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 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 4, Pages 77–84 (Mi ufa449)

On qualitative properties of solutions to quasilinear parabolic equations admitting degenerations at infinity

A. B. Muravnikab

a People's Frendship University, Miklukho-Maklaya str. 6, 117198, Moscow, Russia
b JSC “Concern “Sozvezdie”, Plekhanovskaya str. 14, 394018, Voronezh, Russia

Abstract: 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.

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

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÍØ-4479.2014.1 Russian Foundation for Basic Research 17-01-00401_à The work is supported by the Ministry of Education and Science of Russia in the framework of the Program of Increasing competitiveness of RUDN University “5-100” among leading world scientific and educational centers for 2016–2020 and by a grant of the President of Russian Federation NSh-4479.2014.1 and RFBR grant no. 17-01-00401.

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English version:
Ufa Mathematical Journal, 2018, 10:4, 77–84 (PDF, 350 kB); https://doi.org/10.13108/2018-10-4-77

Bibliographic databases:

UDC: 517.956
MSC: 35K59, 35K65
Received: 21.07.2017

Citation: A. B. Muravnik, “On qualitative properties of solutions to quasilinear parabolic equations admitting degenerations at infinity”, Ufimsk. Mat. Zh., 10:4 (2018), 77–84; Ufa Math. J., 10:4 (2018), 77–84

Citation in format AMSBIB
\Bibitem{Mur18} \by A.~B.~Muravnik \paper On qualitative properties of solutions to quasilinear parabolic equations admitting degenerations at infinity \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 4 \pages 77--84 \mathnet{http://mi.mathnet.ru/ufa449} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 4 \pages 77--84 \crossref{https://doi.org/10.13108/2018-10-4-77} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457367000007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060521708} 

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