rate of type ii blowup for a semilinear heat equation a solution u of a cauchy problem for a semilinear heat equation left begin array ll u t delta u u p quad rm in bf r n times 0 t u x 0 u 0 x geq 0 quad rm in bf r n end array right is said to undergo type ii blowup at t t if lim sup t nearrow t t t 1 p 1 u t infty infty let varphi infty be the radially symmetric singular steady state suppose that u 0 in l infty is a radially symmetric function such that u 0 varphi infty and u0 t change sign at most finitely many times we determine the exact blowup rate of type ii blowup solution with initial data u0 in the case of p pl where pl is the lepin exponent

blow up solutions of inhomogeneous nonlinear schrodinger equations on torus abstract in this paper we study blow up solutions to the cauchy problem of the inhomogeneous nonlinear schrodinger equation t u i f x u f x u k x u 2 u on t 2 we present the l 2 concentration property for general initial data and investigate the l 2 minimality

nondispersive solutions to the l 2 critical half wave equation we consider the focusing l2 critical half wave equation in one space dimension i partial t u d u u 2 u where d denotes the first order fractional derivative standard arguments show that there is a critical threshold m 0 such that all h1 2 solutions with u l 2 0 and the linear momentum p 0 in mathbb r in particular our main result and its proof can be seen as a model scenario of minimal mass blowup for l2 critical nonlinear pdes with nonlocal dispersion

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

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

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

uniqueness of the stationary wave for the extended fisher kolmogorov equation abstract the extended fisher kolmogorov equation ut uxxxx uxx u u3 0 models a binary system near the lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria 1 for the classical case 0 the heteroclinic is u x tanh x 2 and is unique up to the obvious symmetries we prove the conjecture that the uniqueness persists all the way to 1 8 where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur our methods are non perturbative and employ a global cross section to the hamiltonian flow of the stationary fourth order equation on the energy level of 1 we also prove uniform a priori bounds on all bounded stationary solutions valid for any 0

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

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

liouville type results for fourth order elliptic equations liouville type theorems are obtained for the solutions to elliptic equations of the form 2 u q x u p x f u 0 by means of two subharmonic functionals and green type inequalities

dualities for torsion free abelian groups of finite rank two well known dualities have been very useful in the study of torsionfree abelian groups of finite rank warlield duality for locally free groups and arnold duality for quotient divisible groups wa ar in this note we establish a duality on classes of torsion free abelian groups of finite rank which generalizes both warlield and arnold duality our results were inspired by a recent paper of fomin fog who constructed a duality which is also a special case of the one presented here the basic idea is to write a torsion free abelian group of finite rank hereafter group g as a sum g g g where g is locally free and g is quotient divisible a dual for g is then obtained by adding the war field dual of g w g and the arnold dual of gz a g inside of hom g q the idea of breaking up g into a locally free and a quotient divisible part is not a new one it was investigated by murley in mu as noted by murley one easy way to obtain g and gz is to take a full free subgroup f of g and write g f d 0 r where d is a divisible and r is a reduced torsion group then choose subgroups g and g2 of g containing f so that g if r and g f d in general the g obtained in this way is decidedly non unique even up to quasi isomorphism it is therefore somewhat surprising that we can form g w g a g to obtain a group which up to quasi equality is independent of the choice ofg further complexity can be introduced due to the fact that the warlield and arnold duals treat free and divisible localizations differently prompting our definition of px groups definition 2 1 on the category of px groups and quasi homomorphisms in which objects are quasi equality classes g g defines an exact contravariant functor theorem 2 13 the px group g has a special property which was also studied by murley on the px groups with this special property the dualizable px groups determines a duality theorem 3 3 which generalizes the dualities of warlield arnold and fomin proposition 3 5 we remark 474 oo21 8693 90 3 00

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

a note on mixed a reflexive groups abstract this paper presents a warfield type characterization of a reflexive groups in the case that a is a mixed abelian group of torsion free rank 1

