Regularity and geometric character of solution of a degenerate parabolic equation

This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=\Delta{}u^{m}$. Our main objective is to improve the H$\ddot{o}$lder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, the present work will show that: (1) the weak solution $u(x,t)\in{}C^{\alpha,\frac{\alpha}{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$, where $\alpha\in(0,1)$ when $m\geq2$ and $\alpha=1$ when $m\in(1,2)$; (2) the surface $\phi=(u(x,t))^{\beta}$ is a complete Riemannian manifold, which is tangent to $\mathbb{R}^{n}$ at the boundary of the positivity set of $u(x,t)$. (3) the function $(u(x,t))^{\beta}$ is a classical solution to another degenerate parabolic equation if $\beta$ is large sufficiently; Moreover, some explicit expressions about the speed of propagation of $u(x,t)$ and the continuous dependence on the nonlinearity of the equation are obtained. Recalling the older H$\ddot{o}$lder estimate ($u(x,t)\in{}C^{\alpha,\frac{\alpha}{2}}(\mathbb{R}^{n}\times\mathbb{R}^{+})$ with $01$), we see our result (1) improves the older result and, based on this conclusion, we can obtain (2), which shows the geometric characteristic of free boundary.Comment: 18 page

A characterization of covering equivalence

Let A={a_s(mod n_s)}_{s=1}^k and B={b_t(mod m_t)}_{t=1}^l be two systems of residue classes. If |{1\le s\le k: x=a_s (mod n_s)}| and |{1\le t\le l: x=b_t (mod m_t)}| are equal for all integers x, then A and B are said to be covering equivalent. In this paper we characterize the covering equivalence in a simple and new way. Using the characterization we partially confirm a conjecture of R. L. Graham and K. O'Bryant

Strain amplitude response and the microstructure of PA/clay nanocomposites

Polyamide 6/clay nanocomposites (PAn, where n is the mass fraction of clay) with various clay loading were prepared by melt compounding in a twin screw extruder. Exfoliation of clay in a PA matrix was confirmed by X-ray diffraction. Strain amplitude response of PAn in both melt and solution states has been investigated. In the melt state, critical strain amplitude of PAn is sensitive to strain amplitude response and decrease logarithmically with increasing clay loading. The elastic moduli (G′) of PAn are reversible under frequency loop sweeps. Comparisons of strain amplitude response in both melt and solution states have been conducted. Two different responses have been observed: strain thinning in the melt state and weak strain overshoot in the solution state. FTIR studies show that amide II band of PAn shifts toward high wavenumbers, but amide I band and N–H stretching vibration are independent of clay loading. We suggest that two types of strain amplitude response of PAn can be explained: strain thinning which is dominant in PAn caused by physical adsorption and entanglement of PA chains on nanoclays and weak strain overshoot caused by weak bonds between PA chains and nanoclays