On qualitative properties of sign-constant solutions of some quasilinear parabolic problems

A. B. Muravnik

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference on Mathematical Modelling in Applied Sciences ICMMAS-17, St. Petersburg Polytechnic University, July 24-28, 2017, Tome 160 (2019), pp. 85-94.

Résumé

We study the Cauchy problem for quasilinear parabolic inequalities containing squares of the first derivatives of an unknown function (the so-called nonlinearities of the KPZ type). The coefficients of the leading nonlinear terms of the inequalities considered either can be continuous functions (the regular case) or can admit power singularities (the singular case) of degree ho greater than $1$. For the regular case, we prove the damping of global nonnegative solutions to the problem studied. By damping, we mean the boundedness of the support of a solution for each positive $t$, uniform (with respect to $t$) convergence to zero as $|x|\to\infty$, and vanishing (for any $x$) starting with a certain sufficiently large $t$. For the singular case, we proved that the problem considered has no global positive solutions.

Détail

Keywords: parabolic inequalities, quasilinear inequalities, damping of solutions.

Bibliographie

[1] Barenblatt G. I., “Ob avtomodelnykh dvizheniyakh szhimaemoi zhidkosti v poristoi srede”, Prikl. mat. mekh., 16:6 (1952), 679–698 | Zbl

[2] Bitsadze A. V., “K teorii odnogo klassa nelineinykh uravnenii v chastnykh proizvodnykh”, Differ. uravn., 13:11 (1977), 1993–2008 | MR | Zbl

[3] Denisov V. N., Muravnik A. B., “O stabilizatsii resheniya zadachi Koshi dlya kvazilineinykh parabolicheskikh uravnenii”, Differ. uravn., 38:3 (2002), 351–355 | MR | Zbl

[4] Denisov V. N., Muravnik A. B., “Ob asimptotike resheniya zadachi Dirikhle dlya ellipticheskogo uravneniya v poluprostranstve”, Nelineinyi analiz i nelineinye differentsialnye uravneniya, Fizmatlit, M., 2003, 397–417 | Zbl

[5] Kalashnikov A. S., “O ponyatii konechnoi skorosti rasprostraneniya vozmuschenii”, Usp. mat. nauk, 34:2 (1979), 199–200 | MR | Zbl

[6] Kalashnikov A. S., “Nekotorye voprosy kachestvennoi teorii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, Usp. mat. nauk, 42:2 (1987), 135–176 | MR | Zbl

[7] Kalashnikov A. S., “Ob usloviyakh mgnovennoi kompaktifikatsii nositelei reshenii polulineinykh parabolicheskikh uravnenii i sistem”, Mat. zametki, 47:1 (1990), 74–80 | MR | Zbl

[8] Mitidieri E., Pokhozhaev S. I., “Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh”, Tr. Mat. in-ta im. V. A. Steklova, 234, 2001, 3–383

[9] Muravnik A. B., “O stabilizatsii reshenii nekotorykh singulyarnykh kvazilineinykh parabolicheskikh zadach”, Mat. zametki, 74:6 (2003), 858–865 | DOI | MR | Zbl

[10] Muravnik A. B., “On properties of the stabilization functional of the Cauchy problem for quasilinear parabolic equations”, Tr. in-ta mat. NAN Belarusi, 12:2 (2004), 133–137 | MR

[11] Muravnik A. B., “O stabilizatsii reshenii singulyarnykh ellipticheskikh uravnenii”, Fundam. prikl. mat., 12:4 (2006), 169–186

[12] Muravnik A. B., “O razrushenii reshenii nekotorykh sistem vyrozhdayuschikhsya i singulyarnykh kvazilineinykh parabolicheskikh neravenstv”, Chernozemnyi almanakh nauch. issled. Ser. «Fundamentalnaya matematika», 2009, no. 1 (8), 312–323

[13] Muravnik A. B., “O razrushenii reshenii nekotorykh sistem kvazilineinykh parabolicheskikh neravenstv”, Sovr. mat. Fundam. napr., 48, 2013, 84–92

[14] Nikiforov A. F., Uvarov V. B., Spetsialnye funktsii matematicheskoi fiziki, Nauka, M., 1984 | MR

[15] Pokhozhaev S. I., “Ob uravneniyakh vida $\Delta u=f(x,u,Du)$”, Mat. sb., 113:2 (1980), 324–338 | MR | Zbl

[16] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Elementarnye funktsii, Nauka, M., 1981 | MR

[17] Amann H., “Existence and multiplicity theoremes for semi-linear elliptic boundary-value problems”, Math. Z., 150 (1976), 281–295 | DOI | MR | Zbl

[18] Amann H., Crandall M. G., “On some existence theoremes for semi-linear elliptic equations”, Ind. Univ. Math. J., 27:5 (1978), 779–790 | DOI | MR | Zbl

[19] Anh V. V., Leonenko N. N., Sakhno L. M., “Spectral properties of Burgers and KPZ turbulence”, J. Stat. Phys., 122:5 (2006), 949–974 | DOI | MR | Zbl

