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

Abstract

where the output u ∈ R is the state of the perturbed process, the initial perturbed state f ∈ R 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 is a control, and the parameters a, c, α, and λ are real numbers such that ac 6= 0, α > 1, and λ > 0. A function v(x, t), (x, t) ∈ S, belongs to the set V of admissible controls if v is continuous in S, satisfies the conditions (a) |v(x, t)| ≤ u(x, t) if u(x, t) ≥ 1, (b) |v(x, t)| ≤ 1 if u(x, t) ∈ [0, 1], 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 problem