Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

Abstract

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo---Nirenberg's type in generalized Sobolev spaces, we show that u belongs to $$L^2(-a, 0, H^2(B(\sigma), {\mathbb{R}}^N)) \cap H^1(-a, 0, L^2(B(\sigma), {\mathbb{R}}^N))$$ (see Theorem 3).