Global Hölder continuity of solutions to quasilinear equations with Morrey data

We deal with general quasilinear divergence-form coercive operators whose prototype is the [Formula: see text]-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical [Formula: see text]-result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

In this paper, we consider the following indirect signal generation and singular sensitivity n t = Δ n + χ ∇ ⋅ n / φ c ∇ c , x ∈ Ω , t > 0 , c t = Δ c − c + w , x ∈ Ω , t > 0 , w t = Δ w − w + n , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N N = 2 , 3 with smooth boundary ∂ Ω . Under the nonflux boundary conditions for n , c , and w , we first eliminate the singularity of φ c by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.