Boundedness for solutions of weighted forward–backward parabolic equations without assuming higher regularity

We consider the solutions of two evolution equations of the type μ(x,t)ut+Au=0 and (μ(x,t)u)t+Au=0, where μ∈L1 may be positive, null and negative and A a suitable monotone operator. The simple example is A=−Δp with p⩾2. For these functions we prove an unusual local boundedness result using an approximation via the solutions of suitable equations, specifically ɛBu+μ(x,t)ut+Au=0 and ɛBu+(μ(x,t)u)t+Au=0. For A=−Δp, Bu=−(|ut|p−2ut)t.