Phase transition layers with boundary intersection for an inhomogeneous Allen–Cahn equation

We consider the nonlinear problem of inhomogeneous Allen–Cahn equation

${?}^2\mathrm{\Delta }u+V(y)u(1?u^2)=0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{on}?\mathrm{\Omega },$

where Ω is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary, ? is a small positive parameter, ν denotes the unit outward normal of ?Ω, V is a positive smooth function on $\overline{\mathrm{\Omega }}$. Let Γ be a curve intersecting orthogonally with ?Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degenerate conditions with respect to the functional ${\int }_{\mathrm{\Gamma }}{V}^{1/2}$. We can prove that there exists a solution $u_{?}$ such that: as $?\to 0$, $u_{?}$ approaches +1 in one part of Ω, while tends to ?1 in the other part, except a small neighborhood of Γ.     

Hydrogenation of acetylene into ethane–ethene mixtures over modified Pd–alumina catalysts

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Lithium–Lanthanide Bimetallic Metal–Organic Frameworks towards Negative Electrode Materials for Lithium-Ion Batteries

Novel lithium–lanthanide (Ln: cerium and praseodymium) bimetallic coordination polymers with formulas C₁₀H₂LnLiO₈ (Ln: Ce (CeLipma) and Pr (PrLipma)) and C₁₀H₃CeO₈ (Cepma) were prepared through a simple hydrothermal method. The three compounds were characterized by means of FTIR spectroscopy, X-ray diffraction, single-crystal X-ray diffraction, SEM, TEM, and X-ray photoelectron spectroscopy. The results of structural refinement show that they belong to triclinic symmetry and P space group with cerium (or praseodymium) and lithium cations, forming coordination bonds to oxygen atoms from different pyromellitic acid molecules, and leading to the construction of 3D structures. It is interesting to note that the frameworks exclude any coordination water and lattice water. As an electrode material for lithium-ion batteries, CeLipma exhibits a maximum capacity of 800.5 mAh g⁻¹ and a retention of 91.4 % after 50 cycles at a current density of 100 mA g⁻¹. The favorable electrochemical properties of the lanthanide coordination polymers show potential application prospects in the field of electrode materials.