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$${\mathcal L}^{2,\mu}(Q)$$-estimates for parabolic equations and applications. (English) Zbl 0891.35021
Summary: We derive a priori estimates in the Campanato space $${\mathcal L}^{2,\mu}(Q_T)$$ for solutions of the following parabolic equation $u_t-{\partial\over\partial x_i} (a_{ij}(x, t)u_{x_j}+ a_iu)+ b_iu_{x_i}+ cu={\partial\over\partial x_i} f_i+ f_0,$ where $$\{a_{ij}(x,t)\}$$ are assumed to be measurable and satisfy the ellipticity condition. The proof is based on accurate DeGiorgi-Nash-Mosers estimates and a modified Poincaré’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.

##### MSC:
 35D10 Regularity of generalized solutions of PDE (MSC2000) 35K20 Initial-boundary value problems for second-order parabolic equations 35K55 Nonlinear parabolic equations 35B45 A priori estimates in context of PDEs