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

on nonexistence of baras goldstein type for higher order parabolic equations with singular potentials the celebrated result by baras and goldstein 1984 established that the heat equation with singular inverse square potential in a smooth bounded domain n n 3 such that 0 e u t u c x 2 u in 0 t u 0 in the supercritical range c c hardy 1 n 2 2 2 does not have a solution for any nontrivial l 1 initial data uo x 0 in or for a positive measure namely it was proved that a regular approximation of a possible solution by a sequence u n x t of classical solutions of uniformly parabolic equations with bounded truncated potentials given by v x c x 2 v n x min c x 2 n n 1 diverges and as n oo u n x t in x 0 t in the present paper we reveal the connection of this very singular evolution with a spectrum of some limiting operator the proposed approach allows us to consider more general higher order operators for which hardy s inequalities were known since rellich 1954 and initial data that are not necessarily positive in particular it is established that under some natural hypothesis the divergence result is valid for any 2mth order parabolic equation with singular potential u t m u c x 2m u in 0 t where c c h m m 1 with zero dirichlet conditions on and for a wide class of initial data in particular typically the divergence holds for any data satisfying u 0 x is continuous at x 0 and uo 0 0 similar nonexistence i e divergence as e 0 results are also derived for time dependent potentials e 2m q x e t e 2m and nonlinear reaction terms u p e 2m x 2m with p 1 applications to other linear and semilinear schrodinger and wave pdes are discussed

tight subgroups in almost completely decomposable groups a completely decomposable group is a direct sum of groups isomorphic to subgroups of the additive group of rational numbers an almost completely decomposable group x is a finite essential abelian extension of a completely decomposable group a of finite rank in 1974 lady lad74 initiated a systematic theory of such groups based on the fundamental concept of regulating subgroup the regulating subgroups can be defined as the completely decomposable subgroups of least index in an almost completely decomposable group details on the subsequent developments can be found in the survey article mad95 or in the monograph mad99

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

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

liouville type and uniqueness results for a class of elliptic equations abstract liouville type results are obtained for fourth order elliptic equations of the form 2 u q x g u p x f u 0 and q x g u u p x f u 0 by means of two subharmonic functionals defined appropriately on their respective solutions and a green inequality

a critical center stable manifold for schrodinger s equation in three dimensions consider the focusing critical semilinear schrodinger equation in it admits an eight dimensional manifold of special solutions called ground state solitons we exhibit a codimension 1 critical real analytic manifold of asymptotically stable solutions of 0 1 in a neighborhood of the soliton manifold we then show that is center stable in the dynamical systems sense of bates and jones and globally in time invariant solutions in are asymptotically stable and separate into two asymptotically free parts that decouple in the limit a soliton and radiation conversely in a general setting any solution that stays close to the soliton manifold for all time is in the proof uses the method of modulation new elements include a different linearization and an endpoint strichartz estimate for the time dependent linearized equation the proof also uses the fact that the linearized hamiltonian has no nonzero real eigenvalues or resonances this has recently been established in the case treated here of the focusing cubic nls in by the work of marzuola and simpson and costin huang and schlag 2012 wiley periodicals inc

on collapsing ring blow up solutions to the mass supercritical nls we consider the nonlinear schr odinger equation i pa tu delta u u u p 1 0 in dimension n 2 and in the mass super critical and energy subcritical range 1 4n p min n 2n 2 5 for initial data u0 h1 with radial symmetry we prove a universal upper bound on the blow up speed we then prove that this bound is sharp and attained on a family of collapsing ring blow up solutions first formally predicted by gavish fibich and wang

the sharp threshold and limiting profile of blow up solutions for a davey stewartson system abstract the blow up solutions of the cauchy problem for the davey stewartson system which is a model equation in the theory of shallow water waves are investigated firstly the existence of the ground state for the system derives the best constant of a gagliardo nirenberg type inequality and the variational character of the ground state secondly the blow up threshold of the davey stewartson system is developed in r 3 thirdly the mass concentration is established for all the blow up solutions of the system in r 2 finally the existence of the minimal blow up solutions in r 2 is constructed by using the pseudo conformal invariance the profile of the minimal blow up solutions as t t blow up time is in detail investigated in terms of the ground state

