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In this paper, we establish global boundedness and Hölder regularity of the weak solution to the following parabolic p(x,t)‐Laplace equation: ut−diva(x,t,u,∇u)=b(x,t,u,∇u)inQT,u(x,t)=0on∂Ω×(0,T),u(x,0)=u0inΩ,under some proper assumptions on a and b. The Hölder continuity given in the present work extends known results to the equation with a general nondivergence term. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we establish a stability result and an error estimate of weak solutions for the initial-boundary value problem of a generalized thin film equation and also obtain some higher regularity results for weak solutions. [Copyright &y& Elsevier]

In this paper, we investigate the optimal C 1 , α estimates for the elliptic p (⋅) -Laplace equation: div (a (x) | ∇ u | p (x) − 2 ∇ u) = div h (x) + f (x) in Ω with f ∈ L q (⋅) (Ω) and a , h ∈ C σ (Ω ¯). Based on a certain geometric oscillation estimate, the scaling arguments and appropriate localization technique as well as the careful analysis on the variable exponents, we exhibit how the optimal Hölder exponent of ∇ u is influenced by p (⋅) , q (⋅) and σ. This work can be regarded as a natural follow up to the paper by Araújo and Zhang (in press). [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively. [Copyright &y& Elsevier]