We consider the Cahn-Hilliard equation in H 1 (ℝ N ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as |u| → ∞ and logistic type nonlinearities. In both situations we prove the H 2 (ℝ N )-bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J.W. Cholewa, A. Rodriguez-Bernal (2012).
In this study, the adsorption performance of montmorillonite (MMT) was evaluated by Basic Red-5 adsorption experiments considering the influencing factors (initial BR-5 concentration, dosage, time, pH, and temperature). The surface and structural properties were characterized by FT-IR, XRD, XRF, SEM-EDS, AFM, and BET techniques. The adsorption experiments were carried out by batch mode for the evaluation of isotherm, kinetic, and thermodynamic studies. The results of equilibrium adsorption isotherm were interpreted using different isotherm models. The equilibrium data fitted well with the Langmuir isotherm models, and the maximum adsorption capacity was found as 163.93 mg/g. Adsorption data of the BR-5 onto MMT provide well by pseudo-second-order model (R2= 0.999). The Ho, So and Go values were calculated for the nature of the adsorption process. The analysis of the thermodynamic parameters showed spontaneous, exothermic, and viable adsorption of BR-5 under the investigated experimental conditions. A factorial design was applied to examine the effect of three factors initial concentration of dye (50 and 100 mg/L), time (60 and 120 min.) and dosage (0.05 and 1.00 mg/L) on the adsorption process. According to the results, with high efficient adsorption capacity and compatible surface properties are advantageous to be used for uptake of dyes.
The scrambling index of an $n\times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^k(A^{\rm T})^k=J$, where $A^{\rm T}$ denotes the transpose of $A$ and $J$ denotes the $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its Boolean rank $b(M)$ is the smallest positive integer $b$ such that $M=AB$ for some $m\times b$ Boolean matrix $A$ and $b\times n$ Boolean matrix $B$. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b(M)$, and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank.