The Trichotomy of Solutions and the Description of Threshold Solutions for Periodic Parabolic Equations in Cylinders

• Published in: Journal of dynamics and differential equations Vol. 35; no. 4; pp. 3665 - 3689
• Main Authors: Ma, Zhuo, Wang, Zhi-Cheng
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Advection-diffusion equation; Applications of Mathematics; Cylinders; Hyperplanes; Mathematical analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Parameters; Partial Differential Equations; Spreading; Asymptotic behavior; Vanishing; Super- and sub-solutions; 35B40; 35K57; 35C07
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-021-10124-z

In this paper, the nonnegative bounded solutions for reaction–advection–diffusion equations of the form u t - Δ u + α ( t , y ) u x = f ( t , y , u ) in cylinders are studied, where f is a bistable or multistable nonlinearity which is T -periodic in t . We prove that under certain conditions, there are at most three types of solutions for any one-parameter family of initial data: that spread to 1 for large parameters, vanish to 0 for small parameters, and exhibit exceptional behaviors for intermediate parameters. We usually refer to the last as the threshold solutions. It is worth noting that we also give a sufficient condition for solutions to spread to 1 by proving a kind of stability of a pair of diverging traveling fronts. A natural question is what kinds of properties do the threshold solutions have? Under the additional conditions that α ( t , y ) ≡ 0 and that f and u (0,  x ,  y ) are radially symmetric with respect to y around 0 and radially nonincreasing away from 0, by using super- and sub-solutions, Harnack’s inequality and the method of moving hyperplane, we show that any point in the ω -limit set of the threshold solutions is symmetric with respect to x , and exponentially decays to 0 as | x | → ∞ .