On Fractional Pseudo-Parabolic Equations Involving Nonlocal Nonlinearities

In this work, we deal with the analysis of some qualitative properties of solutions to a class of fractional pseudo-parabolic equations involving nonlocal nonlinearity. We first establish the existence of local solutions to the problem. Then, we prove the existence and uniqueness of global solutions, as well as the long-time behavior of these global solutions. Next, we prove the blow-up phenomenon at finite time occurs when the initial datum lies in the unstable set, that is, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>I</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> , without requiring the subcritical initial energy condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo>.</mml:mo> </mml:math> Furthermore, we establish an upper bound for the blow-up time and derive both upper and lower bounds for the decay rate of the solutions. Finally, we show the existence of ground-state solutions to the corresponding stationary equation and investigate the relationship between global solutions and ground-state solutions. Our arguments are based on the Galerkin approximation method, the contraction mapping principle, variational methods combined with the Hardy–Littlewood–Sobolev inequality, and a modified differential inequality.