positive solutions of operator equations on half line abstract this paper is concerned with an operator equation on half line some theorems for the existence of its positive solutions are obtained by using the krasnosel skii guo theorem on cone expansion and compression in a special function space

remarks on the blow up rate for critical nonlinear schrodinger equation with harmonic potential abstract we consider the blow up solutions of the cauchy problem for the critical nonlinear schrodinger equation with a harmonic potential which models the attractive bose einstein condensate we establish the sharp lower and upper bounds of blow up rate as t t blow up time which improve the result of q liu y zhou j zhang upper and lower bound of the blow up rate for nonlinear schrodinger equation with a harmonic potential appl math comput 172 2006 1121 1132

type ii blowup in a doubly parabolic keller segel system in two dimensions abstract this paper is concerned with a parabolic parabolic keller segel system u t u u v in 0 t v t v v u in 0 t u v 0 on 0 t u x 0 u 0 x 0 v x 0 v 0 x 0 in in a smoothly bounded domain r 2 keller and segel proposed it as a model of aggregation of bacteria which is mathematically translated as finite time blowup a solution u v is said to blow up at t t if lim sup t t u t l when u v blows up at t t the blowup is called type i if u t l c t t 1 for t 0 t with some constant c 0 and type ii otherwise in spite of motivation of the modeling it has remained open for the past decades whether or not finite time blowup occurs frequently in the parabolic parabolic system recently it was shown in 18 19 15 that there exists a large class of radial initial data such that the corresponding solutions blow up in finite time however they did not give any information on blowup behavior blowup rate is one of the greatest concerns in investigation of blowup mechanism in this paper we show that each blowup is of type ii in the parabolic parabolic system in radial case

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

multiple positive solutions for singular bvps on the positive half line in this work we are concerned with the existence of multiple positive solutions to a second order nonlinear singular boundary value problem set on the positive half line we mainly use the krasnozels kii and leggett williams fixed point theorems in cones to prove existence of one positive solution two positive solutions and three positive solutions the results complement extend and correct some recent ones

on the singular generalized fisher like equation with derivative depending nonlinearity this work is devoted to the existence of positive nontrivial solutions for a second order nonlinear boundary value problem posed on the positive half line the problem stems from epidemiology and combustion theory the positive nonlinearity depends on the solution and its derivative and may exhibit a singularity at the origin it satisfies general polynomial growth conditions new existence results of multiple positive solutions are proved using recent fixed point theorems on cones in banach spaces examples with numerical computations illustrate the applicability of the existence theorems comparisons with existing results are provided

uniqueness of positive radial solutions of δu f u 0 in ℝ n ii we prove a uniqueness result for the positive solution of u f u 0 in r n which goes to 0 at the result applies to a wide class of nonlinear functions f including the important model case f u u u p 1 p n 2 n 2 the result is proved by reducing to an initial boundary problem for the ode u n 1 r f u 0 and using a shooting method

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

existence and stability of a solution blowing up on a sphere for an l 2 supercritical nonlinear schrodinger equation we consider the quintic two dimensional focusing nonlinear schrodinger equation iut u u 4u which is l2 supercritical even though the existence of finite time blow up solutions in the energy space h1 is known very little is understood concerning the singularity formation numerics suggest the existence of a stable blow up dynamic corresponding to a self similar blowup at one point in space we prove the existence of a different type of dynamic and exhibit an open set among the h1 radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere this is the first result of an explicit description of a blow up dynamic in the l2 supercritical setting

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

on an elliptic equation related to the blow up phenomenon in the nonlinear schrodinger equation this paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in n the equation is derived from the blow up problem in the non linear schrodinger equation

the existence of countably many positive solutions for some nonlinear three point boundary problems on the half line abstract in this paper we study the existence of countably many positive solutions for some nonlinear singular three point boundary problems on the half line u t a t f t u t 0 0 t u 0 b 0 u 0 u 0 where s r r is an increasing homeomorphism and positive homomorphism and 0 0 and 0 a 0 0 and has countably many singularities on 0 by using the fixed point index theorem and a new fixed point theorem in cones the existence of countably many solutions for singular three point boundary value problem are obtained under conditions weaker than those used by liu and zhang b f liu j h zhang the existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions j math anal appl 309 2005 505 516

