Geodesic
On stabilization of solutions of second-order semilinear parabolic equations on closed manifolds
Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 817-834.

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The paper is concerned with problems of existence, uniqueness, and stabilization of weak solutions of one class of semilinear second-order parabolic differential equations on closed manifolds. These equations are inhomogeneous analogues of the Kolmogorov–Petrovskii–Piskunov–Fisher equation, and have significant applied and mathematical value.
Keywords: the Kolmogorov–Petrovskii–Piskunov–Fisher equation, second-order parabolic equation, semilinear equation on manifold, weak solution, stabilization. 
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D. V. Tunitsky. On stabilization of solutions of second-order semilinear parabolic equations on closed manifolds. Izvestiya. Mathematics , Tome 87 (2023) no. 4, pp. 817-834. http://geodesic.mathdoc.fr/item/IM2_2023_87_4_a5/

[1] A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, The investigation of a diffusion equation connected with an increasing amount of substance and its application to a biological problem, Byull. Moskov. Univ., Mat. Mekh., no. 1, 1937 (Russian); French transl. A. Kolmogoroff, I. Petrovsky, N. Piscounoff, “Étude de l'équation de la diffusion avec croissance de la quantite de matière et son application à un problème biologique”, Moscou Univ. Bull. Math., 1:6 (1937), 1–25 | Zbl

[2] R. A. Fisher, “The wave of advance of advantageous genes”, Ann. Eugenics, 7:4 (1937), 335–369 | DOI | Zbl

[3] H. Berestycki, F. Hamel, and L. Roques, “Analysis of the periodically fragmented environment model. I. Species persistence”, J. Math. Biol., 51:1 (2005), 75–113 | DOI | MR | Zbl

[4] B. Perthame, Parabolic equations in biology. Growth, reaction, movement and diffusion, Lect. Notes Math. Model. Life Sci., Springer, Cham, 2015 | DOI | MR | Zbl

[5] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002, arXiv: math/0211159

[6] G. Perelman, Ricci flow with surgery on three-manifoldsя, 2003, arXiv: math/0303109

[7] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifoldsя, 2003, arXiv: math/0307245

[8] D. M. DeTurck, “Deforming metrics in the direction of their Ricci tensors”, J. Diff. Geom., 18:1 (1983), 157–162 | DOI | MR | Zbl

[9] L. I. Nicolaescu, Lectures on the geometry of manifolds, 3rd ed., World Sci. Publ., Hackensack, NJ, 2021 | DOI | MR | Zbl

[10] D. V. Tunitsky, “On solvability of semilinear second-order elliptic equations on closed manifolds”, Izv. RAN. Ser. Mat., 86:5 (2022), 97–115 ; English transl. Izv. Math., 86:5 (2022), 925–942 | DOI | MR | DOI

[11] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Math. Surveys Monogr., 49, Amer. Math. Soc., Providence, RI, 1997 | DOI | MR | Zbl

[12] J. L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Grundlehren Math. Wiss., 111, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1961 | MR | Zbl

[13] L. Hörmander, The analysis of linear partial differential operators, v. II, Grundlehren Math. Wiss., 257, Differential operators with constant coefficients, Springer-Verlag, Berlin, 1983 ; Russian transl. Mir, Moscow, 1986 | DOI | MR | Zbl | MR | Zbl

[14] R. S. Palais, Seminar on the Atiyah–Singer index theorem, With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih, R. Solovay, Ann. of Math. Stud., 57, Princeton Univ. Press, Princeton, NJ, 1965 ; Russian transl. Mir, Moscow, 1970 | MR | Zbl | MR | Zbl

[15] R. O. Wells, Jr., Differential analysis on complex manifolds, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973 ; Russian transl. Mir, Moscow, 1976 | MR | Zbl | MR

[16] G. M. Lieberman, Second order parabolic differential equations, World Sci. Publ., River Edge, NJ, 1996 | DOI | MR | Zbl

[17] Mingxin Wang, Nonlinear second order parabolic equations, CRC Press, Boca Ration, FL, 2021 | DOI | Zbl

[18] L. C. Evans, Partial differential equations, Grad. Stud. Math., 19, 2nd ed., Amer. Math. Soc., Providence, RI, 2010 | MR | Zbl

[19] M. G. Kreĭn and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space”, Uspekhi Mat. Nauk, 3:1(23) (1948), 3–95 (Russian) ; English transl. Amer. Math. Soc. Translation, 26, Amer. Math. Soc., New York, 1950 | MR | Zbl | MR

[20] J. Smoller, Shock waves and reaction-diffusion equations, Grundlehren Math. Wiss., 258, 2nd ed., Springer-Verlag, New York, 1994 | DOI | MR | Zbl

[21] P. Hess, Periodic-parabolic boundary value problems and positivity, Pitman Res. Notes Math. Ser., 247, Longman Scientific Technical, Harlow; John Wiley Sons, New York, 1991 | MR | Zbl

[22] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, 2nd ed., Springer-Verlag, Berlin, 1983 ; Russian transl. Nauka, Moscow, 1989 | DOI | MR | Zbl | MR | Zbl

[23] R. Courant and D. Hilbert, Methods of mathematical physics, v. II, Partial differential equations, (vol. II by R. Courant), Interscience Publishers, a division of John Wiley Sons, New York–London, 1962 ; Russian transl. Mir, Moscow, 1964 | MR | Zbl | MR | Zbl

[24] D. H. Sattinger, “Monotone methods in nonlinear elliptic and parabolic boundary value problems”, Indiana Univ. Math. J., 21:11 (1972), 979–1000 | DOI | MR | Zbl