Weighted L p -type regularity estimates for nonlinear parabolic equations with Orlicz growth

Abstract

In this paper we obtain the following weighted L p -type regularity estimates B ( | f | ) ∈ L q ( ν , ν + T ; L w q ( Ω ) )   locally ⇒ B ( | ∇ u | ) ∈ L q ( ν , ν + T ; L w q ( Ω ) )   locally for any q > 1 of weak solutions for non-homogeneous nonlinear parabolic equations with Orlicz growth u t − div ⁡ ( a ( ( A ∇ u ⋅ ∇ u ) 1 2 ) A ∇ u ) = div ⁡ ( a ( | f | ) f ) under some proper assumptions on the functions a , w , A and f , where B ( t ) = ∫ 0 t τ a ( τ ) d τ . Moreover, we remark that two natural examples of functions a ( t ) are a ( t ) = t p − 2 ( p -Laplace equation) and a ( t ) = t p − 2 log α ⁡ ( 1 + t ) for   α > 0. Moreover, our results improve the known results for such equations.