Qualitative properties of solutions to a fractional pseudo-parabolic equation with singular potential
Xiang-kun Shao, Nan-jing Huang, Xue-song Li
TL;DR
The article investigates a fractional pseudo-parabolic equation with singular potential, establishing a sharp dichotomy between global existence and finite-time blow-up in a low-energy regime defined via the potential-well depth $d$ and the Nehari functional $I$. Using potential-well, energy, and variational methods, it proves global existence with exponential decay under $J(u_0)<d$ and $I(u_0)>0$, as well as finite-time blow-up with rigorous lifespan and rate bounds when $I(u_0)<0$. It further demonstrates the existence of ground-state solutions for the stationary problem $(-\Delta)^s u=|u|^{p-2}u$ and proves that global-in-time solutions converge strongly to stationary states as $t\to\infty$. Collectively, the results extend blow-up techniques to nonnegative energy and elucidate the long-time dynamics of FPPEs with singular potentials, including convergence to stationary solutions and explicit decay and lifespan estimates.
Abstract
This paper investigates the initial boundary value problem for a fractional pseudo-parabolic equation with singular potential. The global existence and blow-up of solutions to the initial boundary value problem are obtained at low initial energy. Moreover, the exponential decay estimates for global solutions and energy functional are further derived, and the upper and lower bounds of both blow-up time and blow-up rate for blow-up solutions are respectively estimated. Specifically, we extend the method for proving blow-up of solutions with negative initial energy in previous literatures to cases involving nonnegative initial energy, which broadens the applicability of this method. Finally, for the corresponding stationary problem, the existence of ground-state solutions is established, and it is proved that the global solutions strongly converge to the solutions of stationary problem as time tends to infinity.