Oscillatory Solutions of Some Autonomous Partial Differential Equations with a Parameter

Abstract

We study a class of evolutionary partial differential equations depending on a parameter τ (stemming from the problems of groundwater flows). The existence of an open interval 𝒯0 of the parameter τ and of a function τ ⟼ Θ(τ), Θ: 𝒯0 ⟼(0, + ∞), is proved with the property that any nonzero global solution u:ℝ+ × Ω → ℝ of the equation cannot remain nonnegative (nonpositive) throughout the set J × Ω; where J ⊂ ℝ+ is any interval whose length is greater than Θ (τ). In other words, these solutions are globally oscillatory and Θ (τ) is the uniform oscillatory time. The interval 𝒯0 and the function Θ are explicitly determined.