blow up and global existence for a nonlocal degenerate parabolic system abstract this paper investigates the blow up and global existence of nonnegative solutions of the system u t u m a v p v t v n b u q x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 1 p q a b 0 and dx it is proved that if pq mn there exist both global and blow up nonnegative solutions when pq mn we show that if the domain is sufficiently small every nonnegative solution is global while if the domain is large enough that is if it contains a sufficiently large ball there exists no global solution in particular when p n q m we show that every positive solution exists globally iff x dx 1 ab where x is the unique positive solution of the linear elliptic problem x 1 x x 0 x

global existence and blowup of a localized problem with free boundary abstract this paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term the local existence and uniqueness of the solution are given by applying the contraction mapping theorem then we present some conditions so that the solution blows up in finite time finally the long time behavior of the global solution is discussed we show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

on the early exercise boundary of the american put option we study the short time behavior of the early exercise boundary for american style put options in the black scholes theory we develop an asymptotic expansion which shows that the simple lower bound of barles et al is a more accurate approximation to the actual boundary than the more complex upper bound our expansion is obtained through iteration using a boundary integral equation this integral equation is derived from the time derivative of the option value function which closely resembles the classical stefan free boundary value problem for melting ice our analytical results are supported by numerical computations designed for very short times

focal sets and real hypersurfaces in complex projective space let m be a real submanifold of cpm and let j denote the complex structure we begin by finding a formula for the location of the focal points of m in terms of its second fundamental form this takes a particularly tractable form when m is a complex submanifold or a real hypersurface on which ji is a principal vector for each unit normal to m the rank of the focal map onto a sheet of the focal set of m is also computed in terms of the second fundamental form in the case of a real hypersurface on which je is principal with corresponding principal curvature u if the map onto a sheet of the focal set corresponding to u has constant rank then that sheet is a complex submanifold over which m is a tube of constant radius theorem 1 the other sheets of the focal set of such a hypersurface are given a real manifold structure in theorem 2 these results are then employed as major tools in obtaining two classifications of real hypersurfaces in cpm first there are no totally umbilic real hypersurfaces in cpm but we show theorem 3 let m be a connected real hypersurface in cpm m 3 with at most two distinct principal curvatures at each point then m is an open subset of a geodesic hypersphere secondly we show that there are no einstein real hypersurfaces in cpm and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the ricci tensor in theorem 4 one of the first results in the geometry of submanifolds is that an umbilic hypersurface m in euclidean space must be an open subset of a hyperplane or sphere the proof goes as follows assume that the shape operator is a scalar multiple of the identity a ax and use the codazzi equation to show that x is constant then either x 0 in which case m lies on a hyperplane or the focal points fx x x l x t the unit normal all coincide and m lies on the sphere of radius 1 x centered at the unique focal point this simple idea suggests a plan of attack for classifying hypersurfaces in terms of the nature of the principal curvatures under fairly general conditions the set of focal points corresponding to a principal curvature x can be given a differentiable received by the editors december 19 1980 presented to the society at its annual meeting in san francisco january 10 1981 1980 mathematics subject classificatiom primary 53b25 53c40

blow up for a porous medium equation with a localized source in this paper we investigate the blow up properties of the positive solutions to a localized porous medium equation v t dv m av p 1v q 1 x 0 t subject to homogeneous dirichlet condition and positive initial datum v 0 x under appropriate hypotheses we establish the local existence and obtain that in the case of p 1 q 1m the solution of the above problem blows up for large initial datum while it admits a global solution for small initial datum moreover for the special case p 1 0 q 1 m and a is large under an additional hypothesis on the initial datum we can also obtain the asymptotic behavior of the blow up solution

blowing up of solution for a nonlocal reaction diffusion problem in combustion theory abstract in a nonlocal reaction diffusion model in combustion theory the reaction function involves a physical parameter which is a measure of the strength of the reaction mechanism the purpose of this paper is to show the existence of a critical value such that for a unique global time dependent solution exists and converges to a steady state solution as t and for the solution blows up in finite time a characterization as well as upper and lower bounds of are given

total blow up versus single point blow up traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials kassoy and poland 4 significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period during this induction period there is a spatially homogeneous pressure rise in the system which causes a com pressive heating effect in the constant volume container mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0 x t this model is given by e ak 1 j e x t dx y x 2 0 x 0 d x 2 0

