Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\Delta u+V(x) |u|^{p-1}u$ in $\Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=M\vf (x) as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.Comment: 29 page

A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

This paper can be thought of as a remark of \cite{llw}, where the authors studied the eigenvalue distribution $\mu_{X_N}$ of random block Toeplitz band matrices with given block order $m$. In this note we will give explicit density functions of $\lim\limits_{N\to\infty}\mu_{X_N}$ when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of $m\times m$ Gaussian unitary ensemble (GUE for short). The series $\{\lim\limits_{N\to\infty}\mu_{X_N}|m\in\mathbb{N}\}$ can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.Comment: 6 page

Proteinaceous Resin and Hydrophilic Encapsulation: A Self-Healing-Related Study

Inspired by living organisms, self-healing materials have been designed as smart materials. Their automatic healing nature is achieved through the use of capsule in which the healing agent is encapsulated. The occurrence of cracks leads to ripping of the capsule, along with crack propagation and release of the healing agent that wets the crack surface to eventually heal (bond) the crack. Such automatic repair of the crack significantly extends the service life of the material. A vast majority of existing self-healing systems have been designed for the epoxy matrix - the most common commercially used thermoset - that possesses low crack resistance. Currently, self-healing systems have not yet been introduced for fully protein-based materials, despite their great potential to replace currently used synthesis precursors for the latter and the eco-friendly nature of self-healing materials. This has been probably due to two major obstacles: poor mechanical properties of the protein-based matrix, and extreme difficulty associated with the encapsulation of hydrophilic healing agents suitable for the protein-based matrix. This study provides possible solutions towards addressing both these obstacles. To improve the inherent mechanical properties of protein-based resin, soy protein isolate (SPI) was chosen as the model in this study. Dialdehyde carboxymethyl cellulose (DCMC) was synthesized and used as the crosslinking agent to modify the SPI film. As-synthesized DCMC - a fully bio-based material - exhibited high mechanical strength, excellent thermal stability, and reduced moisture sensitivity. Good compatibility and effective crosslinking were believed to be the key reasons for such property enhancements. However, these were accompanied by poor crack resistance, where self-healing is a pertinent solution. A novel healing system for the protein matrix was designed in this work via the use of formaldehyde as a healing agent. Subsequently, the well-acknowledged challenge, e.g. hydrophilic agent encapsulation, was addressed through the development of novel polyurethane-Poly(melamine-formaldehyde) (PU-PMF) dual-component capsules. Remarkably, the external PU insulation layer was fabricated through interfacial polymerization based on a water-in-oil-in-oil (W/O/O) emulsion template. Surface tension was identified as the main driving factor for the formation of the external oil phase. The internal PMF layer was observed to strongly influence the internal morphology of the capsule. A protocol was developed, and a typical capsule with dense and neat shell morphology with a shell/capsule diameter (around 3 %) was fabricated. This study provides solutions for the two aforementioned obstacles related to the development of the healing system for the protein-based materials