Considered herein is the initial–boundary value problem for a fractional pseudo-parabolic equation with memory term and logarithmic nonlinearity given by ut+(−Δ)su+(−Δ)sut=∫0tg(t−τ)(−Δ)su(τ)dτ+uln|u| 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 (u0∈Wδ,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 (u0∈Wδ,2) by introducing some new approximation methods and techniques. Here, the sets Wδ,i(i=1,2) defined in Section 2.2 denote potential well families involving the parameter δ>0. Finally, we give a series of numerical examples used to illuminate above theoretical results.

中文翻译：

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具有记忆项和对数非线性的分数阶伪抛物线方程：适定性、爆炸和渐近稳定性


这里考虑的是分数阶伪抛物线方程的初始边界值问题，该方程具有记忆项和对数非线性，由 ut+（−Δ）su+（−Δ）sut=∫0tg（t−τ）（−Δ）su（τ）dτ+uln|u|在不同的初始能级下。首先，利用任意初始能级下的 Galerkin 近似和收缩映射原理建立弱解的局部适定性。其次，利用修正势阱理论、Galerkin 近似、扰动能法、微积分不等式技术等，研究了低初始能级 （u0∈Wδ，1） 下的全局适定性、多项式和指数能量衰减估计、有限时间爆炸。随后，基于上述低初始能的结论，通过引入一些新的近似方法和技术，在临界初始能级 （u0∈Wδ，2） 处还推导出了全局存在、多项式和指数能量衰减估计以及有限时间爆炸。在这里，第 2.2 节中定义的集合 Wδ，i（i=1,2） 表示涉及参数 δ>0 的潜在井族。最后，我们给出了一系列用于阐明上述理论结果的数值示例。