Boundedness for solutions of weighted forward–backward parabolic equations without assuming higher regularity. (July 2022) - Paronetto, Fabio
Abstract: We consider the solutions of two evolution equations of the type μ ( x, t ) u t + A u = 0 and ( μ ( x, t ) u ) t + A u = 0, where μ ∈ L 1 may be positive, null and negative and A a suitable monotone operator. The simple example is A = − Δ p with p ⩾ 2 . For these functions we prove an unusual local boundedness result using an approximation via the solutions of suitable equations, specifically ɛ B u + μ ( x, t ) u t + A u = 0 and ɛ B u + ( μ ( x, t ) u ) t + A u = 0 . For A = − Δ p, B u = − ( | u t | p − 2 u t ) t .