going beyond the threshold scattering and blow up in the focusing nls equation we study the focusing nonlinear schrodinger equation i partial t u delta u u p 1 u 0 x in mathbb r n in the l2 supercritical regime with finite energy and finite variance initial data we investigate solutions above the energy or mass energy threshold in our first result we extend the known scattering versus blow up dichotomy above the mass energy threshold for finite variance solutions in the energy subcritical and energy critical regimes obtaining scattering and blow up criteria for solutions with arbitrary large mass and energy as a consequence we characterize the behavior of the ground state initial data modulated by a quadratic phase our second result gives two blow up criteria which are also applicable in the energy supercritical nls setting we finish with various examples illustrating our results

scattering versus blow up beyond the threshold for the focusing choquard equation abstract this note investigates the choquard equation i u u i u p u p 2 u 0 indeed above the mass energy threshold a scattering versus finite time blow up dichotomy is obtained in the mass super critical and energy sub critical regime

method of characteristics analysis of one dimensional members abstract the method of characteristics moc is used to simulate axial torsional and flexural vibrations in single structural members supporting discrete masses it is shown that the method is capable of predicting natural frequencies of vibration in dissipative as well as non dissipative systems numerical limitations of the analysis are identified but shown to be unimportant when members are represented by a sufficient but acceptably small number of elements the method is shown to have useful advantages over conventional mass spring analyses including the ability to simulate wave action in members and to predict true stiffnesses it is argued that there is scope for future developments leading to the large deflection analysis of complex structures

a genuinely multidimensional high resolution scheme for the elastic plastic wave equation in this paper we present a numerical method of high order for solving the multidimensional elastic plastic wave equation the basic idea is to decompose the hyperbolic pde into advection equations which can be solved numerically furthermore the occurrence of plasticity makes it necessary to solve an ode for the stress strain relationship at evey point

sharp threshold of global well posedness vs finite time blow up for a class of inhomogeneous choquard equations it is the purpose of this work to obtain a sharp threshold of global existence vs blow up dichotomy for mass super critical and energy subcritical solutions to an inhomogeneous choquard equation it is the purpose of this work to obtain a sharp threshold of global existence vs blow up dichotomy for mass super critical and energy subcritical solutions to an inhomogeneous choquard equation

global dynamics above the ground state energy for the focusing nonlinear klein gordon equation abstract the analysis of global dynamics of nonlinear dispersive equations has a long history starting from small solutions in this paper we study the focusing cubic nonlinear klein gordon equation in r 3 with large radial data in the energy space this equation admits a unique positive stationary solution q called the ground state in 1975 payne and sattinger showed that solutions u t with energy e u u strictly below that of the ground state are divided into two classes depending on a suitable functional k u if k u 0 then one has finite time blow up if k u 0 global existence moreover these sets are invariant under the flow recently ibrahim masmoudi and the first author 22 improved this result by establishing scattering to zero for k u 0 by means of a variant of the kenig merle method kenig and merle 2006 2008 25 26 in this paper we go slightly beyond the ground state energy and we give a complete description of the evolution in that case for example in a small neighborhood of q one encounters the following trichotomy on one side of a center stable manifold one has finite time blow up for t 0 on the other side scattering to zero and on the manifold itself one has scattering to q both as t in total the class of data with energy at most slightly above that of q is divided into nine disjoint non empty sets each displaying different asymptotic behavior as t which includes solutions blowing up in one time direction and scattering to zero on the other the analogue of the solutions found by duyckaerts and merle 2009 2008 13 14 for the energy critical wave and schrodinger equations appear here as the unique one dimensional stable unstable manifolds approaching q exponentially as t or t respectively the main technical ingredient in our proof is a one pass theorem which excludes the existence of almost homoclinic orbits between q as well as q and almost heteroclinic orbits connecting q with q in a companion paper nakanishi and schlag 2010 31 we establish analogous properties for the nls equation

solution of the implicitly discretised reacting flow equations by operator splitting abstract a non iterative method for handling the coupling of the implicitly discretised time dependent fluid flow equations is described the method is based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations the main feature of the technique is the splitting of the solution process into a series of steps whereby operations on pressure are decoupled from those on velocity at each step with the split sets of equations being amenable to solution by standard techniques at each time step the procedure yields solutions which approximate the exact solution of the difference equations the accuracy of this splitting procedure is assessed for a linearised form of the discretised equations and the analysis indicates that the solution yielded by it differs from the exact solution of the difference equations by terms proportional to the powers of the time step size by virtue of this it is possible to dispense with iteration thus resulting in an efficient implicit scheme while retaining simplicity of implementation relative to contemporary block simultaneous methods this is verified in a companion paper which presents results of computations carried out using the method

spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect abstract based on biological meaning a kind of diffusive intraguild predation igp resource ig prey and ig predator model with delay effect is investigated in this paper the model has holling type i functional response between resource ig prey and resource ig predator holling type ii functional response between ig prey and ig predator we first give sufficient conditions on the stability of possible nonnegative constant steady state solutions for the proposed model which give us a complete picture of the global dynamics then we investigate hopf bifurcation near the unique positive constant steady state solution by taking delay as bifurcation parameter and derive the hopf bifurcation threshold it is shown that the delay can induce three types of bistability node node bistability node cycle bistability and cycle cycle bistability periodic oscillations and irregular oscillations triggering spatiotemporal chaos in the diffusive igp model numerical simulations are performed to illustrate our theoretical results and suggest that delay can even trigger the emergence of self organised spatiotemporal patterns which evolve from spiral patterns to irregular spatial patterns via spatiotemporal hopf bifurcation in addition the impact of diffusion on the model s dynamics under certain time delay are also explored

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

a finite element based level set method for fluid elastic solid interaction with surface tension summary a numerical method for simulating fluid elastic solid interaction with surface tension is presented a level set method is used to capture the interface between the solid bodies and the incompressible surrounding fluid within an eulerian approach the mixed velocity pressure variational formulation is established for the global coupled mechanical problem and discretized using a continuous linear approximation in both velocity and pressure three ways are investigated to reduce the spurious oscillations of the pressure that appear at the fluid solid interface first two stabilized finite element methods are used the mini element and the algebraic subgrid method second the surface integral corresponding to the surface tension term is treated either by the continuum surface force technique or by a surface local reconstruction algorithm finally besides the direct evaluation method proposed by bruchon et al an alternative method is proposed to avoid the explicit computation of the surface curvature which may be a source of difficulty these different issues are addressed through various numerical examples such as the two incompressible fluid flow the elastic inclusion embedded into a newtonian fluid or the study of a granular packing copyright 2012 john wiley sons ltd

a finite volume procedure to solve elastic solid mechanics problems in three dimensions on an unstructured mesh a finite volume fv procedure is described for solving the elastic solid mechanics equations in three dimensions on an unstructured mesh for bodies undergoing thermal or mechanical loads the fv procedure is developed in parallel with the conventional fe galerkin procedure so that the differences in each approach may be clearly distinguished the matrix form of the fv procedure is described and is implemented in parallel with the fe procedure both for two dimensional quadrilateral and three dimensional brick meshes the fv and fe procedures are then compared against a range of benchmark problems that test the basic capability of the fv technique it is shown to be approximately as accurate as the fe procedure on similar meshes though its system matrix set up time is twice as long for a node by node set up procedure

on the defect of compactness for the strichartz estimates of the schrodinger equations abstract in this paper we prove that every sequence of solutions to the linear schrodinger equation with bounded data in h1 r d d 3 can be written up to a subsequence as an almost orthogonal sum of sequences of the type h d 2 2nv t tn h2n x xn hn where v is a solution of the linear schrodinger equation with a small remainder term in strichartz norms using this decomposition we prove a similar one for the defocusing h1 critical nonlinear schrodinger equation assuming that the initial data belong to a ball in the energy space where the equation is solvable this implies in particular the existence of an a priori estimate of the strichartz norms in terms of the energy

weak concentration and wave operator for a 3d coupled nonlinear schrodinger system reported in this paper are results concerning the cauchy problem and the dynamics for a cubic nonlinear schrodinger system arising in nonlinear optics a sharp criterion is given concerned with the dichotomy global existence versus finite time blow up when a radial solution blows up in finite time we prove the concentration in the critical lebesgue space sufficient condition for the scattering and the construction of the wave operator in the energy space is also provided