[20] Barral J., Jin X., Rhodes R., Vargas V., “Gaussian multiplicative chaos and KPZ duality”, Commun. Math. Phys., 323:2 (2013), 451–485 | DOI | MR | Zbl

[21] Benjamini I., Schramm O., “KPZ in one dimensional random geometry of multiplicative cascades”, Commun. Math. Phys., 289:2 (2009), 653–662 | DOI | MR | Zbl

[22] Bernardin C., Gonçalves P., Sethuraman S., “Occupation times of long-range exclusion and connections to KPZ class exponents”, Probab. Th. Related Fields, 166:1-2 (2016), 365–428 | DOI | MR | Zbl

[23] Corwin I., Ferrari P. L., Péché S., “Universality of slow decorrelation in KPZ growth”, Ann. Inst. H. Poincaré. Probab. Stat., 48:1 (2012), 134–150 | DOI | MR | Zbl

[24] Denisov V. N., Muravnik A. B., “On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations”, Electr. Res. Ann. Am. Math. Soc., 9 (2003), 88–93 | DOI | MR | Zbl

[25] Duplantier B., “Liouville quantum gravity and the KPZ relation: A rigorous perspective”, XVIth Int. Congr. on Math. Phys., World Scientific, Hackensack, New Jersey, 2010, 56–85 | DOI | MR | Zbl

[26] Evans L. C., Knerr B. F., “Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities”, Ill. J. Math., 23:1 (1979), 153–166 | DOI | MR | Zbl

[27] Funaki T., Hoshino M., “A coupled KPZ equation, its two types of approximations, and existence of global solutions”, J. Funct. Anal., 273:3 (2017), 1165–1204 | DOI | MR | Zbl

[28] Galaktionov V. A., Mitidieri E. L., Pohozaev S. I., Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion, and Schrödinger Equations, CRC Press, Boca Raton, FL, 2015 | MR | Zbl

[29] Ginelli F., Hinrichsen H., “Mean field theory for skewed height profiles in KPZ growth processes”, J. Phys. A, 37:46 (2004), 11085–11100 | DOI | MR | Zbl

[30] Gladkov A., Guedda M., Kersner R., “A KPZ growth model with possibly unbounded data: correctness and blow-up”, Nonlin. Anal., 68:7 (2008), 2079–2091 | DOI | MR | Zbl

[31] Guedda M., Kersner R., “Self-similar solutions to the generalized deterministic KPZ equation”, Nonlin. Differ. Equ. Appl., 10:1 (2003), 1–13 | DOI | MR | Zbl

[32] Kardar M., Parisi G., Zhang Y.-C., “Dynamic scaling of growing interfaces”, Phys. Rev. Lett., 19 (1986), 889–892 | DOI

[33] Kazdan I. L., Kramer R. I., “Invariant criteria for existence of solutions to secondorder quasilinear elliptic equations”, Commun. Pure Appl. Math., 31:7 (1978), 619–645 | DOI | MR | Zbl

[34] Medina E., Hwa T., Kardar M., Zhang Y.-C., “Burgers equation with correlated noise: Renormalization group analysis and applications to directed polymers and interface growth”, Phys. Rev., A39 (1989), 3053–3075 | DOI | MR

[35] Muravnik A. B., “On stabilization of solutions of singular quasi-linear parabolic equations with singular potentials”, Fluid Mech. Appl., 71 (2002), 335–340 | MR | Zbl

[36] Muravnik A. B., “On stabilization of solutions of elliptic equations containing Bessel operators”, Integral Methods in Science and Engineering. Analytic and Numerical Techniques, Birkhäuser, Boston–Basel–Berlin, 2004, 157–162 | DOI | MR | Zbl

[37] Muravnik A. B., “On a quasilinear analog of Gidas—Spruck theorem”, Nonlin. Bound. Value Probl., 14 (2004), 105–111. | Zbl

[38] Muravnik A. B., “On local blow-up of solutions of quasilinear elliptic and parabolic inequalities”, Nonlin. Bound. Value Probl., 16 (2006), 86–95 | MR | Zbl

[39] Muravnik A. B., “On nonexistence of global solutions of the Cauchy problem for quasilinear parabolic inequalities”, Analytic Methods of Analysis and Differential Equations, Cambridge Sci. Publ., Cottenham, 2006, 183–197 | MR | Zbl

[40] Quastel J., “KPZ universality for KPZ”, XVIth Int. Congr. on Math. Phys., World Scientific, Hackensack, New Jersey, 2010, 401–405 | DOI | MR | Zbl

[41] Schehr G., “Extremes of $N$ vicious walkers for large $N$: application to the directed polymer and KPZ interfaces”, J. Stat. Phys., 149:3 (2012), 385–410 | DOI | MR | Zbl

[42] Spohn H., “Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals”, Phys. A, 369:1 (2006), 71–99 | DOI | MR

[43] Spohn H., “KPZ scaling theory and the semidiscrete directed polymer model”, Math. Sci. Res. Inst. Publ., 65 (2014), 483–493 | MR | Zbl