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

geometric treatments and a common mechanism in finite time singularities for autonomous odes abstract geometric treatments of blow up solutions for autonomous ordinary differential equations and their blow up rates are concerned our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces and dynamics on their center stable manifolds in particular we show that dynamics on center stable manifolds of invariant sets at infinity with appropriate time scale desingularizations as well as blowing up of singularities characterize dynamics of blow up solutions as well as their rigorous blow up rates similarities for characterizing finite time extinction and asymptotic behavior of compacton traveling waves to blow up solutions are also shown

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

numerical validation of blow up solutions with quasi homogeneous compactifications we provide a numerical validation method of blow up solutions for finite dimensional vector fields admitting asymptotic quasi homogeneity at infinity our methodology is based on quasi homogeneous compactifications containing quasi parabolic type and directional type compactifications divergent solutions including blow up solutions then correspond to global trajectories of associated vector fields with appropriate time variable transformation tending to equilibria on invariant manifolds representing infinity we combine standard methodology of rigorous numerical integration of differential equations with lyapunov function validations around equilibria corresponding to divergent directions which yields rigorous upper and lower bounds of blow up time as well as rigorous profile enclosures of blow up solutions

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

construction of solutions with exactly k blow up points for nonlinear schrodinger equation with a harmonic potential abstract in this paper we consider the nonlinear schrodinger equation with a harmonic potential which describes bose einstein condensate for arbitrarily given k points x 1 x 2 x k in r n we construct a solution which blows up exactly at x 1 x 2 x k furthermore the precise behavior of the solution is described both at those blow up points and in r n x 1 x 2 x k

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

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

stochastic nonlinear schrodinger equations abstract this paper is devoted to the well posedness of stochastic nonlinear schrodinger equations in the energy space h 1 r d and is a continuation of our recent work barbu et al 2014 we allow general complex coefficients in the noise thus covering both the conservative and non conservative case the latter being important to include quantum measurement effects in the underlying physical model we consider both focusing and defocusing nonlinearities and prove global well posedness in h 1 r d including also the pathwise continuous dependence on initial conditions with exponents of the nonlinearity in exactly the same sharp range as in the deterministic case in particular in the special conservative case this work improves earlier results in de bouard and debussche 2003 where the critical exponents could not be reached for d 6 in the defocusing and for d 4 in the focusing case respectively moreover the local existence uniqueness and blowup alternative are also established for the energy critical case the approach presented here is mainly based on the rescaling approach already used in barbu et al 2014 to study the l 2 case and also on the strichartz estimates established in marzuola et al 2008 for large perturbations of the laplacian

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

liouville type and uniqueness results for a class of sixth order elliptic equations on considere des equations de la forme 3 u p 2 u q u ru 0 ou p q et r sont des fonctions positives et c 2 dans e n et on demontre des resultats de type liouville a l aide des fonctions sous harmoniques definies sur les solutions et des inegalites de type green

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

standing ring blow up solutions to the n dimensional quintic nonlinear schrodinger equation we consider the quintic nonlinear schr odinger equation in dimension n 3 this problem is energy critical in dimension n 3 and energy super critical for n 4 we prove the existence of a radially symmetric blow up mechanism with l2 concentration along the unit sphere of rn this singularity formation is moreover stable by smooth and radially symmetric perturbation of the initial data this result extends the result obtained for n 2 in 29 and is the first result of description of a singularity formation in the energy supercritical class for nls type problems our main tool is the proof of the propagation of regularity outside the blow up sphere in the presence a so called log log type singularity

