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Photocatalytic reduction of carbon dioxide into useful feedstocks has attracted increasing attention.

Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model.

This paper is devoted to a new integrable two-component Novikov equation with Lax pairs and bi-Hamiltonian structures. Ons the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. On the other hand, we prove that the strong solutions maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Organic anions possess various functional properties; however, their presence in wastewater causes environmental pollution. Thus, coupling the separation of such species with the resultant function could be highly desirable. Herein, we propose a "killing two birds with one stone" strategy for highly efficient separation of organic pollutant anions from water at room temperature through direct construction of functional cationic metal-organic frameworks (CMOFs) based on the organic anions as charge-balancing anions. To illustrate this strategy, 2,4,6-trinitrophenolate anion (PA-) is chosen as a typical anion, while 4,4'-azo-triazole (atrz) is strategically chosen as a suitable neutral ligand. The resultant positive framework exhibits a high adsorption capacity and selectivity for PA-. Remarkably, its adsorption capacity is 869.6 mg g-1, which is more than 30 times that of multiwalled carbon nanotubes and 15 times that of activated carbon. Its capacity is even higher than that of BUT-13 (865 mg g-1), the highest adsorbent ever known. 1H NMR and single-crystal X-ray diffraction show that the high capacity is attributed to strong electrostatic interaction between the positive framework and PA-, which leads to all the pores being completely occupied by PA- anions. 1H NMR titration reveals that the selectivity comes from stronger hydrogen-bonding interaction between the ligand of the positive framework and PA-, which is confirmed from the eight times length of the shifted signal of atrz due to the addition of PA- compared with the competing anions. The stronger interaction is further confirmed from the high stability of the resultant CMOF in high-concentration salt solutions containing the competing anions, particularly in 100-fold molar NaNO3 and Na2SO4 solutions. Meanwhile, first-principles simulation shows that the high binding energy between the positive framework and PA- contributes to enhancing the selectivity. Moreover, the resultant CMOF is a potential energetic material with an improved oxygen balance, high heat of formation, and heat of detonation.

This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic and the nonperiodic case is well-posed in Sobolev spaces in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous.

A symmetrical pyrene derivative chemical structure was prepared by a classical synthetic method such as the Sonogashira cross-coupling reaction. The molecular structure of the product was characterised in detail by nuclear magnetic resonance (NMR), mass spectrometry (MS) and other methods. Furthermore, the optical properties of the novel products were studied by UV-vis and photoluminescence spectroscopy. The electrochemical properties of the molecules were fully characterised by comparison of electrochemical experiments and DFT simulation. Scanning electron microscope (SEM) observed that the product successfully formed a regular self-assembly structure. The product verifies the role of the molecular structure of the disc-mounted molecules on the optical and self-assembly properties, and is of reference value in the field of organic optoelectronic molecules

In present paper, we study the Cauchy problem for a generalized Camassa-Holm equation, which was discovered by Novikov. Our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted spaces.

This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood–Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

In this paper, we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force. We first establish the local well-posedness in a range of the Besov spaces, as well as the local well-posedness in the critical space. Then, we study the Gevrey regularity of the shallow-water model by using a generalized Cauchy-Kovalevsky theorem, which implies that the shallow-water model admits analytical solutions locally in time and globally in space. Moreover, we obtain a precise lower bound of the lifespan and the continuity of the solution. Finally, working with moderate weight functions that are commonly used in time-frequency analysis, some persistence results to the shallow-water model are illustrated.

Precise control over signal amplification provides unparalleled opportunities for diverse applications. However, spatiotemporally controlled amplification has not been realized because of the lack of a design methodology. The aim of this study was thus to develop a conceptual approach for remote control over signal amplification at a chosen time and site in living cells. This system was constructed by re-engineering the functional units of the hybridization chain reaction (HCR) and combination with upconversion photochemistry, thus resulting in an activatable HCR with the high spatial and temporal precision of near-infrared (NIR) light. As a proof of concept, we demonstrate the spatially and temporally resolved amplified imaging of messenger RNA (mRNA) with ultrahigh sensitivity in vitro and in vivo. Furthermore, by using a system targeting subcellular sites we have developed a new technique for NIR-initiated amplified imaging of mRNA exclusively within a specific organelle.

