Relative stabilization of solutions of a degenerating parabolic equation with nonlinear lower-order terms

Abstract

For a nonlinear degenerate parabolic equation of a perturbed process, we use a nonlinear feedback control to solve the stabilization problem for solutions with initial functions in some class to be constructed.Consider the Cauchy problemwhere the output u ∈ R 1 is the state of the perturbed process, the initial perturbed state f ∈ R 1 is a continuous function (according to the theory in [1, pp. 180-185], it is meaningful to consider nonnegative functions u and f ), the input v ∈ R 1 is a control, and the parameters a, c, α, and λ are real numbers such that ac = 0, α > 1, and, and produces only nonnegative solutions of problem (1), (2). Conditions (a) and (b) imply that the output v and the input u of Eq. (1) are comparable in magnitude: condition (a) implies the inequality |v|/u ≤ 1 for u ≥ 1, and condition (b) gives |u − |v|| ≤ 1 for u ∈ [0, 1].Parabolic equations with gradient nonlinearities have been comprehensively studied [2] from various viewpoints. The first boundary value problemin domains D bounded and unbounded in R n was considered in [2], and for a wide class of such problems, it was shown that global solutions are absent for sufficiently large initial data. Simplest sufficient conditions for the existence and absence of a global solution were given in [3] for the first boundary value problem (c)-(e) with the function F = λu p − |∇u| q , where D is a bounded domain in R n , n ≥ 1, with smooth boundary, λ > 0, p > 1, and q > 1. The lifespan (finite or infinite) of a solution of problem (c)-(e) was found in [4] in the case of the function F = u p − µ|∇u| q , µ > 0, and an unbounded domain D for various ranges of the parameters p, q, and µ. In [5], problem (c)-(e) with the function F = u|u| p−1 − µ|∇u| q , µ > 0, was treated as a model of population dynamics describing the evolution of the population density of a species. The present paper is a continuation of [6]. It is known (e.g., see [7, p. 37]) that, owing to the degeneration of Eq. (1) for u = 0, problem (1), (2) cannot have a classical solution even for smooth initial data. It is reasonable [8] to define a solution of the equation as follows.Definition 1. A nonnegative continuous function u(x, t) inS is called a solution of Eq. (1) for a given control v(x, t) if u(x, t) has a continuous derivative u x (x, t) and satisfies the integral identity t2 t1 R uh t + u α h xx + a |u x | λ h + cvh dx dt − R uh t2 t1 dx = 0 ( * )