Fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity: Well-posedness, blow up and asymptotic stability

Abstract

Considered herein is the initial–boundary value problem for a fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity given by $u_t+{(-\Delta )}^{s}u+{(-\Delta )}^s{u}_t={\int }_0^{t}g(t-\tau ){(-\Delta )}^{s}u\left(\tau \right)d\tau +uln\left|u\right|$ under different initial energy levels. The local well-posedness of weak solution is firstly established by using Galerkin approximation and contraction mapping principle at arbitrary initial energy level. Secondly, the global well-posedness, polynomial and exponential energy decay estimates, finite time blow up are investigated at low initial energy level ($u_0\in W_{\delta ,1}$) by utilizing modified potential well theory, Galerkin approximation, perturbed energy method, differential–integral inequality technique etc. Subsequently, based on the above conclusions of low initial energy, the global existence, polynomial and exponential energy decay estimates and finite time blow up are also derived at critical initial energy level ($u_0\in W_{\delta ,2}$) by introducing some new approximation methods and techniques. Here, the sets $W_{\delta ,i}$ $(i=1,2)$ defined in Section 2.2 denote potential well families involving the parameter $\delta >0$. Finally, we give a series of numerical examples used to illuminate above theoretical results.