^{}School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.

Receive Date: 26 November 2015,
Revise Date: 12 June 2016,
Accept Date: 16 July 2016

Abstract

In this paper we focus on the Cauchy problem for the generalized IBq equation with damped term in $n$-dimensional space. We establish the global existence and decay estimates of solution with $L^q(1\leq q\leq 2)$ initial value, provided that the initial value is suitably small. Moreover, we also show that the solution is asymptotic to the solution $u_L$ to the corresponding linear equation as time tends to infinity. Finally, asymptotic profile of the solution $u_L$ to the linearized problem is also discussed.

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