the blow up rate for a degenerate parabolic equation with a non local source abstract in this paper we establish the local existence of the solution and the finite time blow up result for the equation v v m xx a l l v q dx x l l 0 moreover it is proved that the solution has global blow up and there exist constants c 1 c 2 such that c 1 t 1 1 q 1 max x l l v x c 2 t 1 1 q 1 where t 1 is the blow up time

blow up for a degenerate parabolic equation with a nonlocal source abstract in this paper existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained for the degenerate parabolic problem x q u t u xx 0 a u m x t d x ku n x t in 0 a 0 t where 0 t a 0 m n 1 k 0 are constants

the diffusive logistic model with a free boundary in heterogeneous environment abstract in this paper we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition to understand the effect of the dispersal rate d and the parameter the ratio of the expansion speed of the free boundary and the population gradient at the expanding front on the dynamics of this model we divide the heterogeneous environment into two cases strong heterogeneous environment and weak heterogeneous environment by choosing d and as variable parameters we derive sufficient conditions for species spreading resp vanishing in the strong heterogeneous environment while in the weak heterogeneous environment we obtain sharp criteria for the spreading and vanishing moreover when spreading happens we give an estimate for the asymptotic spreading speed of the free boundary these theoretical results may have important implications for prediction and prevention of biological invasions

on a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium the main result of the paper is the uniqueness of nonnegative solutions of the cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut um xx un x where m 1 and n 0 in particular n is allowed to be smaller than one the proof is based on a refined test function argument the condition that u be nonnegative is crucial but the restriction to one space variable is not

weight sum formulae in lie algebra representations abstract many formulae in the representation theory of lie algebras involve sums of weights the most celebrated being that of h weyl math z 24 1926 377 395 in this note we will derive some elementary but very useful formulae for summing subsets of weights of a lie algebra representation we also give a selection of applications of these formulae

nonholonomic modules over enveloping algebras of semisimple lie algebras abstract we prove that the enveloping algebras of all semisimple lie algebras over the complex numbers which are not of type a 1 admit nonholonomic simple modules

global existence and blowup of a nonlocal problem in space with free boundary abstract this paper concerns a double fronts free boundary problem for the reaction diffusion equation with a nonlocal nonlinear reaction term in space for such a problem we mainly study the blowup property and global existence of the solutions our results show that if the initial value is sufficiently large then the blowup occurs while the global fast solution exists for a sufficiently small initial data and the intermediate case with a suitably large initial data gives the existence of the global slow solution

uniform blow up rate for a nonlocal degenerate parabolic equations abstract in this paper we introduce a new method for investigating the rate of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms in some cases we prove that the solutions have global blow up and the rate of blow up is uniform in all compact subsets of the domain in each case the blow up rate of u t is precisely determined

well posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors abstract in this paper we study a free boundary problem modelling the growth of nonnecrotic tumors the main trait of this free boundary problem is that it is essentially multidimensional so that its well posedness is hard to establish by using the usual methods in the classical theory of free boundary problems in this paper we use the functional analysis method based on the theory of analytic semigroups to prove that this problem has a unique local solution in suitable function spaces continuous dependence of the solution on the initial data and regularities of the solution can also be easily obtained by using the argument of this paper

a quasilinear reaction diffusion system coupled via nonlocal sources in this paper we establish the critical exponent and the blow up rate for a quasilinear reaction diffusion system with nonlocal nonlinear sources and inner absorptions subject to homogeneous dirichlet conditions and nonnegative initial data it is found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities while the blow up rate is independent of the nonlinear diffusion exponents due to the effects of the coupled nonlocal sources two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow up rate respectively we prove moreover that the blow up could be global due to the nonlocality of the nonlinear sources

stable actions and central extensions a probability measure preserving action of a countable group is called stable if its transformation groupoid absorbs the ergodic hyperfinite equivalence relation of type text ii 1 under direct product we show that for a countable group g and its central subgroup c if g c has a stable action then so does g combining a previous result of the author we obtain a characterization of a central extension having a stable action

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity abstract we prove convergence of positive solutions to u t u u u u 2 u 0 u 0 u 0 in a bounded domain r n n 1 with smooth boundary in the case of u 0 1 and identify the w 0 1 2 limit of u t as t as the solution of the corresponding stationary problem this behaviour is different from the cases of u 0 1 and u 0 1 which are known to result in convergence to zero or blow up in finite time respectively the proof is based on a monotonicity property of u 2 along trajectories and the analysis of an associated constrained minimization problem

