In this paper, we discuss the definition of measurable value solution for quasilinear degenerate parabolic equation ∂ t u + ∂ x f ( u )= ∂ xx A ( u ) , ( x , t ) ∈ R + 2 = R × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ R . It is shown that if the equation is with the weak degeneracy, than there exists a L 2 -entropy solution .