Gradient bounds for a widely degenerate orthotropic parabolic equation

In this paper, we consider the following nonlinear parabolic equation $$ ∂ ₜu\,=\,∑ ᵢ₌₁ⁿ∂ _{xᵢ}[ ( u_{xᵢ} -δ ᵢ)₊ᵖ⁻¹{u_{xᵢ}}{ u_{xᵢ} }] \,\,\,\,\,\,\,\,\,\,in\,\,\,Ω × I, $$ ∂ t u = ∑ i = 1 n ∂ x i ( | u x i | - δ i ) + p - 1 u x i | u x i | in Ω × I , where $$Ω $$ Ω is a bounded open subset of $$Rⁿ$$ R n for $$n≥ 2$$ n ≥ 2 , $$I⊂ R$$ I ⊂ R is a bounded open interval, $$p≥ 2$$ p ≥ 2 , $$δ ₁,… ,δ ₙ$$ δ 1 , … , δ n are non-negative numbers and $$( \,· \,) ₊$$ · +