Hölder Continuity of Weak Solutions to Certain Nonlinear Parabolic Systems in Two Space Dimensions

Abstract

In the present paper we prove the Hölder continuity of weak solutions to a nonlinear parabolic system in two space dimensions

$$ \frac{{\partial u^i }} {{\partial t}} - D_\alpha a_i^\alpha (x,t,\nabla u) = B_i (x,t,u,\nabla u) in Q (i = 1,...,N) $$

(Q = Ω × (0, T), Ω ⊂ ℝ²) where the coefficients a ^α _i (x,t,ξ)(α = 1,2;i = 1,...,N) are measurable in x, continuous in t, and Lipschitz continuous in ξ whereas the right hand side B_i satisfies the controlled growth condition.