We are concerned with divergence type quasilinear parabolic equation with measurable coefficients and lower order terms model of which is a doubly nonlinear anisotropic parabolic equations with absorption term. This class of equations has numerous applications which appear in modeling of electrorheological fluids, image precessing, theory of elasticity, theory of non-Newtonian fluids with viscosity depending on the temperature. But the qualitative theory doesn't construct for these anisotropic equations. So, naturally, that during the last decade there has been growing substantial development in the qualitative theory of second order anisotropic elliptic and parabolic equations. The main purpose is to obtain the pointwise upper estimates in terms of distance to the boundary for nonnegative solutions of such equations. This type of estimates originate from the work of J. B. Keller, R. Osserman, who obtained a simple upper bound for any solution, in any number of variables for Laplace equation. These estimates play a crucial role in the theory of existence or nonexistence of so called large solutions of such equations, in the problems of removable singularities for solutions to elliptic and parabolic equations. Up to our knowledge all the known estimates for large solutions to elliptic and parabolic equations are related with equations for which some comparison properties hold. We refer to I.I. Skrypnik, A.E. Shishkov, M. Marcus , L. Veron, V.D. Radulescu for an account of these results and references therein. Such equations have been the object of very few works because in general such properties do not hold. The main ones concern equations only in the precise choice of absorption term \(f(u)=u^q\). Among the people who published significative results in this direction are I.I. Skrypnik, J. Vetois, F.C. Cirstea, J. Garcia-Melian, J.D. Rossi, J.C. Sabina de Lis. The main result of the paper is a priori estimates of Keller-Osserman type for nonnegative solutions of a doubly nonlinear anisotropic parabolic equations with absorption term that have been proven despite of the lack of comparison principle. To obtain these estimates we exploit the method of energy estimations and De Giorgy iteration techniques.
A priori estimates and unique continuation theorems for second order parabolic equations