Anomalous pseudo-parabolic Kirchhoff-type dynamical model

Xiaoqiang Dai,Qiang Lin,Xueteng Tian,Jiangbo Han

doi:10.1515/anona-2021-0207

Abstract

Abstract In this paper, we study an anomalous pseudo-parabolic Kirchhoff-type dynamical model aiming to reveal the control problem of the initial data on the dynamical behavior of the solution in dynamic control system. Firstly, the local existence of solution is obtained by employing the Contraction Mapping Principle. Then, we get the global existence of solution, long time behavior of global solution and blowup solution for J(u 0) ⩽ d, respectively. In particular, the lower and upper bound estimates of the blowup time are given for J(u 0)<d. Finally, we discuss the blowup of solution in finite time and also estimate an upper bound of the blowup time for high initial energy.

Highlights

Introduction and main resultThe paper is devoted to the study of an anomalous pseudo-parabolic Kirchho -type dynamical model as follows ut+ M([u]s )(−∆)s u + (−∆)s ut = |u|q−u, in Ω × R+, u(x, ) = u (x), in Ω, (1.1)u(x, t) =, in (RN \ Ω) × R+, where s ∈ (, ), N > s, Ω ⊂ RN is a bounded domain with Lipschitz boundary ∂Ω
We discuss the blowup of solution in nite time and estimate an upper bound of the blowup time for high initial energy
Sun et al [34] obtained the nite time blowup results for (1.5) provided that the initial energy satis es J(u ) < d(∞), where d(∞) is a nonnegative constant, and derive the estimates of the lower bound and upper bound for the blowup time

Summary

Introduction

For the initial boundary value problem (1.4), Ding and Zhou [6] proved the global existence and nite time blowup of solution when J(u ) ≤ d for the case M(σ) = m σλ− . We prove a threshold result of global existence and nonexistence of solutions for problem (1.1) with the sub-critical initial energy J(u ) < d. By the similar argument, we derive the lower bound estimate for blowup time of solution to problem (1.1) with J(u ) < d.

Objectives

Results

Conclusion