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

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

blow up rate of type ii and the braid group theory a solution u of a cauchy problem or a cauchy dirichlet problem for a semilinear heat equation u t u u p with nonnegative initial data u 0 is said to undergo type ii blow up at t t if lim sup t t 1 p 1 u t let be the radially symmetric singular steady state of the cauchy problem suppose that u 0 l is a radially symmetric function such that u 0 and uo t change sign at most finitely many times by application of the braid group theory we determine the exact blow up rate of solution with initial data u 0 which undergoes type ii blow up in the case of p p jl where p jl is the exponent of joseph and lundgren

duality for almost completely decomposable groups we establish a duality for near isomorphism categories of almost completely decomposable groups through a combination of butler duality and warfield duality we are then able to dualize results in the literature to obtain classifications and structure theories for new classes of almost completely decomposable groups for example known results on p local crq groups those groups x that contain a completely decomposable subgroup b with x b a cyclic p group can be dualized to results on p local co crq groups those groups y that embed in a completely decomposable group b so that the quotient b y is a cyclic p group a variety of other applications is given

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

blow up for the 1d nonlinear schrodinger equation with point nonlinearity i basic theory abstract we consider the 1d nonlinear schrodinger equation nls with focusing point nonlinearity 0 1 i t x 2 p 1 0 where x is the delta function supported at the origin in this work we show that 0 1 shares many properties in common with those previously established for the focusing autonomous translationally invariant nls 0 2 i t p 1 0 the critical sobolev space h c for 0 1 is c 1 2 1 p 1 whereas for 0 2 it is c d 2 2 p 1 in particular the l 2 critical case for 0 1 is p 3 we prove several results pertaining to blow up for 0 1 that correspond to key classical results for 0 2 specifically we 1 obtain a sharp gagliardo nirenberg inequality analogous to weinstein 44 2 apply the sharp gagliardo nirenberg inequality and a local virial identity to obtain a sharp global existence blow up threshold analogous to weinstein 44 glassey 17 in the case c 0 and duyckaerts holmer roudenko 12 guevara 18 and fang xie cazenave 13 for 0 c 1 3 prove a sharp mass concentration result in the l 2 critical case analogous to tsutsumi 43 merle tsutsumi 36 and 4 show that minimal mass blow up solutions in the l 2 critical case are pseudoconformal transformations of the ground state analogous to merle 28

renormalization and blow up for wave maps from s 2 times mathbb r to s 2 we construct a one parameter family of finite time blow ups to the co rotational wave maps problem from s 2 x r to s 2 parameterized by nu is an element of 1 2 1 the longitudinal function u t alpha which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from r 2 to s 2 the domain of this harmonic map is identified with a neighborhood of the north pole in the domain s 2 via the exponential coordinates alpha theta in these coordinates u t alpha q lambda t alpha r t alpha where q r 2arctan r is the standard co rotational harmonic map to the sphere lambda t t 1 nu and r t alpha is the error with local energy going to zero as t 0 blow up will occur at t alpha 0 0 due to energy concentration and up to this point the solution will have regularity h1 nu

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

type ii blow up mechanisms in a semilinear heat equation with critical joseph lundgren exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant it is well known that type ii blow up does occur if n 11 and p p jl where p jl stands for the joseph lundgren exponent p jl n 10 1 4 n 4 2 n 1 n 11 on the other hand it is also known that if p p jl type ii blow up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned it has long remained open whether or not type ii blow up occurs for the borderline case p p jl n 11 in the present article we prove constructively the existence of type ii blow up solutions for the joseph lundgren critical case thus giving an answer to this question

the transition from diffusion to blow up for a nonlinear schrodinger equation in dimension 1 we consider the time dependent one dimensional nonlinear schrodinger equation with a pointwise singular potential we prove that if the strength of the nonlinear term is small enough then the solution is well defined for any time regardless of the choice of initial data in contrast if the nonlinearity power is larger than a critical value for some initial data a blow up phenomenon occurs in finite time in particular if the system is initially prepared in the ground state of the linear part of the hamiltonian then we obtain an explicit condition on the parameters for the occurrence of the blow up

