L2,μ(Q)-estimates for Parabolic Equations and Applications

Abstract

In this paper we derive a priori estimates in the Campanato space L^{2,\mu}(Q_T) for solutions of tbe following parabolic equation u_t - \frac{∂}{∂x_i}(a_{ij}(x,t)u_x_j+a_iu) + b_iu_x_i + cu = \frac{∂}{∂_x_i}f_i + f_0 where {a_{ij}(x, t)} are assumed to be measurable and satisfy the ellipticity condition. The proof is based on accurate DeGiorgi-Nash-Moser's estimate and a modified Poincare's inequality. These estimates are very useful in the study of the regularity of solutions for some nonlinear problems. As a concrete example, we obtain the classical solvability for a strongly coupled parabolic system arising from the thermistor problem.