Decay of Nonnegative Solutions of Singular Parabolic Equations with KPZ-Nonlinearities
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L DIFFERENTIAL EQUATIONS
Decay of Nonnegative Solutions of Singular Parabolic Equations with KPZ-Nonlinearities A. B. Muravnika,b,* a
JSC Concern “Sozvezdie,” Voronezh, 394018 Russia b RUDN University, Moscow, 117198 Russia *e-mail: [email protected]
Received February 15, 2020; revised February 15, 2020; accepted April 9, 2020
Abstract—The Cauchy problem for quasilinear parabolic equations with KPZ-nonlinearities is considered. It is proved that the behavior of the solution as t → ∞ can change substantially as compared with the homogeneous case if the equation involves zero-order terms. More specifically, the solution decays at infinity irrespective of the behavior of the initial function and the rate and character of this decay depend on the conditions imposed on the lower order coefficients of the equation. Keywords: parabolic equations, quasilinear equations, KPZ-nonlinearities, lower-order terms, behavior at infinity DOI: 10.1134/S0965542520080126
1. INTRODUCTION Differential equations with nonlinearities involving the scalar square of the gradient of the unknown function (known as Kardar–Parisi–Zhang (KPZ) nonlinearities) arise in various applications (see, e.g., [1–16]). Such equations are also of interest from a purely theoretical point of view, since they contain the squared first derivative of the unknown function: it is well known (see, e.g., [17–19]) that this is the highest power exponent at which Bernstein-type conditions for the corresponding elliptic problem ensure a priori estimates for the L∞ -norms of the first derivatives of the solution via the L∞ -norm of the solution itself. This paper addresses parabolic equations with KPZ-nonlinearities. The influence of lower-order terms is investigated. It is well known from the classical theory (see [20]) that, in contrast to, for example, the elliptic case, lower-order terms can fundamentally change the nature of solutions. For example, a zeroorder term added to the heat equation can make the well-known Repnikov–Eidelman criterion for solution stabilization completely inapplicable (see [21]): as soon as such a term is added, the behavior of the solution of the Cauchy problem as t → ∞ depends not on the behavior of the ball means of the initial function, but rather on the properties of the coefficient at the unknown function in the equation. In the linear case, this phenomenon has been studied fairly well (see [22–24] and references therein). In this work, we investigate it in the case of quasilinear equations with KPZ-terms in which the coefficient of the nonlinearity admits a nonintegrable singularity with respect to the unknown function. The interest in such 2 coefficients is motivated by the fact that even the case of a constant coefficient at ∇u corresponds to applications that are not covered by classical models of mathematical physics (see [1, 2]). Adding feedback relationships to a model, we substantially enrich it. 2. RESULTS Consider the Cauchy problem
∂u = Δu + β ∇u 2 + C ( x, t )u, ∂t u u t =0 = u0( x),
x ∈ » n,
x∈» ,
1375
n
t > 0,
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