. A free boundary problem for a reaction-diffusion equation in biology, Indian J. Pure Appl.Math. 2019, Vol.50, pp.95-112.
    
24. 
    Wang R., Wang L and Wang Zh. Free boundary problem of a reaction-diffusion equation with nonlinear convection term, J. Math.Anal.Appl.2018. 467, pp.1233-1257.
    
25. 
    Wang M., Zhao J. Free Boundary Problems for a Lotka-Volterra Competition System, Jour. Dyn. Diff. Eq.,2014, Vol.26, pp.1-21.
    
26. 
    Wang M., Zhang Y. Dynamics for a two-species competitive Keller-Segel chemotaxis system with a free boundary, J. Math. Anal. Appl, 2021, Vol 502. pp.73-82.
    
27. 
    Zhou L. and etc. A reaction-diffusion-advection equation with a free boundary and sign-changing coefficient. Acta Appl. Math. 2016, Vol.243, pp.189-216.