In this paper, by establishing the boundedness of singular integral operators with variable kernels and their commutators with BMO functions on Morrey spaces of homogeneous groups, we prove a local a prior estimate in Sobolev-Morrey space for solutions to the nondivergence parabolic equation with discontinuous coefficients.

In this paper, we study the asymptotic behaviors of solution for stochastic non-Newtonian fluid with white noise in two-dimensional domain. In particular, we will prove the existence of random attractors in H.

We study capillary spreadings of thin films of liquids of power-law rheology. These satisfy u_t+(u^{λ+2}|u_{xxx}^{λ-1}u_{xxx})_x=0, where u(x,t) represents the thickness of the one-dimensional liquid and λ > 1. We look for traveling wave solutions so that u(x,t)=g(x+ct) and thus g satisfies g'''=\frac{|g-ε|^\frac1λ}{g^{1+\frac2λ}}sgn(g-ε). We show that for each ε > 0 there is an infinitely oscillating solution, g_ε, such that lim_{t→∞}g_ε=ε and that g_ε→g_0 as ε→ 0, where g_0 ≡ 0 for t ≥ 0 and g_0=c_λ|t|^{\frac{3λ}{2λ+1}} for t < 0  for some constant c_λ.

Let (M,g) be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation u_t=Δu-∇φ·∇u-au\log u-bu, where φ is a C^2 function, and a ≠ 0 and b are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).

We consider the following nonlinear problem -Δu=u^{\frac{N+2}{N-2}}, u > 0, in R^N\Ω, u(x)→ 0, as |x|→+∞, \frac{∂u}{∂n}=0, on ∂Ω, where Ω⊂R^N N ≥ 4 is a smooth and bounded domain and n denotes inward normal vector of ∂Ω. We prove that the above problem has infinitely many solutions whose energy can be made arbitrarily large when Ω is convex seen from inside (with some symmetries).