the existence of countably many positive solutions for one dimensional p laplacian with infinitely many singularities on the half line we study the existence of countably many positive solutions of boundary value problems on the half line for differential equations of second order the fixed point index theory and a new fixed point theorem in cones are used

existence and iteration of positive solutions for multi point boundary value problems on a half line in this study using fixed point index theorems and monotone iterative techniques we present iterative schemes as well as the existence of positive solutions to the m point boundary value problem on a half line

blowup dynamics for smooth data equivariant solutions to the critical schrodinger map problem we consider the energy critical schrodinger map problem with the 2 sphere target for equivariant initial data of homotopy index k 1 we show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm which generates finite time blowup solutions we give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

error estimates for hermite and even tempered gaussian approximations in quantum chemistry atomic like basis functions provide a natural physically motivated description of electronic states among which gaussian type orbitals are the most widely used basis functions in molecular simulations this paper aims at developing a systematic analysis of numerical approximations based on linear combinations of gaussian type orbitals we derive a priori error estimates for hermite type gaussian bases as well as for even tempered gaussian bases numerical results are presented to support the theory

boundary value problems on infinite intervals and semiconductor devices the nonlinear differential equation y f x y y 0 x f satisfying bernstein type growth conditions we also examine an important application which occurs in the theory of semiconductor devices

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

optimal condition for blow up of the critical lq norm for the semilinear heat equation abstract we shed light on a long standing open question for the semilinear heat equation u t u u p 1 u namely without any restriction on the exponent p 1 nor on the smooth domain we prove that the critical l q norm blows up whenever the solution undergoes type i blow up a similar property is also obtained for the local critical l q norm near any blow up point in view of recent results of existence of type ii blow up solutions with bounded critical l q norm which are counter examples to the open question our result seems to be essentially the best possible result in general setting this close connection between type i blow up and critical l q norm blow up appears to be a completely new observation our proof is rather involved and requires the combination of various ingredients it is based on analysis in similarity variables and suitable rescaling arguments combined with backward uniqueness and unique continuation properties for parabolic equations as a by product we obtain the nonexistence of self similar profiles in the critical l q space such properties were up to now only known for p p s and in radially symmetric case for p p s where p s is the sobolev exponent

fixed point theorem of leggett williams type and its application we establish a norm type cone expansion and compression fixed point theorem for completely continuous operators our theorem is then applied to prove the existence of positive solution of second order three point boundary value problem

existence of positive solutions for a semipositone boundary value problem on the half line in this paper we consider the boundary value problem on the half line x t k 2 t x t lm t f t x t 0 t 0 x 0 0 limt x t 0 where k 0 0 and f 0 x 0 r are continuous we show the existence of positive solutions by using a fixed point theorem in cones

on the approximation of electronic wavefunctions by anisotropic gauss and gauss hermite functions the electronic schrodinger equation describes the motion of n electrons under coulomb interaction forces in a field of clamped nuclei the solutions of this equation the electronic wavefunctions depend on 3n variables three spatial dimensions for each electron we study the approximability of these wavefunctions by linear combinations of anisotropic gauss functions or more precisely gauss hermite functions products of polynomials and anisotropic gauss functions in the narrow sense we show that the original singular wavefunctions can up to given accuracy and a negligibly small residual error be approximated with only insignificantly more such terms than their convolution with a gaussian kernel of sufficiently small width and that basically arbitrary orders of convergence can be reached

the existence of countably many positive solutions for nonlinear singular m point boundary value problems on the half line in this paper by introducing a new operator improving and generating a p laplace operator for some p 1 we study the existence of countably many positive solutions for nonlinear boundary value problems on the half line f u a t f u t 0 0r is the increasing homeomorphism and positive homomorphism and f 0 0 we show the sufficient conditions for the existence of countably many positive solutions by using the fixed point index theory and a new fixed point theorem in cones

a centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

the instability of bourgain wang solutions for the l2 critical nls we consider the two dimensional l 2 critical nonlinear schr odinger equation i partial tu delta u uu 2 0 in their pioneering 1997 work bourgain and wang have constructed smooth solutions which blow up in finite time t nabla u t l 2 sim 1 over t t and which display some decoupling between the regular and the singular part of the solution at blow up time we prove that this dynamic is unstable more precisely we show that any such solution with small super critical l 2 mass lies on the boundary of both h 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log log regime exhibited in work by f merle and p rapha e l we moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in f merle s 1992 it comm pure appl math paper in the critical mass case

