On qualitative properties of solutions to quasilinear parabolic

A. B. Muravnik

Ufa mathematical journal, Tome 10 (2018) no. 4, pp. 77-84

Résumé

We consider the Cauchy problem for a quasilinear parabolic equations $\rho(x)u_t=\Delta u + g(u)|\nabla u|^2$, where the positive coefficient $\rho$ degenerates at infinity, while the coefficient $g$ either is a continuous function or have singularities of at most first power. These nonlinearities called Kardar–Parisi–Zhang nonlinearities (or KPZ-nonlinearities) arise in various applications (in particular, in modelling directed polymer and interface growth). Also, they are of an independent theoretical interest because they contain the second powers of the first derivatives: this is the greatest exponent such that Bernstein-type conditions for the corresponding elliptic problem ensure apriori $L_\infty$-estimates of first order derivatives of the solution via the $L_\infty$-estimate of the solution itself. Earlier, the asymptotic properties of solutions to parabolic equations with nonlinearities of the specified kind were studied only for the case of uniformly parabolic linear parts. Once the coefficient $\rho$ degenerates (at least at infinity), the nature of the problem changes qualitatively, which is confirmed by the presented study of qualitative properties of (classical) solutions to the specified Cauchy problem. We find conditions for the coefficient $\rho$ and the initial function guaranteeing the following behavior of the specified solutions: there exists a (limit) Lipschitz function $A(t)$ such that, for any positive $t$, the generalized spherical mean of the solution tends to the specified Lipschitz function as the radius of the sphere tends to infinity. The generalized spherical mean is constructed as follows. First, we apply a monotone function to a solution; this monotone function is determined only by the coefficient at the nonlinearity (both in regular and singular cases). Then we compute the mean over the $(n-1)$-dimensional sphere centered at the origin (in the linear case, this mean naturally is reduced to a classical spherical mean). To construct the specified monotone function, we use the Bitsadze method allowing us to express solutions of the studied quasilinear equations via solutions of semi-linear equations.

Détail

Keywords: KPZ-nonlinearities, long-time behavior, degeneration at infinity.
Mots-clés : parabolic equations

Bibliographie

[1] S. Kamin, M.A. Pozio, A. Tesei, “Admissible conditions for parabolic equations degenerating at infinity”, St. Peter. Math. J., 19:2 (2008), 239–251 | DOI | MR | Zbl

[2] M. Kardar, G. Parisi, Y.-C. Zhang, “Dynamic scaling of growing interfaces”, Phys. Rev. Lett., 56 (1986), 889–892 | DOI | Zbl

[3] E. Medina, T. Hwa, M. Kardar, Y.-C. Zhang, “Burgers equation with correlated noise: Renormalization group analysis and applications to directed polymers and interface growth”, Phys. Rev. A, 39 (1989), 3053–3075 | DOI | MR

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

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

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

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

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

[9] J. Quastel, “KPZ universality for KPZ”, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, 401–405 | DOI | MR | Zbl

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

[11] G. Schehr, “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

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

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

[14] T. Funaki, M. Hoshino, “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

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

[16] S.I. Pokhozhaev, “On equations of the form $\Delta u = f(x, u,Du)$”, Math. USSR-Sb., 41:2 (1982), 269–280 | DOI | MR | Zbl | Zbl

[17] V. N. Denisov, A. B. Muravnik, “On stabilization of the solution of the Cauchy problem for quasilinear parabolic equations”, Diff. Equat., 38:3 (2002), 369–374 | DOI | MR | Zbl

[18] A. B. Muravnik, “On stabilization of solutions of singular quasi-linear parabolic equations with singular potentials”, Tubes, Sheets and Singularities in Fluid Dynamics, Fluid Mech. Appl., 71, Springer, Dordrecht, 2002, 335–340 | MR

[19] A. B. Muravnik, “Stabilization of solutions of certain singular quasilinear parabolic equations”, Math. Notes, 74:6 (2003), 812–818 | DOI | DOI | MR | Zbl

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

[21] V. N. Denisov, A. B. Muravnik, “On the asymptotic behavior of the solution of the Dirichlet problem for an elliptic equation in a half-space”, Nonlinear analysis and nonlinear differential equations, Fizmatlit, M., 2003, 397–417 (in Russian) | Zbl

[22] A.B. Muravnik, “On properties of the stabilization functional of the Cauchy problem for quasilinear parabolic equations”, Trudy Inst. Matem. NAN Belarus, 12:2 (2004), 133–137 | MR

[23] A.B. Muravnik, “On stabilization of solutions of elliptic equations containing Bessel operators”, Integral methods in science and engineering. Analytic and numerical techniques, Birkhäuser, Boston, MA, 2004, 157–162 | DOI | MR | Zbl

[24] A.B. Muravnik, “On stabilization of solutions of singular elliptic equations”, J. Math. Sci., 150:5 (2008), 2408–2421 | DOI | MR | Zbl

[25] A. V. Bitsadze, “Theory of a class of nonlinear partial differential equations”, Differ. Equat., 13 (1977), 1388–1399 | MR | Zbl | Zbl

[26] H. Brezis, S. Kamin, “Sublinear elliptic equations in ${\mathbb R}^n$”, Manuscripta Math., 74 (1992), 87–106 | DOI | MR | Zbl