SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.

BOUNDARY BLOW-UP RATE OF THE LARGE SOLUTION FOR AN ELLIPTIC COOPERATIVE SYSTEM

In this article we consider positive large solution of cooperative systems of the form -△u1=λ1u1 + a1u1u2^q1 -b1(x)u1^p1+1,-△u2=λ2u2 + a2u1^q2u2 -b2(x)u2^p2+1 in a bounded smooth domainΩ■ R^N(λi ∈ R, ai, bi>0, 0 < qi < pj, i, j ∈{1, 2}, i ≠ j), Based on the construction of certain sup and sub-solution, we show existence, uniqueness and blow-up rate of the large solution.

BIOLOGICAL INVASION PROBLEM WITH FREE BOUNDARY NONLOCAL DIFFUSION EQUATION

In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondly,the existence and uniqueness of local solutions are proved by the Sobolev embedding theorem and the comparison principle.Finally,according to the relevant research data and contents of red fire ants,the diffusion area and nest number of red fire ants were simulated without external disturbance.This paper mainly simulates the early diffusion process of red fire ants.In the early diffusion stage,red fire ants grow slowly and then spread over a large area after reaching a certain number.

Transition-layer Solutions of Quasilinear Elliptic Boundary Blow-up Problems and Dirichlet Problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation：where ω 〉 0, a（x） is a continuous function satisfying 0 〈 a（x） 〈 1 for x ∈Ω, Ω is a bounded smooth domain in R^N. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO A NONLOCAL EVOLUTION p-LAPLACE SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.

THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.

Global and Blow-up Solutions to a p-Laplace Equation with Nonlocal Source and Nonlocal Boundary Condition

This paper deals with an evolution p-Laplace equation with nonlocal source subject to weighted nonlocal Dirichlet boundary conditions. We give sufficient conditions for the existence of global and non-global solutions.

How to Trace out Impact-Triggered Effects Globally Scattered around Formation Boundaries: Case Uhry, North Germany (Eocene/Oligocene Boundary)

By focusing on impact-triggered phenomena having occurred synchronously with or shortly prior to formation boundaries, two glass sand pits (Upper Maastrichtian) located near Uhry, North Germany have been studied in regard to the K/T boundary throughout the last 40 years during progressive exploitation of glass sand. However, a clastic sequence of sand, mass flow and pelite deposited in a deep channel of about 10 - 12 m in depth, eroded into the glass sand, surprisingly shows an Upper Eocene/Lower Oligocene age, well defined by a Dinocyst assemblage (Chiripteridium c. galea, Enneado cysta arcuata, Areoligera tauloma = D 12na - D 14na) from a 0.5 meter thick pelite that marks the Rupelian transgression within an estuarian system running northwest/southeastward. The section exposes a high energy mass flow and formerly solid frozen angular glass sand blocks of up to a meter-size embedded in fluvial sand of the channel base. Furthermore, erratic clastics of up to 0.4 meter in diameter appear at the pelite base. The “unusual” Dinocyst assemblage is of autochthonous origin and comprises the fresh water alga Pediastrum Kawraiskyias indicator for cold climate, hitherto only known from Quaternary. Missing pollen indicate a vegetation-less hinterland. Thus, there cannot be any doubt that around the E/O b. at least one “rare event” has happened as verified by short tremendous flooding and significant temperature fall (“cosmic winter”). According to the attitude of the global impact scientific community, these phenomena belong to the spectrum of “indirect effects” of major impacts. Radiometric ages of relevant major impact events underline that both impact craters of Popigai, Russia (100 Kilometer in diameter, 35.7 Ma) and Chesabreake, USA (85 Kilometer in diameter, 35.5 Ma) happened shortly before the E/O b.(33.75 Ma). In addition, a tektite strewn field along the eastern coast of the USA and micro-tektites (Gulf of Mexico, Caribbean Sea, Barbados) yield an age of ~34.4 Ma, close to the E/O b. Consequently, there does exist an extremely high probability that Uhry site hosts impact-triggered products at the E/O b. It should be stressed that the Upper Eocene Epoch comprises an amazingly high number of impact events during the time-span 34.2 - 37.0 Ma.

ASYMPTOTIC NON-STABILITY AND BLOW-UP AT BOUNDARY FOR SOLUTIONS OF A FILTRATION EQUATION

For a class of nonlinear filtration equation with nonlinear second-third boundary value condition, it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region. The priori estimate of the solutions was established by iterative method. By using this estimate the solutions may blow-up on the boundary of the region and thus it may have asymptotic non-stability.