caratheodory s method for a class of second order differential equations on the half line the method of equivalent variational methods originally due to caratheodory for free problems in the calculus of variations is extended to investigate boundary value problems for a class of second order differential equations on the half line some applications are presented to illustrate the potential of this method

some remarks on the nonlinear schrodinger equation with fractional dissipation we consider the cauchy problem for the l2 critical focussing nonlinear schrodinger equation with a fractional dissipation according to the order of the fractional dissipation we prove the global existence or the existence of finite time blowup dynamics with the log log blow up speed for u t l2

existence of self similar blow up solutions for zakhrov equation in dimension two part i we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right we prove the existence of blow up solutions stable self similar blow up solutions for this problem and we study various properties of these solutions

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

determination of blowup type in the parabolic parabolic keller segel system this paper is concerned with a parabolic parabolic keller segel system begin aligned left begin array ll u t nabla cdot nabla u u nabla v quad text in omega times 0 t v t delta v alpha v u quad text in omega times 0 t end array right end aligned with a constant alpha ge 0 and nonnegative initial data in a smoothly bounded domain omega subset mathbb r 2 under the neumann boundary condition or in omega mathbb r 2 it was introduced as a model of aggregation of bacteria which is mathematically translated as finite time blowup a solution u v is said to blow up at t t 0 and type ii otherwise it was shown in mizoguchi j funct anal 271 3323 3347 2016 that each blowup is type ii in radial case in this paper we obtain the conclusion in general case

multiple solutions of two point boundary value problems of ordinary differential equations in banach spaces soit p un cone normal de l espace de banach reel e on note par n la constante normale de p on considere le probleme aux deux points limites x f t x t i x 0 x 1 ou i 0 1 et f une application continue de i p dans p on etudie l existence de solutions positives multiples

triple positive solutions for boundary value problems on infinite intervals abstract this paper deals with the existence of triple positive solutions for sturm liouville boundary value problems of second order nonlinear differential equation on a half line by using a fixed point theorem in a cone due to avery peterson we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

positive solutions of singular boundary value problems on the half line in this paper by using the fixed point theory in the cone with a special norm and space the existence of positive solutions for a class of singular boundary value problems on the half line is established our results improve many known results including singular and non singular cases

positive bound states of systems of nonlinear schrodinger equations abstract we study the existence of positive bound states of non autonomous systems of nonlinear schrodinger equations both the singular case and the regular case are discussed the proof is based on a nonlinear alternative principle of leray schauder

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

positive solutions for three point boundary value problem on the half line in this paper we consider the existence of positive solutions for the following boundary value problem on the half line r t x t f t x t x t 0 email protected 0 x 0 email protected x limt x t 0 by applying fixed point theorems we obtain a variety of existence results in particular the nonlinear term is involved with the first order derivative

rough blowup solutions to the l 2 critical nls we study the singularity formation for the cubic focusing l 2 critical nonlinear schrodinger equation on mathbb r 2 in a series of recent works merle and raphael have completely described the so called log log blowup regime and proven its stability in the energy space h 1 our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space l 2 by blending the merle raphael techniques with the quantitative i method developed by colliander keel staffilani takaoka and tao for the study of the cauchy problem for rough data we obtain the stability of the log log regime in h s for all s 0

determination of the blow up rate for a critical semilinear wave equation in this paper we determine the blow up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance

blow up behavior of solutions to a degenerate parabolic parabolic keller segel system qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic parabolic keller segel system the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one m c 2 n 1 n in both cases it is shown that only type ii blow up is possible that is blow up at the same rate as backward self similar solutions never occurs further information on the generation of singularities induced by mass concentration are given in the critical case

