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About the triangular Shigesada–Kawasaki–Teramoto reaction cross diffusion system

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Abstract

We show how the use of a refined version of the duality method together with the use of more traditional multiplicators enables to quickly get regularity results for the so-called “triangular” Shigesada–Kawasaki–Teramoto system. This system is the coupling of a reaction diffusion equation with a reaction cross diffusion equation. It comes out of the study of segregation in biological populations and involves only linear and quadratic terms. This work provides an alternative to the estimates provided by Lou, Ni, Wu (Discret Contin Dyn Syst 4(2):193-203, 1998).

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References

  1. Amann, Herbert: Dynamic theory of quasilinear parabolic equations II. React. Diff. Syst. Diff. Int. Eq. 3(1), 13–75 (1990)

    Google Scholar 

  2. Amann, Herbert: Dynamic theory of quasilinear parabolic systems III. Global existence. Math. Z. 202(2), 219–250 (1989)

    Article  MathSciNet  Google Scholar 

  3. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), 9–126, Teubner-Texte Math., 133, Teubner, Stuttgart, (1993)

  4. Breden, M., Desvillettes, L., Fellner, K.: Smoothness of moments of the solutions of discrete coagulation equations with diffusion. Monatschefte Math. 123(3), 437–463 (2017)

    Article  MathSciNet  Google Scholar 

  5. Cañizo, José A., Desvillettes, Laurent: Fellner, Klemens Improved duality estimates and applications to reaction-diffusion equations. Commun. Partial Diff. Equ. 39(6), 1185–1204 (2014)

    Article  Google Scholar 

  6. Chen, L., Jüngel, A.: Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Diff. Equ. 224(1), 39–59 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  7. Choi, Y.S., Lui, R., Yamada, Y.: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete Contin. Dyn. Syst. 9(5), 1193–1200 (2003)

    Article  MathSciNet  Google Scholar 

  8. Choi, Y.S., Lui, R., Yamada, Y.: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discret. Contin. Dyn. Syst. 10(3), 719–730 (2004)

    Article  MathSciNet  Google Scholar 

  9. Desvillettes, L., Lepoutre, Th., Moussa, A.: Entropy, duality and cross diffusion. SIAM J. Math. Anal. 46, 820–853 (2014)

    Article  MathSciNet  Google Scholar 

  10. Desvillettes, L., Lepoutre, Th., Moussa, A., Trescases, A.: On the entropic structure of reaction-cross diffusion systems. Comm. Partial Diff. Equ. 40(9), 1705–1747 (2015)

    Article  MathSciNet  Google Scholar 

  11. Desvillettes, L., Trescases, A.: New results for triangular reaction cross diffusion systems. J. Math. Anal. Appl. 430(1), 32–59 (2015)

    Article  MathSciNet  Google Scholar 

  12. Hoang, Luan T., Nguyen, Tuoc V., Phan, Truyen V.: Self-diffusion and cross-diffusion equations: \(W^{1,p}\)-estimates and global existence of smooth solutions. ArXiv: 1311.6828

  13. Iida, Masato, Mimura, Masayasu, Ninomiya, Hirokazu Diffusion: Cross-diffusion and competitive interaction. J. Math. Biol. 53(4), 617–641 (2006)

    Article  MathSciNet  PubMed  Google Scholar 

  14. Izuhara, H., Mimura, M.: Reaction-diffusion system approximation to the cross-diffusion competition system. Hiroshima Math. J. 38(2), 315–347 (2008)

    Article  MathSciNet  Google Scholar 

  15. Lou, Y., Ni, W., Wu, Y.: On the global existence of a cross-diffusion system. Discrete Contin. Dynam. Syst. 4(2), 193–203 (1998)

    Article  MathSciNet  Google Scholar 

  16. Matano, H., Mimura, M.: Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. 19(3), 1049–1079 (1983)

    Article  MathSciNet  Google Scholar 

  17. Mimura, M.: Stationary pattern of some density-dependent diffusion system with competitive dynamics. Hiroshima Math. J. 11(3), 621–635 (1981)

    Article  MathSciNet  Google Scholar 

  18. Shim, S.: Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion. J. Math. Anal. Appl. 279(1), 1–21 (2003)

    Article  MathSciNet  Google Scholar 

  19. Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theoret. Biol. 79(1), 83–99 (1979)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  20. Tuoc, P.: Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions. Proc. Amer. Math. Soc. 135(12), 3933–3941 (2007)

    Article  MathSciNet  Google Scholar 

  21. Tuoc, P.: On global existence of solutions to a cross-diffusion system. J. Math. Anal. Appl. 343(2), 826–834 (2008)

    Article  MathSciNet  Google Scholar 

  22. Yagi, A.: Global solution to some quasilinear parabolic system in population dynamics. Nonlinear Anal. 21(8), 603–630 (1993)

    Article  MathSciNet  Google Scholar 

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  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité and Sorbonne Université, CNRS and IUF, 75013, Paris, France

    L. Desvillettes

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  1. L. Desvillettes
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Desvillettes, L. About the triangular Shigesada–Kawasaki–Teramoto reaction cross diffusion system. Ricerche mat 73 (Suppl 1), 105–114 (2024). https://doi.org/10.1007/s11587-023-00805-w

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