THE BLOW-UP PROPERLIES OF SOLUTIONS TO SEMILINEAR HEAT EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.

Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source

This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, we prove that the unique solution has global blow-up property whenever blow-up occurs.

Blow-up of the Solutions of the Initial Boundary Value Problem of Camassa-Holm Equation

Any classical non-null solution to the initial boundary value problem of Camassa-Holm equation on finite interval with homogeneous boundary condition must blow up in finite time. An initial boundary value problem of CamassaHolm equation on half axis is also investigated in this paper. When the initial potential is nonnegative,then the classical solution exists globally; if the derivative of initial data on zero point is nonpositire, then the life span of nonzero solution nmst be finite.

BLOW-UP IN A FRACTIONAL LAPLACIAN MUTUALISTIC MODEL WITH NEUMANN BOUNDARY CONDITIONS

In this paper,a fractional Laplacian mutualistic system under Neumann boundary conditions is studied.Using the method of upper and lower solutions,it is proven that the solutions of the fractional Laplacian strong mutualistic model with Neumann boundary conditions will blow up when the intrinsic growth rates of species are large.

Blow-up Results for the Nonclassical Diffusion Equations with Dynamical Boundary Condition

In this article,under suitable conditions on the internal term f(·)and the bound-ary nonlinear term g(·),we prove blow-up results for a class of nonclassical diffusion equations with nonlinear boundary condition.

Estimates on Blow-up Rate of Nonlinear Parabolic Systems with Nonlinear Boundary Conditions

In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.

Elastic one-return boundary element method and hybrid elastic thin-slab propagator for strong-contrast media with sharp boundaries

In this paper, we developed the theory and algorithm of an elastic one-way boundary element method（BEM） and a corresponding hybrid elastic thin-slab propagator for earth media with sharp boundaries between strong contrast media. This approach can takes the advantage of accurate boundary condition of BEM and completely overcomes the weak contrast limitation of the perturbationtheory based one-way operator approach. The one-way BEM is a smooth boundary approximation, which avoids huge matrix operations in exact full BEM. In addition, the one-way BEM can model the primary-only transmitted and reflected waves and therefore is a valuable tool in elastic imaging and inversion. Through numerical tests for some simple models,we proved the validity and efficiency of the proposed method.

Blow-up vs.Global Finiteness for an Evolution p-Laplace System with Nonlinear Boundary Conditions

In this paper, the authors consider the positive solutions of the system of the evolution p-Laplacian equationswith nonlinear boundary conditionsand the initial data （u0, v0）, where Ω is a bounded domain in Rn with smooth boundary δΩ, p 〉 2, h（·,·） and s（·,· ） are positive C1 functions, nondecreasing in each variable. The authors find conditions on the functions f, g, h, s that prove the global existence or finite time blow-up of positive solutions for every （u0, v0）.

Delineation of urban growth boundary based on FLUS model under the perspective of land use evaluation in hilly mountainous areas

With rapid economic development,the size of urban land in China is expanding dramatically.The Urban Growth Boundary(UGB)is an expandable spatial boundary for urban construction in a certain period in order to control the urban sprawl.Reasonable delineation of UGB can inhibit the disorderly spread of urban space and guide the normal development of the city.It is of practical significance for the construction of green urban space.The study utilizes GIS technology to establish a land construction suitability evaluation system for Nankang city,which is experiencing rapid urban expansion,and outlines the preliminary UGB under the future land use simulation(FLUS)model.At the same time,considering the coupled coordination of"Production-Living-Ecological Space",and based on the suitability evaluation,we revised the preliminary UGB by combining the advantages of the patch-generating land use simulation(PLUS)model and the convex hull model to delineate the final UGB.The results show that:1)the comprehensive score of the evaluation of the suitability of the construction of land from high to low shows the distribution of the center of the city to the surrounding circle type spread,the center of the city has the highest suitability score.The results of convex hull model show that the urban expansion type of Nankang is epitaxial.In the future,the urban expansion will mainly occur in the northern part of the city.The PLUS model predicts an increase of 3359.97 hm^(2)of construction land in Nankang by 2035,of which 2022.97 hm^(2)is urban construction land.2)The FLUS model has a prediction accuracy of 86.3%and delineates a preliminary UGB area of 9215.07 hm^(2).3)We used the results of the construction suitability evaluation,PLUS model simulation results,and convex hull model predictions to revise the originally delineated UGB.The final delineated UGB area is 8895.67 hm^(2)and it is capable of meeting the future development of the study area.The results of the delineation can promote sustainable urban development,and the delineation methodology can provide a reference basis for the preparation of territorial spatial planning.