In the present study an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force is derived from the governing equation in two dimensional flows. The transport equation theory is then applied to investigate the local well-posedness and wave breaking phenomena for this model. The nonexistence of the Camassa-Holm-type peaked solution and classification of various traveling-wave solutions to the new system are also established. Moreover it is shown that all the symmetric waves to this model are traveling waves.

Consideration in the present paper is a mathematical model proposed as an equation of long-crested shallow water waves propagating in one direction with the effect of Earth’s rotation. This model equation is analogous to the Camassa–Holm approximation of the two-dimensional incompressible and irrotational Euler equations, and its solution corresponding to physically relevant initial perturbations is more accurate on a much longer timescale. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on the blow-up criteria and wave-breaking phenomena in the periodic setting are investigated. Moreover, working with moderate weight functions that are commonly used in time–frequency analysis, some persistence results to the equation are illustrated.

In this paper, we are concerned with the Cauchy problem of the multi-component Novikov equation. We establish the local well-posedness in a range of the Besov spaces by using Littlewood–Paley decomposition and transport equation theory. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.

DNA-based molecular circuits able to perform complex information processing in biological systems are highly desirable. However, conventional DNA circuits are constitutively always in an ON state and immediately operate when they meet the biomolecular inputs, precluding precise molecular computation at a desired time and in a desired site. In this work, we report a conceptual methodology for the construction of photonic nanocircuits that enable DNA molecular computation in vitro and in vivo with high spatial precision. Upon remote activation by spatially restricted NIR-light input, two types of cancer biomarker inputs can sequentially trigger conformational changes of the DNA circuit through a structure-switching aptamer and toehold-mediated strand exchange, leading to release of a signaling output. Of note, the NIR-light-gated nanocircuit allows for intended control over the specific timing and location of DNA computation, providing spatial and temporal capabilities for multiplexed imaging. Furthermore, an OR-AND-gated nanocircuit of higher complexity was designed to illustrate the versatility of our approach. The present work illustrates the potential of the use of upconversion nanotechnology as a regulatory tool for spatial and temporal control of DNA computation in cells and animals.

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well posed in Sobolev...

A novel upconversion nanoarchitecture is designed, whose 980 nm light photosensitized activity could be activated by 808 nm light., The precise control of singlet oxygen (1O2) generation is in great demand for biological studies and precision medicine. Here, a nanoarchitecture is designed and synthesized for generating 1O2 in a dual NIR light-programmable manner, while shifting to the therapeutic window. The nanoarchitecture is constructed by controlled synthesis of mesoporous silica-coated upconversion nanoparticles (UCNPs), wherein the porphyrin photosensitizers (PSs) are covalently embedded inside the silica walls while NIR (808 nm)-responsive diarylethene (DAE) photochromic switches are loaded in the nanopores. Upon irradiation with 980 nm NIR light, the UCNP core absorbs low energy photons and transfers energy to the PSs in the silica wall, leading to efficient 1O2 generation. Furthermore, this 980 nm NIR light photosensitized activity can be remotely controlled by irradiation with a distinct NIR wavelength (808 nm). The 1O2 generation is inhibited when the DAE installed in the nanopores is in the closed form, whereas irradiation of the nanoconstruct with 808 NIR light leads to the transformation of DAE to the open form, and thus enabling full recovery of the 980 nm NIR light excited 1O2 generation capability. The NIR light-mediated on-demand “activation” of the nanoarchitecture for bioimaging and controllable photodynamic therapy is further demonstrated in vitro and in vivo.

In this paper, we are concerned with \begin{document}$N$\end{document} -Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Holder continuous in $H^{s}$.

This paper deals with the Cauchy problem for the generalized Camassa–Holm equation. The existence of weak solutions for the generalized Camassa–Holm equation in lower order Sobolev space with is obtained.