long time behavior of solutions of fisher kpp equation with advection and free boundaries we consider fisher kpp equation with advection ut uxx ux f u for x g t h t where g t and h t are two free boundaries satisfying stefan conditions this equation is used to describe the population dynamics in advective environments we study the influence of the advection coefficient on the long time behavior of the solutions we find two parameters c0 and with c0 0 which play key roles in the dynamics here c0 is the minimal speed of the traveling waves of fisher kpp equation more precisely by studying a family of the initial data 0 where is some compactly supported positive function we show that 1 in case 0 c0 there exists 0 such that spreading happens when i e u t 1 locally uniformly in r and vanishing happens when 0 i e g t h t remains bounded and u t 0 uniformly in g t h t 2 in case c0 there exists 0 such that virtual spreading happens when i e u t 0 locally uniformly in g t and u t ct 1 locally uniformly in r for some c c0 vanishing happens when 0 and in the transition case u t o t v c0 t uniformly the latter is a traveling wave with a big head near the free boundary x c0 t and with an infinite long tail on the left 3 in case c0 there exists 0 such that virtual spreading happens when and u t 0 uniformly in g t h t when 0 4 in case vanishing happens for any solution

minimal mass blow up solutions for the l2 critical nls with inverse square potential abstract we study minimal mass blow up solutions of the focusing l 2 critical nonlinear schrodinger equation with inverse square potential i t u u c x 2 u u 4 n u 0 with n 3 and 0 c n 2 2 4 we first prove a sharp global well posedness result all h 1 solutions with a mass i e l 2 norm strictly below that of the ground states are global note that unlike the equation in free space we do not know if the ground state is unique in the presence of the inverse square potential nevertheless all ground states have the same minimal mass we then construct and classify finite time blow up solutions at the minimal mass threshold up to the symmetries of the equation every such solution is a pseudo conformal transformation of a ground state solution

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

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

global existence and blowup for a class of the focusing nonlinear schrodinger equation with inverse square potential abstract we consider a class of the focusing nonlinear schrodinger equation with inverse square potential i t u u c x 2 u u u u 0 u 0 h 1 t x r r d where d 3 4 d 4 d 2 and c 0 satisfies c d d 2 2 2 in the mass critical case 4 d we prove the global existence and blowup below ground states for the equation with d 3 and c d in the mass and energy intercritical case 4 d 4 d 2 we prove the global existence and blowup below the ground state threshold for the equation this extends similar results of 18 and 22 to any dimensions d 3 and a full range c d we finally prove the blowup below ground states for the equation in the energy critical case 4 d 2 with d 3 and c d 2 4 d d 2 2 d

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

the minimal habitat size for spreading in a weak competition system with two free boundaries abstract in this paper we focus on the dynamics for a lotka volterra type weak competition system with two free boundaries where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species respectively in the weak competition case the dynamics of this model can be classified into four cases which forms a spreading vanishing quartering the notion of the minimal habitat size for spreading is introduced to determine if species can always spread some sufficient conditions for spreading and vanishing are established also when spreading occurs some rough estimates for spreading speed and the long time behavior of solutions are established

a note on the 2d generalized zakharov kuznetsov equation local global and scattering results abstract we consider the generalized two dimensional zakharov kuznetsov equation u t x u x u k 1 0 where k 3 is an integer number for k 8 we prove local well posedness in the l 2 based sobolev spaces h s r 2 where s is greater than the critical scaling index s k 1 2 k for k 3 we also establish a sharp criteria to obtain global h 1 r 2 solutions a nonlinear scattering result in h 1 r 2 is also established assuming the initial data is small and belongs to a suitable lebesgue space

a nonlocal diffusion model with free boundaries in spatial heterogeneous environment abstract this paper is concerned with the nonlocal diffusive model with double free boundaries in spatial heterogeneous environment where the spatial heterogeneity is described by the sign indefinite coefficients such a model can be used to illustrate the spreading or vanishing of a new or invasive species due to the lack of comparison principle in the nonlocal reaction diffusion equation many classical methods cannot be used directly to this nonlocal problem this motivates us to find new techniques we first establish the spreading vanishing dichotomy as well as some criteria that ensure the species spreading or vanishing by principal eigenvalues of associated scalar elliptic eigenvalue problems and then we determine the spreading speed when spreading occurs