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

blow up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition abstract in this paper we consider a semilinear heat equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove global existence theorem for max p l 1 some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given

uniform blow up profile for a degenerate parabolic system with nonlocal source in this paper we establish the local existence of the solution and the finite time blow up result for the following system ut ur1 u a updx forx ohm t 0 ut ur2 u b uqdx forx ohm t 0 where p q 1 and 0 r1 r2 1 moreover it is proved that the solution has global blow up and limt t t t p 1 r2 yu x t 1 r1 11 r1 a a y p 1 r2 y q 1 r1 py xa p 1 r2 b1 r2 y limt t t t q 1 r1 yu x t 1 r2 11 r2 b a y q 1 r1 y p 1 r2 qy xb q 1 r1 a1 r1 y uniformly on compact subsets of where pq 1 r1 1 r2 and t is the blow up time

blow up for a degenerate parabolic equation with a nonlocal source abstract in this paper we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation v x v m xx 0 l v p 1 dx kv q 1 with homogeneous dirichlet boundary conditions in the interval 0 l where 0 m 1 we first establish the local existence and uniqueness of its classical solutions then we show that the positive solution blows up in finite time if the initial datum is sufficient large finally we prove that the blow up set is the whole interval and we also obtain the estimates of the blow up rate

asymptotic behavior of the solutions of the dead core problem u u pup 0 1 1 this equation originates from the study of a class of models for the reaction diffusion processes of a gas inside a chemical reactor 2 81 from the mathematical point of view the major interest of equation 1 1 lies in the competition of the linear diffusion and the nonlinear fast p 0 t 0 with initial and boundary conditions

equations for biological evolution in this paper we consider mathematical models inspired by the mechanisms of biological evolution we take populations which are subject to interaction and mutation in the cases we consider the interaction is through competition or through a prey predator relationship the models consider the specific characteristics as taking values in real intervals and the equations are of the integro differential type in the case of competition thanks to the fact that some of the equations have solutions which are quite explicit we succeed in proving the existence of attracting stationary solutions in the case of prey and predator using techniques of dynamical systems in infinite dimensional spaces we succeed in showing the existence of a global attractor which in some instances reduces to a point our analysis takes into account having distributions corresponding to all individuals having the same characteristics as possible populations

analysis on a coupled parabolic system with free boundary abstract the purpose of this paper is to investigate a parabolic problem with coupled reaction terms and free boundary including the existence uniqueness regularity and long time behavior of positive solutions firstly by the contraction mapping theorem we establish the local existence and uniqueness of positive solutions then we prove the solution blows up in finite time with large initial data by comparison principle and give more details for these large initial data by exhibiting an energy condition the simultaneous blow up result of the maximum of u and v is obtained and the blow up set of blow up solutions is a compact subset of 0 h 0 furthermore there is a global and fast solution which decays uniformly at an exponential rate if the initial datum is small while there is a global and slow solution provided that the initial datum is suitably large finally for initial data x x we obtain a trichotomy conclusion by considering the size of parameter

a class of ii _1 factors with a unique mcduff decomposition we provide a fairly large class of ii 1 factors n such that m n bar otimes r has a unique mcduff decomposition up to isomorphism where r denotes the hyperfinite ii 1 factor this class includes all ii 1 factors n l infty x rtimes gamma associated to free ergodic probability measure preserving p m p actions gamma curvearrowright x mu such that either a gamma is a free group mathbb f n for some n ge 2 or b gamma is a non inner amenable group and the orbit equivalence relation of the action gamma curvearrowright x mu satisfies a property introduced in jones and schmidt am j math 109 1 91 114 1987 on the other hand settling a problem posed by jones and schmidt in 1985 we give the first examples of countable ergodic p m p equivalence relations which do not satisfy the property of jones and schmidt 1987 we also prove that if mathcal r is a countable strongly ergodic p m p equivalence relation and mathcal t is a hyperfinite ergodic p m p equivalence relation then mathcal r times mathcal t has a unique stable decomposition up to isomorphism finally we provide new characterisations of property gamma for ii 1 factors and of strong ergodicity for countable p m p equivalence relations

global solutions and blow up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources this paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources subjecting to homogeneous dirichlet boundary condition the main aim of this paper is to study conditions on the global existence and or blow up in finite time of solutions and give the estimates of blow up rates of blow up solutions

the nef value and defect of homogeneous line bundles formulas for the nef value of a homogeneous line bundle are derived and applied to the classification of homogeneous spaces with positive defect and to the classification of complete homogeneous real hypersurfaces of projective space