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In this paper, we study the boundary blow-up problem related to the infinity Laplacian $$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

Potato tubers (250 g) were washed with distilled water and smashed in 500 mL of a mixture containing 50% alcohol and 1.6% citric acid. The resulting mixture was filtrated twice and centrifuged at 4000 rpm. The supernatant was transferred into a column filled with the pretreated macroporous resin AB-8 for 6 h, and the pigments absorbed in AB-8 were eluted with 95% alcohol. The eluate was concentrated with rotary evaporator at 30 °C and dried in vacuo for 12 h. Two biological replicates were used in this study.

Cells can acquire a stem-like cell phenotype through epithelial-mesenchymal transition (EMT). However, it is not known which of the stem-like cancer cells are generated by these phenotype transitions. We studied the EMT-inducing roles of SNAILs (the key inducers for the onset of EMT) in selected cancer cells (lung cancer cell line with relatively stable genome), in order to provide more implications for the investigation of EMT-related phenotype transitions in cancer. However, SNAILs fail to induce completed EMT. In addition, we proved that Snail accelerates the early G1 phase whereas Slug accelerates the late G1 phase. Blocking G1 phase is one of the basic conditions for the onset of EMT-related phenotype transitions (e.g., metastasis, acquiring stemness). The discovery of this unexpected phenomenon (promoting G1 phase) typically reveals the heterogeneity of cancer cells. The onset of EMT-related phenotype transitions in cancer needs not only the induction and activation of SNAILs, but also some particular heredity alterations (genetic or epigenetic alterations, which cause heterogeneity). The new connection between heredity alteration (heterogeneity) and phenotype transition suggests a novel treatment strategy, the heredity alteration-directed specific target therapy. Further investigations need to be conducted to study the relevant heredity alterations.