Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation （ODE）. Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.

Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources

In this paper,we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball.We are interested in the critical global exponent q_o and the critical Fujita exponent q_c for the problem considered,and show that q_o=q_c for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources,which is quite different from the known results that q_o〈q_c for the onedimensional case;moreover,the value is different from the slow case.

Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations

Numerical algorithms for stiff stochastic differential equations are developed using lin-ear approximations of the fast diffusion processes,under the assumption of decoupling between fast and slow processes.Three numerical schemes are proposed,all of which are based on the linearized formulation albeit with different degrees of approximation.The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems.Convergence analysis is conducted for one of the schemes,that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1.Approximations arriving at the other two schemes are discussed.Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.

Hybrid ionic/electronic interphase enabling uniform nucleation and fast diffusion kinetics for stable lithium metal anode

Lithium(Li)dendrite issue,which is usually caused by inhomogeneous Li nucleation and fragile solid electrolyte interphase(SEI),impedes the further development of high-energy Li metal batteries.However,the integrated construction of a high-stable SEI layer that can regulate uniform nucleation and facilitate fast Li-ion diffusion kinetics for Li metal anode still falls short.Herein,we designed an artificial SEI with hybrid ionic/electronic interphase to regulate Li deposition by in-situ constructing metal Co clusters embedded in LiF matrix.The generated Co and LiF both enable fast Li-ion diffusion kinetics,meanwhile,the lithiophilic properties of Co clusters can serve as Li-ion nucleation sites,thereby contributing to uniform Li nucleation and non-dendritic growth.As a result,a dendrite-free Li deposition with a low overpotential(16.1 mV)is achieved,which enables an extended lifespan over 750 h under strict conditions.The full cells with high-mass-loading LiFePO_(4)(11.5 mg/cm^(2))as cathodes exhibit a remarkable rate capacity of 84.1 mAh/g at 5 C and an improved cycling performance with a capacity retention of 96.4%after undergoing 180 cycles.

Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds

In the paper,the authors provide a new proof and derive some new elliptic type(Hamilton type)gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function.These results generalize earlier results in the literature.And some parabolic type Liouville theorems for ancient solutions are obtained.

Existence of Weak Solutions to a Class of Semiconductor Equations with Fast Diffusion Term

In this paper, we consider the transient drift-diffusion model with fast diffusion term. This problem is not only degenerate but also singular. We present the existence result for the Neumann boundary value problem with general nonlinear diffusivities.

BLOW-UP AND EXISTENCE FOR FAST DIFFUSION EQUATIONS WITH GENERAL NONLINEARITIES

We give conditions on the function f so that the Cauchy problem to the fast diffusionequations with max has or no global solutions. Asa special case we recover the results to the case f(u) and prove that, if f(u) is convex,on a neighborhood of u=o with logm denoting the logarithm iterated m times, and f satisfies acertain growth condition at infinity, the global solutions exist if q>1.

Fast ion diffusion alloy layer facilitating 3D mesh substrate for dendrite-free zinc-ion hybrid capacitors

Although aqueous zinc ion hybrid capacitors have advantageous integration of batteries and supercapacitors,they still suffer from the inherent problems of dendrite growth and interfacial side reactions on Zn anodes.Herein,a universal fast zinc-ion diffusion layer on a three-dimensional(3 D)mesh structure model is demonstrated to effectively improve Zn plating/stripping reversibility.The fast ion diffusion alloy layer accelerates the Zn^(2+)migration in an orderly manner to homogenize Zn^(2+)flux and overcomes the defects of the commercial mesh substrate,effectively avoiding dendrite growth and side reactions.Consequently,the proof-of-concept silver-zinc alloy modified stainless steel mesh delivers superb reversibility with the high coulombic efficiency over 99.4%at 4 mA cm^(-2)after 1600 cycles and excellent reliability of over 830 h at 1 mA cm^(-2),Its feasibility is also evidenced in commercial zinc ion hybrid capacitors with activated carbon as the cathode.This work enriches the fundamental comprehension of fast zinc-ion diffusion layer combined with a 3 D substrate on the Zn deposition and opens a universal approach to design advanced host for Zn electrodes in zinc ion hybrid capacitors.

