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

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.