To investigate photoelectric properties of meso-extended porphyrin derivatives with electron-donating or electron-withdrawing substituents, a series of functionalized porphyrin materials have been designed and synthesized by Suzuki coupling reaction. The meso-extended structures were fully characterized by 1H NMR, IR spectroscopy and mass spectrometry. The photophysical properties of porphyrin derivatives were carefully examined by UV-Visible and fluorescence spectra, and the solvatochromic effect was observed and discussed. In particular, Z-scan technique was employed to characterize the third-order nonlinear optical (NLO) properties of the products such as nonlinear absorption and refraction, the third-order nonlinear refractive indexes (𝔵(3)-value) of these porphyrin derivatives achieved 3.9×10−12 esu. In addition, the compounds could be self-assembled into highly organized morphologies through phase-exchange method. All the results indicated that the discotic materials have the potential for optoelectronic applications.

Typical asymmetric donor-π-bridge-acceptor(D-π-A) zinc porphyrin dyes have been synthesized, and further modified by reacting the ethynyl groups of them with click reagent through a formal [2+2] click reaction. The electronic interaction between the push and pull electronic groups and the click reaction process were investigated by UV-Vis absorption spectroscopy. The nonlinear optical(NLO) properties of the dyes were studied by the Z-scan technique and clearly reverse saturable absorption to saturable absorption(RSA-SA) transition could be observed through click reaction of the dyes with TCNE. Furthermore, the photovoltaic properties of these porphyrin dyes were reasonably explained by the J-V curve fitting based on the equivalent-circuit model as well as the comparison between the absorption and incident-photon-to-current-conversion efficiency(IPCE) spectra. Besides, these dyes with different donors or acceptors could self-assembly into different microstructures by phase transfer methodology.

In this paper, we are concerned with the Cauchy problem of the new integrable four-component system with cubic nonlinearity. We establish the local well-posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.

Two pyrene derivatives were successfully synthesized via an efficient copper(I)-catalyzed azide alkyne 1,3-dipolar cycloaddition. The photophysical and electrochemical properties were characterized using ultraviolet-visible absorption spectra, fluorescence spectra, cyclic voltammograms and density functional theory modulations. These results showed that the symmetry structure of these derivatives formed an electron-delocalized organic system, which have larger effects in achieving a third-order nonlinear optical (NLO) response. The third-order nonlinear properties including the nonlinear absorption and the nonlinear susceptibilities investigated by Z-scan technique indicate that the title compounds can serve as a promising candidate for third-order NLO applications.

We successfully synthesized a series of gold nanoparticles capped with 11-(4-(pyren-1-yl)phenoxy)undecane-1-thiol (A-SH) and with 1-dodecanethiol. The homodispersed gold nanoparticles were fully verified by TEM, UV-Vis and PL spectroscopy. TEM images of the nanoparticles showed similar particle sizes at approximately 2.5 nm also evidenced by the absorption spectra. By solvent-exchange co-self-assembly of a pyrene derivative composited with these functionalized nanoparticles, the different arrangements of Au nanoparticles were observed. Compared to Au nanoparticles functionalized by 1-dodecanethiol (GNP), the pyrene–thiol-capped gold nanoparticles (GNP-A) showed better dispersity in the pyrene derivative matrix because of their π–π interactions.

This paper deals with a fully parabolic chemotaxis-growth system with generalized volume-filling effect and sublinear secretion $$\begin{aligned} \left\{ \begin{array}{ll} u_t=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (\psi (u)\nabla v)+ru-\mu u^{2}, &{}\quad (x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta v-v+g(u), &{}\quad (x,t)\in \Omega \times (0,\infty ), \end{array} \right. \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^{2}\), where \(\varphi (u)\) is a nonlinear diffusion function, \(\psi (u)\) is a chemotactic sensitivity and g(u) is a production rate of the chemoattractant. Under some suitable assumptions on the nonlinearities \(\varphi (u)\), \(\psi (u)\) and g(u), we study the global boundedness and decay of solutions for the system.

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation. The local well-posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces Hs) by using Littlewood-Paley decomposition and transport equation theory. Moreover, the local well-posedness in critical case (with ) is considered.

