Global strong solutions to nonlocal Benjamin-Bona-Mahony equations with exponential nonlinearities

Abstract

In the current work, we consider the Cauchy problem for a class of adjusted Benjamin-Bona-Mahony (BBM) equations. These equations are modified by considering the time-fractional Caputo derivative of order $\alpha \in (0,1)$ (instead of the classical one) and an additional nonlinearity of exponential type. The first main result includes the unique global existence of strong solutions. The approach for this goal can be summarized as follows. First of all, we use the standard contraction arguments to prove the local existence and uniqueness of a mild solution. Next, apply a weak version of Grönwall's inequality to improve the temporal regularity of the solutions. Using this regularity, we deduce energy estimates for solutions which helps us to obtain the global boundedness. The second aim of the study is about the behavior of solutions according to the fractional order α. Precisely, we show that our solutions converge to those of the classical model (with integer order derivative) as α approaches $1^{-}$. The desired result is derived by some singular integral estimates which is the combination of some essential basic inequalities.