p-Laplacian boundary value problems on exterior regions

This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?R^N$U?{\mathbb{R}}^N$ with dimension N?3$N?3$. Existence-uniqueness results when p∈(1,N)$p\in (1,N)$ are provided in a space E^1,p(U)$E^{1,p}(U)$ of functions that contains W^1,p(U)$W^{1,p}(U)$. Functions in E^1,p(U)$E^{1,p}(U)$ are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an L^p$L^p$-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.     

Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities

This is a subsequent work of our previous one in [25]. Let the nonlinear terms of non-autonomous parabolic problems with singular initial data satisfy subcritical and critical growth conditions. We first establish the existence of uniform attractors in W^1,r(Ω)$W^{1,r}(\Omega )$, 1<r<N$1<r<N$, for the family of processes corresponding to the equations with external forces being translation bounded but not translation compact. Then, we prove the existence of pullback attractors in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively, for the process corresponding to the equation with the weaker assumption on the external force than previous one. Finally, we investigate the robust of attractors and establish the existence of pullback exponential attractors for the process acting in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively.     

Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H^s(T)×H^r(T)$H^s(\mathbb{T})\times H^r(\mathbb{T})$ to C([0,1],H^s(T)×H^r(T))$C([0,1],H^s(\mathbb{T})\times H^r(\mathbb{T}))$ when s?1$s?1$ and r?0$r?0$. We also demonstrate the nonuniform continuous property in the continuous function space C¹(T)×C¹(T)$C^1(\mathbb{T})\times C^1(\mathbb{T})$.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.