A novel optical rhodamine–quinoline type chemosensor (REQ) was successfully synthesized and characterized as a reversible molecular switch. It displayed high selectivity toward Al3+ among 19 metal ions and the resultant complex [REQ–Al3+] acted as a sequential chemosensor toward F− among 16 anions in neutral aqueous media (ethanol–water, v/v = 3/7, Tris–HNO3, 20 mM, pH 7.0). Moreover, detecting fluoride at concentrations below the maximum contaminant level in drinking water defined by the U. S. Environmental Protection Agency (4.0 mg L−1, 211 μM) could be achieved with a detection limit (LOD) estimated to be down to 8.0 μM. The reversible ring-open mechanism of the rhodamine spirolactam induced by Al3+ binding and the 1 : 1 stoichiometric structure between REQ and Al3+ were adequately supported by Job plot evaluation, optical titration, and 1H NMR study. Finally, it could be conveniently used for the determination of Al3+ and F− with the “naked eye” due to the obvious color change of the testing solution, showing its potential practical application.

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev (Comm Part Diff Equ 30:761–781, 2005). Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s},s>3/2\) for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Finally, it is shown that the solution map for the generalized Camassa–Holm equation is Holder continuous in \(H^{r}\)-topology.

We are concerned with the Cauchy problem of the new integrable three-component system with cubic nonlinearity. We establish the local well-posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.

A novel optical Rhodamine-Furan type chemosensor (RF) was easily synthesized and characterized as a revisable switch. It dispalyed high selectivity toward Al 3+ among 19 metal ions and the resultant complex [RF–Al 3+ ] as a sequential chemosensor toward F − among 16 anions in neutral aqueous media (ethanol/water, v/v = 2/3, Tris–HNO 3 , 10 mM, pH 7.0). Moreover, detecting fluoride at concentration below the maximum contaminant level in drinking water defined by the U.S. Environmental Protection Agency (4.0 mg L −1 , 211 μM) could be achieved with a detection limit (LOD) estimated down to 1.6 μM. Both sensors were very easily to construct and only required simple synthesis. The reversible ring-open mechanism of the rhodamine spirolactam induced by Al 3+ binding and the 1:1 stoichiometric structure between RF and Al 3+ were adequately supported by job plot evaluation, optical titration, and 1 H NMR study. Finally, it could be conveniently used for the determination of fluorides in dental products such as toothpastes with “naked eye” due to the obvious color change of testing solution, showing its potentially practical application.

In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

We study the Cauchy problem associated with a new integrable two-component system with cubic nonlinearity, which was recently proposed by Song, Qu and Qiao. We establish the local well-posedness in a range of the Besov spaces. Moreover, for analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend a result by Danchin, and Himonas et al. to more complex equations.

A series of asymmetric triphenylene derivatives containing typical D-π-A structures is successfully synthesized by means of [2+2] cycloaddition-cycloreversion click reactions. The photophysical and electrochemical properties, as well as the click reactions, are characterized by means of UV/Vis absorption spectroscopy, cyclic voltammetry, and DFT modulations. In addition, the third-order nonlinear properties, including the nonlinear absorption and the nonlinear susceptibilities, are investigated by using Z-scan techniques. A typical reverse saturable absorption-saturable absorption behavior is observed for the third-order nonlinear absorption, with the third-order nonlinear susceptibilities of the compounds being 1.05×10(-12) , -1.50×10(-12) , and -0.52×10(-12) esu, respectively.

This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces B p , r s , p , r ∈ [ 1 , ∞ ] , s > max { 3 2 , 1 + 1 p } (which generalize the Sobolev spaces H s ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with s = 3 2 , p = 2 , r = 1 ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated.

In this paper, we study the Cauchy problem of the periodic two-component hyperelastic-rod wave equation. We first establish the local well-posedness and precise blow-up scenario. Then we obtain several blow-up results and the blowup rate of strong solutions. Moreover, we present two global existence results for strong solutions. Finally, we consider analyticity of solutions and the initial–boundary value problem.

In this paper, we are concerned with the Cauchy problem for the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev [S. Hakkaev, K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation, Comm. Partial Differential Equations 30 (2005) 761–781]. We establish the local well-posedness for a range of Besov spaces. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extends some results of Danchin [R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988] and Himonas and Misiolek [A. Himonas, G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575–584] to more general equations.