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

on a free boundary problem for a reaction diffusion advection logistic model in heterogeneous environment abstract in this paper we investigate a reaction diffusion advection equation with a free boundary which models the spreading of an invasive species in one dimensional heterogeneous environments we assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal and the spreading mechanism of species is determined by a stefan type condition investigating the sign of the principal eigenvalue of the associated linearized eigenvalue problem under certain conditions we obtain the sharp criteria for spreading and vanishing via system parameters also we establish the long time behavior of the solution and the asymptotic spreading speed finally some biological implications are discussed

the diffusive logistic equation with a free boundary and sign changing coefficient abstract this short paper concerns a diffusive logistic equation with a free boundary and sign changing coefficient which is formulated to study the spread of an invasive species where the free boundary represents the expanding front a spreading vanishing dichotomy is derived namely the species either successfully spreads to the right half space as time t and survives persists in the new environment or it fails to establish itself and will extinct in the long run the sharp criteria for spreading and vanishing are also obtained when spreading happens we estimate the asymptotic spreading speed of the free boundary

a diffusive logistic equation with a free boundary and sign changing coefficient in time periodic environment abstract this paper concerns a diffusive logistic equation with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment in which the variable intrinsic growth rate may be very negative in a suitable large region see conditions h1 h2 4 3 such a model can be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front in the case of higher space dimensions with radial symmetry and when the intrinsic growth rate has a positive lower bound this problem has been studied by du guo peng 11 they established a spreading vanishing dichotomy the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed in the present paper we show that the above results are retained for our problem

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

fluid structure interaction with large structural displacements the calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain this paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium to be studied in a fixed reference configuration the resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions this approach treats the structure in a fully lagrangian way and uses an associated arbitrary lagrangian eulerian ale formulation for the fluid such a strategy has three advantages which are detailed in the present paper it can be discretised in time by implicit stable energy conserving time integration schemes and solved by simple iterative uncoupled algorithms the fluid and the structural problems can be approximated by independent finite element spaces the most recent geometrically exact nonlinear shell models can be used for the structures the full strategy has been implemented into an industrial cfd code results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper

a finite volume format for structural mechanics summary a general finite volume method fvm for the analysis of structural problems is presented it is shown that the fvm can be considered to be a particular case of finite elements with a non galerkin weighting for structural analysis this can readily be interpreted as equivalent to the unit displacement method which involves mainly surface integrals both displacement and mixed fv formulations are presented for static and dynamic problems the finite volume method fvm evolved in the early seventies via finite difference approxima tions on non orthogonal grids the popularity of the fvm has been extensive in the field of computational fluid dynamics cfd and heat transfer on the contrary in the field of computational solid mechanics csm the use of the fvm has never achieved such acceptance an early attempt to use fv concepts in csm is due to wilkins6 as an alternative approximation to derivatives in a cell in this he defines the average gradient of a function u in a volume l2 as using the well known divergence theorem such a definition of gradients can be written entirely in terms of function values at the boundary of a volume and has been used in the early hydrocodes of the lawrence livermore laboratory the reasons for the unpopularity of the fvm in structural mechanics is understandable as finite volumes are well known to be less accurate than galerkin based finite elements for self adjoint elliptic problems a comparison between fvm and fem has been recently pre sented by zienkiewicz and oiiate here the authors show that fvm and fem share concepts such as mesh discretization and interpolation giving precisely the same discretized systems of equations for some particular cases this coincidence also shows clearly for 2 d and 3 d structural problems as detailed in this work here however surface integrals are mostly involved and the number of computations can be shown to be proportional to the number of sides in the mesh this leads to an overall solution cost very similar to that of fe computations note that for a refined mesh of three node triangles the number of sides is 1 5 times that of elements this fact suggests that computational speed is not a prior one of the keys to the possible success of fvm in structural problems however the possibility of obtaining the element matrices and vectors in terms of computations along the element sides opens new possibilities for the solution of some structural problems which in the authors opinion may be worth exploring in detail indeed it

buckling of thick cylindrical shells under external pressure a new analytical expression for the critical load and comparison with elasticity solutions abstract in this paper a set of stability equations for thick cylindrical shells is derived and solved analytically the set is obtained by integration of the differential stability equations across the thickness of the shell the effects of transverse shear and the non linear variation of the stresses and displacements are accounted for with the aid of the higher order shell theory proposed by voyiadjis g z and shi g 1991 a refined two dimensional theory for thick cylindrical shells international journal of solids and structures 27 3 261 282 for a thick shell under external hydrostatic pressure the stability equations are solved analytically and yield an improved expression for the buckling load reference solutions are also obtained by solving numerically the differential stability equations both the full set that contains strains and rotations as well as the simplified set that contains rotations only were solved numerically the relative magnitude of shear strain and rotation was examined and the effect of thickness was quantified differences between the benchmark solutions and the analytic expressions based on the refined theory and the classical shell theory are analysed and discussed it is shown that the new analytic expression provides significantly improved predictions compared to the formula based on thin shell theory

