searching an asymptotic solution of an nonlinear ODE

I am finding an asymptotic solution for the following ODE:

(eq1) dy/dt = y^{p} [1 + log^{-a}(2 + y^{2})] with p > 1, a > 0, y(0) = y_{0} > 0 .

We already know the solution of the following equation

(eq2) dy/dt = y^{p} with p > 1, y(0) = y_{0} > 0

can solved by taking v = y^{1-p} ==> y(t) = ((1-p)t + c)^{(-1/(p-1))} tends to +infinity as t -> T_{0 . } A question concerns with the solution of (eq1) that: u(t) ~~> ? as u(t) --> + infinity ( or t --> T_{1})

Should it be asymptotic as the solution of (eq2)?

Thank you anyone who can help to solve this problem!