A series of pyrene derivatives featuring nonplanar structures of N,N-didodecylanilino-substituted donor and TCNE/TCNQ adduct acceptors have been efficiently synthesized via formal [2+2] cycloaddition. As the efficient click by TCNE and TCNQ, the products show strong charge-transfer (CT) bands in the visible (near-IR region), potent redox activities, and related photophysical properties. UV/vis spectra and electrochemical studies show that the CT properties of these systems are readily tunable by strong cyano acceptor introduction on the nucleophile, which have a larger effect on the lowest unoccupied molecular orbital (LUMO). In particular, compared with the adduct of TCNE, the product clicked with TCNQ possessed a stronger D–A conjugation and bulkier π spacers. The TCNQ product also has a larger third-order nonlinear optical property which was characterized by Z-scan experiments. This could point to potentially interesting applications of the new pyrene derivatives in optoelectronic devices and develop the new modification process for the design of different pyrene-based molecular electronic devices.

In this article we give a complete description of the set of blow-up points of solutions of the problem We obtain that the blow-up set B(u) satisfies . The proof is based on the analysis of the asymptotic behaviour of self-similar representation and on the comparison methods.

A series of triphenylene derivatives with six symmetric substituents was synthesized from hexabromotriphenylene. The synthesis was conducted by six-fold palladium-catalyzed Hagihara-Sonogashira cross- coupling reactions to yield the hexa-alkynyl substituted triphenylene derivatives of HTP1, HTP2, HTP3 and HTP4. The six symmetric substituents can not only endow the triphenylene the longer π-conjugated range, but also increase the solubility of the compounds. Their photophysical, electrochemical, thermal properties were investigated respectively. With the comparison of their properties, the structure-property relationships were established which demonstrated the influences of different substituents on the electronic nature and the mesomorphic phase of these disk-shaped molecules. In addition, with the scanning electron microscopy(SEM) and polarized optical microscopy( POM) characterization, the self-assembly behaviors of the compounds were also investigated.

A series of novel asymmetrical pyrene derivatives with aralkyl and aldehyde groups have been successfully synthesized, and their spectroscopic parameters were investigated detailedly. The effects of aralkyl substituent amounts on the electronic absorption and emission spectra have been explored in evaluation of their potential as optical materials components. Moreover, the electrochemical properties of the new pyrene derivatives have been studied for the molecules' HOMO and LUMO levels by cyclic voltammograms. In particular, titration of anhydrous ethanol into dilute CH 2 Cl 2 solution of above compounds, an unusual change of the fluorescence value was observed. The change is believed to correlate strongly to intermolecular charge transfer. The morphological evolution of self-assembled pyrene derivatives with (arylethynyl)aldehyde substituents, from tamen-distinct-like morphologies to vesicular textures, was obtained by increasing aralkyl-substituted fragments. These asymmetrical discotic compounds represent an interesting set of candidates for optoelectronic device components.

Making use of the Karamata regular variation theory and the López-Gómez localization method, we establish the uniqueness and asymptotic behaviour near the boundary ∂Ω for the large solutions of the singular boundary-value problemwhere Ω is a smooth bounded domain in ℝN. The weight function b(x) is a non-negative continuous function in the domain, which can vanish on the boundary ∂Ω at different rates according to the point x0 ∊ ∂Ω. f(u) is locally Lipschitz continuous such that f(u)/u is increasing on (0, ∞) and f(u)/up = H(u) for sufficiently large u and p > 1, here H(u) is slowly varying at infinity. Our main result provides a sharp extension of a recent result of Xie with f satisfying limu→f(u)/up = H for some positive constants H > 0 and p > 1.

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

In this paper, we deal with the global existence and blow-up of solutions to a doubly degenerative parabolic system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of non-negative solutions, which extend the recent results of Zheng, Song and Jiang (S. N. Zheng, X. F. Song and Z. X. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), 308–324), Xiang, Chen and Mu (Z. Y. Xiang, Q. Chen, C. L. Mu, Critical curves for degenerate parabolic equations coupled via nonlinear boundary flux, Appl. Math. Comput. 189 (2007), 549–559) and Zhou and Mu (J. Zhou and C. L Mu, On critical Fujita exponents for degenerate parabolic system coupled via nonlinear boundary flux, Pro. Edinb. Math. Soc. 51 (2008), 785–805) to more general equations.