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

coupled axial and flexural vibration of 1 d members abstract the method of characteristics is used to model vibrations of one dimensional members in the presence of coupling between axial and flexural waves only small deflections are considered and so the couple fx y is large only when the axial force fx is large alternative numerical schemes are presented for the integration of terms describing coupling and a suitable explicit scheme is identified the analysis is used to illustrate the influence of axial loads on frequencies and amplitudes of lateral oscillations of cantilever structures strong parametric behaviour is reproduced for a single storey shear structure including unstable resonance at double the linear natural frequency

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

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

a weak competition system with advection and free boundaries abstract in order to study the influence of advection speed on the competitive outcomes of two invading species a reaction diffusion advection weak competition system with four free boundaries in one dimensional space is proposed and investigated in the case of small advection speed the explicit classification of the competitive outcomes the estimation of the spreading speed the long time behavior of the solutions and the minimal habitat size which determines whether the species can always spread or not are obtained the results are similar as the case with no advection in the case of large advection speed both two species cannot spread successfully but may virtual spread downstream in the case of medium sized advection speed some competitive outcomes and the long time behaviors of the solutions are also obtained mathematical results suggest that the competitive outcomes which depend on the advection speed and moving parameters are very complicated some criteria for spreading vanishing and virtual spreading are also established in all cases

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

a diffusive logistic model with a free boundary in time periodic environment abstract we study the diffusive logistic equation with a free boundary in time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for time independent environment in the cases of one space dimension and higher space dimensions with radial symmetry this free boundary problem has been studied in du and lin 2010 12 du and guo 2011 9 in both cases a spreading vanishing dichotomy was established and when spreading occurs the asymptotic spreading speed was determined in this paper we show that the spreading vanishing dichotomy is retained in time periodic environment and we also determine the spreading speed the former is achieved by further developing the earlier techniques and the latter is proved by introducing new ideas and methods

the spreading front of invasive species in favorable habitat or unfavorable habitat abstract spatial heterogeneity and habitat characteristic are shown to determine the asymptotic profile of the solution to a reaction diffusion model with free boundary which describes the moving front of the invasive species a threshold value r 0 fr d t is introduced to determine the spreading and vanishing of the invasive species we prove that if r 0 fr d t 0 1 for some t 0 0 the spreading must happen while if r 0 fr d 0 1 the spreading is also possible our results show that the species in the favorable habitat can establish itself if the diffusion is slow or the occupying habitat is large in an unfavorable habitat the species dies out if the initial value of the species is small however big initial number of the species is benefit for the species to survive when the species spreads in the whole habitat the asymptotic spreading speed is given some implications of these theoretical results are also discussed

sharper criteria for the wave collapse abstract sharper criteria for three dimensional wave collapse described by the nonlinear schrodinger equation nlse are derived the collapse threshold corresponds to the ground state soliton which is known to be unstable thus for nonprefocusing distributions this represents the separatrix between collapsing and noncollapsing sectors numerical results support the theoretical results generalizations of the criteria for the nlse with arbitrary power nonlinearity are also presented

scattering for the radial 3d cubic focusing inhomogeneous nonlinear schrodinger equation abstract the purpose of this work is to study the 3d focusing inhomogeneous nonlinear schrodinger equation i u t u x b u 2 u 0 where 0 b 1 2 let q be the ground state solution of q q x b q 2 q 0 and s c 1 b 2 we show that if the radial initial data u 0 belongs to h 1 r 3 and satisfies e u 0 s c m u 0 1 s c e q s c m q 1 s c and u 0 l 2 s c u 0 l 2 1 s c q l 2 s c q l 2 1 s c then the corresponding solution is global and scatters in h 1 r 3 our proof is based in the ideas introduced by kenig merle 1 in their study of the energy critical nls and holmer roudenko 2 for the radial 3d cubic nls

scattering for nls with a delta potential abstract we prove h 1 scattering for the mass supercritical defocusing nonlinear schrodinger equation perturbed by a repulsive delta potential