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

type ii blowup of solutions to a simplified keller segel system in two dimensional domains abstract we consider a parabolic elliptic system which is introduced as a simplified version of the so called keller segel system in particular we consider the system in a bounded domain of two dimensional euclidean space in that situation we can find solutions to the system blowing up in finite time then these solutions become the sum of an l 1 function and delta functions at the blowup time as regards the blowup speed there exists a radial solution whose blowup speed is faster than that of backward self similar solutions we refer to that blowup as type ii blowup in this paper we investigate whether finite time blowup solutions to the system exhibit type ii blowup or not

the hamiltonian structure of the nonlinear schrodinger equation and the asymptotic stability of its ground states in this paper we prove that ground states of the nls which satisfy the sufficient conditions for orbital stability of m weinstein are also asymptotically stable for seemingly generic equations the key issue is to prove that a certain coefficient is non negative because is a square power we assume that the nls has a smooth short range nonlinearity we assume also the presence of a very short range and smooth linear potential to avoid translation invariance the basic idea is to perform a birkhoff normal form argument on the hamiltonian as in a paper by bambusi and cuccagna on the stability of the 0 solution for nlkg but in our case the natural coordinates arising from the linearization are not canonical so we need also to apply the darboux theorem with some care though in order not to destroy some nice features of the initial hamiltonian

on positive solutions of boundary value problems on the half line we study the existence of positive solutions of boundary value problems on the half line for differential equations of second order the krasnoselskii fixed point theorem on cone compression and expansion is used

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

existence of three positive solutions for m point boundary value problems on infinite intervals abstract in this paper by using the avery peterson fixed point theorem on a cone we establish the existence of three positive solutions for m point boundary value problems on infinite intervals p x t t f t x t x t 0 0 t x 0 i 1 m 2 i x i lim t x t 0 where p s s p 2 s p 1 r r f t u v r 3 r is a continuous function r 0 i 0 and 0 1 2 m 2 are given

successive iteration and positive solutions for a second order multi point boundary value problem on a half line this paper deals with the existence of positive solutions for some second order multi point boundary value problem on the half line our approach is based on the fixed point theorem and the monotone iterative technique without the assumption of the existence of lower and upper solutions we obtain not only the existence of positive solutions for the problems but also establish iterative schemes for approximating the solutions

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

on the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process

positive solutions for singular semipositone boundary value problems on infinite intervals by using the fixed point theory on a cone with a special norm and space we discuss the existence of positive solutions for a class of semipositone boundary value problems on infinite intervals the work improves many known results including singular and non singular cases

the existence of multiple positive solutions for multi point boundary value problems on the half line in this paper we consider the existence of multiple positive solutions for some nonlinear m point boundary value problems on the half line f u a t f t u t 0 0r is an increasing homeomorphism and positive homomorphism and f 0 0 using a fixed point theorem for operator on a cone we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem

positive solutions for multi point boundary value problem on the half line this paper presents a variety of existence results for nonlinear multi point boundary value problem on the half line in particular we consider x t px t qx t f t x t 0 t 0 x 0 x 0 i 1nkix i a limt x t ert b r 0 p p2 4q2 existence results are established using fixed point theorems on cone

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

type ii blow up for the four dimensional energy critical semi linear heat equation abstract we consider the energy critical four dimensional semi linear heat equation t u u u 3 0 we show the existence of type ii finite time blow up solutions and give a sharp description of the corresponding singularity formation these solutions concentrate a universal bubble of energy in the critical topology u t r 1 q r t u in h 1 where the blow up profile is given by the talenti aubin soliton q r 1 1 r 2 8 and with speed t t t log t t 2 as t t our approach uses a robust energy method approach developed for the study of geometrical dispersive problems raphael and rodnianski 2012 18 merle et al 2011 15 and lies in the continuation of the study of the energy critical harmonic heat flow raphael and schweyer 2011 19 and the energy critical four dimensional wave equation hillaret and raphael 2010 5

on positive solutions of boundary value problems for second order functional differential equations on infinite intervals we study the existence of positive solutions for the following boundary value problem on infinite interval for second order functional differential equations x t px t qx t f t xt 0 t 0 x t x t t t 0 limt x t 0 and x t px t qx t f t xt x t 0 t 0 x t x t t t 0 limt x t 0 where p 0 2 2 0 and q 0 the fixed point theorem on cone is used

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

application of the virial theorem for improving eigenvalue calculations of multiparticle systems abstract the virial theorem is a fundamental property for multiparticle systems in quantum mechanics and offers an elegant relationship between kinetic and potential energies of quantum states in this paper we study applications of the virial theorem for improving eigenvalue calculations of multiparticle systems we propose the virial steepest descent scheme and the two point linear extrapolation to improve the accuracy of such eigenvalue calculations through post processing eigenvalue data obtained from standard methods mathematical analysis of both methods is presented examples are included to illustrate the efficiency of both methods

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star