Local vanishing properties of solutions of elliptic and parabolic quasilinear equations

Source abstract

We use a local energy method to study the vanishing property of the weak solutions of the elliptic equation − div A ( x , u , D u ) + B ( x , u , D u ) = 0 - \operatorname {div}\;A(x,u,Du) + B(x,u,Du) = 0 and of the parabolic equation ∂ ψ ( u ) / ∂ t − div A ( t , x , u , D u ) + B ( t , x , u , D u ) = 0 \partial \psi (u)/\partial t - \operatorname {div}\;\mathcal {A}(t,x,u,Du) + \mathcal {B}(t,x,u,Du) = 0 . The results are obtained without any assumption of monotonicity on A A , B B , A \mathcal {A} and B \mathcal {B} .