Periodic boundary value problem for nonlinear Sobolev-type equations

Abstract

The large-time asymptotic behavior of solutions to the periodic boundary value problem for a nonlinear Sobolev-type equation is studied. In particular, the case where the initial perturbations are not small is considered. In this case, the large-time behavior of solutions is dichotomous.

Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

“…Thus, the equation of the form (1.1) is also called a Sobolev type equation. Since the last century, pseudo-parabolic equations have been studied in different aspects, such as the integral representations of solutions [12], long-time behavior of solutions [17], Riemann problem and Riemann-Hilbert problem [8], nonlocal boundary value problems [4], and periodic problems [6,18,24,25]. Cao et al [5] investigated the blow-up theorems of the Cauchy problem…”

Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

“…Kaikina et al [10] discussed the periodic boundary value problem of (3) under some assumption forms of nonlinear function . Cao et al [11] investigated a class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources.…”

Global existence and nonexistence of solutions for the nonlinear pseudo‐parabolic equation with a memory term

“…In the absence of the memory term ( λ ( t ) = 0), β = γ = 1, the model reduces to the semilinear pseudo‐parabolic equation It describes nonstationary processes in crystalline semiconductors. Kaikina, Naumkin, and Shishmarev discussed the periodic boundary value problem of the equation , where f ( u ) =− λ | u | p − 1 , λ > 0 and p > 1. They proved that if the initial data are small, then there exists a unique solution and found that the solution exhibit power‐law decay in time or dichotomous large‐time behavior.…”

“…The existence, nonexistence, asymptotic behavior, and regularities of nonlinear pseudo-parabolic equations were investigated by many authors. We also refer the reader to see [11][12][13][14][15][16][17][18] and the papers cited therein.…”

“…It describes nonstationary processes in crystalline semiconductors. Kaikina, Naumkin, and Shishmarev [14] discussed the periodic boundary value problem of the equation (1.4), where f .u/ D juj p 1 , > 0 and p > 1. They proved that if the initial data are small, then there exists a unique solution and found that the solution exhibit power-law decay in time or dichotomous large-time behavior.…”