Renormalized solutions to a parabolic equation with mixed boundary constraints

Abstract

We establish the existence and uniqueness of a renormalized solution to the parabolic equation ∂b(u)∂t−div(a(x,t,u,∇u))=f in Ω×(0,T) subject to a mixed boundary condition. Here b(u) is a real function of u, −div(a(x,t,u,∇u)) is of Leray–Lions type and f is an L1-function. Then we compare the renormalized solution to two other notions of solution: distributional solution and weak solution.