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In this paper, we investigate the Cauchy problem for the generalized IBq equation with damping in one dimensional space. When σ = 1, the nonlinear approximation of the global solutions is established under small condition on the initial value. Moreover, we show that as time tends to infinity, the solution is asymptotic to the superposition of nonlinear diffusion waves which are given explicitly in terms of the selfsimilar solution of the viscous Burgers equation. When σ ≥ 2, we prove that ourdoi:10.22436/jnsa.009.12.08 fatcat:mibina6vlvhm7lfwtq7uspnw7e