Existence results for some anisotropic degenerate parabolic problems with lower-order term and source

In this study, we establish the existence and regularity of weak solutions for the anisotropic degenerate parabolic equation, namely. ∂tu − ΣNi=1 (Di(ai(t, x, u)|Di u|pi−2 Di u) + |u|τpi−2 u|Di u|pi ) = |u|r−2 u, if 1 ≤ r < τp− + 1 and τ ≥ 1/p− with p− = min1 ≤ i ≤ N pi, then there exists a non-negative weak solution for every positive initial data in L1.