Sn-doped BiOCl nanosheet with synergistic H^(+)/Zn^(2+)co-insertion for“rocking chair”zinc-ion battery

The development of insertion-type anodes is the key to designing“rocking chair”zinc-ion batteries.However,there is rare report on high mass loading anode with high performances.Here,{001}-oriented Bi OCl nanosheets with Sn doping are proposed as a promising insertion-type anode.The designs of cross-linked CNTs conductive network,{001}-oriented nanosheet,and Sn doping significantly enhance ion/electron transport,proved via experimental tests and theoretical calculations(density of states and diffusion barrier).The H^(+)/Zn^(2+)synergistic co-insertion mechanism is proved via ex situ XRD,Raman,XPS,and SEM tests.Accordingly,this optimized electrode delivers a high reversible capacity of 194 m A h g^(-1)at 0.1 A g^(-1)with a voltage of≈0.37 V and an impressive cyclability with 128 m A h g^(-1)over 2500 cycles at 1 A g^(-1).It also shows satisfactory performances at an ultrahigh mass loading of 10 mg cm^(-2).Moreover,the Sn-Bi OCl//MnO_(2)full cell displays a reversible capacity of 85 m A h g^(-1)at 0.2 A g^(-1)during cyclic test.

INVESTIGATION ON MECHANICALLY ALLOYED SUPERSATURATED NANOCRYSTALLINE Al-Ti ALLOYS

The Al_(100-x)Ti_(x)(x = 5,10,15,35) mixed powders were mechanically alloyed in a planetary bull mill. It was found that the initial composition strongly affects the final productsi a mixture of supersaturated solid solution Al(Ti) and a fee phase for x = 5 and 10; and a single supersaturated solid solution Al(Ti)for x=15 and 35. With inereasing Ti contents from 5at% to 35at%, the grain size of solid solution Al(Ti) decreased from 40nm to 10nm. The results suggest that fast diffusion along nanocrystalline grain boundary is the main alloying process. DSC curves and TEM observation indicate that the solute atom segregation can thermally stabilize the nanocrvstalline solid solutions.

Properties of Solutions of Kolmogorov-Fisher Type Biological Population Task with Variable Density

In this paper, we discussed population model of two competing populations with non-linear double diffusion and variable density which described by nonlinear system of competing individuals. We identify new properties, such as finite speed of propagation, and localization of the outbreaks in a specific area.

Controllable synthesis of high loading LiFePO_4/C nanocomposites using bimodal mesoporous carbon as support for high power Li-ion battery cathodes

Mesoporous LiFePO4/C composites containing 80 wt% of highly dispersed LiFePO4 nanoparticles(4-6 nm) were fabricated using bimodal mesoporous carbon(BMC) as continuous conductive networks. The unique pore structure of BMC not only promises good particle connectivity for LiFePO4, but also acts as a rigid nano-confinement support that controls the particle size. Furthermore, the capacities were investigated respectively based on the weight of LiFePO4 and the whole composite. When calculated based on the weight of the whole composite, it is 120 mAh·g-1at 0.1 C of the high loading electrode and 42 mAh·g-1at 10 C of the low loading electrode. The electrochemical performance shows that high LiFePO4 loading benefits large tap density and contributes to the energy storage at low rates, while the electrode with low content of LiFePO4 displays superior high rate performance, which can mainly be due to the small particle size, good dispersion and high utilization of the active material, thus leading to a fast ion and electron diffusion.

Some Notes on Reaction Diffusion Systems with Nonlinear Boundary Conditions

Abstract This paper deals with the existence and nonexistence of global positive solution to a semilinear reaction-diffusion system with nonlinear boundary conditions. For the heat diffusion case, the necessary and sufficient conditions on the global existence of all positive solutions are obtained. For the general fast diffusion case, we get some conditions on the global existence and nonexistence of positive solutions. The results of this paper fill the same gaps which were left in this field.

The Moser-Trudinger-Onofri Inequality

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods(in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality.In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally,a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.

Harnack Differential Inequalities for the Parabolic Equation u_t= LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.

Self-assembled α-MnO2 urchin-like microspheres as a high-performance cathode for aqueous Zn-ion batteries

Aqueous Zn-ion batteries(AZIBs)are one of the promising battery technologies for the green energy storage and electric vehicles.As one attractive cathode material for AZIBs,α-MnO2 materials exhibit superior electrochemical properties.However,their long-term reversibility is still in great suspense.Considering the decisive effect of the structure and morphology on theα-MnO2 materials,hierarchicalα-MnO2 materials would be promising to improve the cycle performance of AZIB.Here,we synthesized theα-MnO2 urchin-like microspheres(AUM)via a self-assembled method.The porous microspheres composed of one-dimensionalα-MnO2 nanofibers with high crystallinity,which improved the surface area and active sites for Zn2+intercalation.The AUM-based AZIB realized a high initial capacity of 308.0 mA hg-1,and the highest energy density was 396.7 W hkg-1.The kinetics investigation confirmed the high capacitive contribution and fast ion diffusion of the AUM.Ex-situ XRD measurement further verified the synergistic insertion/extraction of H+and Zn2+ions during the charge/discharge process.The superiority of the AUM guaranteed good electrochemical performance and reversible phase evolution,and this application would promote the follow-up research on the advanced AZIB.