Based on the existence of smooth solution of smooth initial date for non-uniformly parabolic equation, we obtain the existence of smooth solution of the equation \(u(t)=(a(u_{x}))_{x}+b(x,u),\; (x,t)\in \mathbb{R}\times (0,T)\) under general initial date u0 ∈ L loc p (ℝ), where p > 1. In the case a′ = (1 + s2)−m/2(m > 1), the same result is established for u0 ∈ W loc 1,p (ℝ) and p ≥ m − 1, m > 2. We also show the existence of weak solution of this equation under conditions u0 ∈ W loc 1,p (ℝ) and p > 1. The method depends on the prior estimate of the smooth solution and its derivative which allow us to use the standard parabolic theory to prove the existence of solutions. Moreover, an example is established to show that there may not be a classical solution of this equation for p ∈ (1, m − 1) under the condition u0 ∈ W loc 1,p (ℝ).