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

sharp upper and lower bounds on the blow up rate for nonlinear schrodinger equation with potential abstract we consider the blow up solutions of the cauchy problem for critical nonlinear schrodinger equation with a repulsive harmonic potential in terms of merle and raphael s recent arguments as well as carles transform the sharp upper and lower bounds of the blow up rate are obtained

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 loglog blow up in higher order reaction diffusion and wave equations a formal geometric approach abstract in 1934 petrovskii s boundary regularity study of the heat equation gave the first appearance of the ln ln t t blow up factor in pde theory we discuss the origin of analogous non self similar blow up in higher order reaction diffusion parabolic or wave hyperbolic equations of the form u t u 2 u x x x x u or u t t u 2 u x x x x u in l l 0 t with zero dirichlet boundary conditions at x l where l l 0 2 we present formal arguments that the standard similarity blow up rate 1 t t acquires an extra universal ln ln t t factor the explanation is based on a geometric matching with the so called logarithmic travelling waves as group invariant solutions of the pdes we also discuss connections with log log blow up factors occurring in earlier studies of plasma physics parabolic equations and the nonlinear critical schrodinger equation

system design for nonlinear plants using operator based robust right coprime factorization and isomorphism the nonlinear system design problem is investigated concerning with robust stability and output tracking problems for nonlinear perturbed feedback control systems using operator based robust right coprime factorization and isomorphism first a compensator is designed to factorize a nonlinear perturbed plant using isomorphism then the right factorization of the plants can be realized further by guaranteeing an output tracking property the existence of two stable controllers is proved so that a bezout identity for the right factorization of the plant can be satisfied meanwhile the robust stability can be guaranteed by the two controllers therefore the right factorization is exactly to be a robust right coprime factorization that is by the proposed design scheme the robust right coprime factorization is realized and the plant output can also track to the reference input a numerical example is given to show the effectiveness of the proposed design scheme

on the blow up solutions for the nonlinear fractional schrodinger equation abstract this paper is dedicated to the blow up solutions for the nonlinear fractional schrodinger equation arising from pseudorelativistic boson stars first we compute the best constant of a gg n inequality by the profile decomposition theory and variational arguments then we find the sharp threshold mass of the existence of finite time blow up solutions finally we study the dynamical properties of finite time blow up solutions around the sharp threshold mass by giving a refined compactness lemma

construction of a minimal mass blow up solution of the modified benjamin ono equation we construct a minimal mass blow up solution of the modified benjamin ono equation mbo begin aligned u t u 3 d 1 u x 0 end aligned mbo which is a standard mass critical dispersive model let q in h frac 1 2 q 0 be the unique ground state solution of d 1 q q q 3 constructed using variational arguments by weinstein commun part differ equations 12 1133 1173 1987a j differ equations 69 192 203 1987b and albert et al proc r soc lond a 453 1233 1260 1997 and whose uniqueness was recently proved by frank and lenzmann acta math 210 261 318 2013 we show the existence of a solution s of mbo satisfying vert s vert l 2 vert q vert l 2 and begin aligned s t frac 1 lambda frac 1 2 t q left frac cdot x t lambda t right rightarrow 0 quad text in h frac 1 2 mathbb r text as t downarrow 0 end aligned where begin aligned lambda t sim t quad x t sim ln t quad hbox and quad vert s t vert dot h frac 1 2 sim t frac 1 2 vert q vert dot h frac 1 2 quad hbox as t downarrow 0 end aligned this existence result is analogous to the one obtained by martel et al j eur math soc 17 1855 1925 2015 for the mass critical generalized korteweg de vries equation however in contrast with the gkdv equation for which the blow up problem is now well understood in a neighborhood of the ground state s is the first example of blow up solution for mbo the proof involves the construction of a blow up profile energy estimates as well as refined localization arguments developed in the context of benjamin ono type equations by kenig et al ann inst h poincare anal non lin 28 853 887 2011 due to the lack of information on the mbo flow around the ground state the energy estimates have to be considerably sharpened in the present paper

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

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