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

a sharp condition for scattering of the radial 3d cubic nonlinear schrodinger equation we consider the problem of identifying sharp criteria under which radial h 1 finite energy solutions to the focusing 3d cubic nonlinear schrodinger equation nls i t u u u 2 u 0 scatter i e approach the solution to a linear schrodinger equation as t the criteria is expressed in terms of the scale invariant quantities u 0 l 2 nabla u 0 l 2 and m u e u where u 0 denotes the initial data and m u and e u denote the conserved in time mass and energy of the corresponding solution u t the focusing nls possesses a soliton solution e it q x where q is the ground state solution to a nonlinear elliptic equation and we prove that if m u e u q l 2 nabla q l 2 then the solution blows up in finite time the technique employed is parallel to that employed by kenig merle 17 in their study of the energy critical nls

sharp threshold of blow up and scattering for the fractional hartree equation abstract we consider the fractional hartree equation in the l 2 supercritical case and find a sharp threshold of the scattering versus blow up dichotomy for radial data if m u 0 s s c s c e u 0 m q s s c s c e q and m u 0 s s c s c u 0 h s 2 m q s s c s c q h s 2 then the solution u t is globally well posed and scatters if m u 0 s s c s c e u 0 m q s s c s c e q and m u 0 s s c s c u 0 h s 2 m q s s c s c q h s 2 the solution u t blows up in finite time this condition is sharp in the sense that the solitary wave solution e i t q x is global but not scattering which satisfies the equality in the above conditions here q is the ground state solution for the fractional hartree equation

solution techniques for the fully discretized equations in computation of fluid structure interactions with the space time formulations we provide an overview of the solution techniques we have developed for the fully discretized equations encountered at every time step in computation of fluid structure interactions with the space time techniques these coupled nonlinear equations are generated from the finite element discretization of the governing equations for the fluid mechanics structural mechanics and the motion of the fluid mechanics mesh the fluid mechanics equations are discretized with the deforming spatial domain stabilized space time formulation the mesh motion is governed by the equations of elasticity with the smaller elements stiffened in the finite element formulation the coupled fully discretized equations are solved with the block iterative quasi direct and direct coupling methods we present numerical examples with incompressible flows and membrane and cable structures 2005 elsevier b v all rights reserved

global existence and scattering for rough solutions of a nonlinear schr odinger equation on r 3 we prove global existence and scattering for the defocusing cubic nonlinear schrodinger equation in h s r 3 for s 4 the main new estimate in the argument is a morawetz type inequality for the solution this estimate bounds k x t k l 4 r3 r whereas the well known morawetz type estimate of lin strauss controls r 1 0 r r3 x t 4 x dxdt

on some free boundary problems of the prey predator model abstract in this paper we investigate some free boundary problems for the lotka volterra type prey predator model in one space dimension the main objective is to understand the asymptotic behavior of the two species prey and predator spreading via a free boundary we prove a spreading vanishing dichotomy namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment or they fail to establish and die out in the long run the long time behavior of solution and criteria for spreading and vanishing are also obtained finally when spreading successfully we provide an estimate to show that the spreading speed if exists cannot be faster than the minimal speed of traveling wavefront solutions for the prey predator model on the whole real line without a free boundary

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

fsi analysis of liquid filled pipes most reported work on transient fluid structure interaction fsi in liquid filled pipes has been carried out in the time domain when needed information in the frequency domain e g frequency responses has been deduced by discrete fourier transforms in this paper the analysis is undertaken directly in the frequency domain and has the advantage of enabling linear dispersive terms to be included in a fully coupled manner in principle time domain results e g pressure histories can be obtained by numerical inverse laplace transforms in both domains the mathematical model has one pair of equations for each mode of wave propagation e g pressure waves in the liquid flexural waves in the pipe axial fsi coupling exists in the equations and also in boundary conditions the development used herein highlights common features between analysis in the frequency domain and analysis by the method of characteristics moc in the time domain a general formulation not restricted to pipe systems is presented the method is validated by comparison with an alternative exact analytical solution with results obtained by discrete fourier transform from an moc analysis and by comparison with measured data from a laboratory apparatus

nondispersive radial solutions to energy supercritical non linear wave equations with applications in this paper we establish optimal pointwise decay estimates for non dispersive compact radial solutions to non linear wave equations in 3 dimensions in the energy supercritical range as an application we show for the full energy supercritical range in the defocusing case that if the scale invariant sobolev norm of a radial solution remains bounded in its maximal interval of existence then the solution must exist for all times and scatter

asymptotic dynamic for dipolar quantum gases below the ground state energy threshold abstract we consider the gross pitaevskii equation describing a dipolar bose einstein condensate without external confinement we first consider the unstable regime where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist we prove that under the energy threshold given by the ground state all global in time solutions behave as free waves asymptotically in time the ingredients of the proof are variational characterization of the ground states energy a suitable profile decomposition theorem and localized virial estimates enabling to carry out a concentration compactness and rigidity scheme as a byproduct we show that in the stable regime where standing states do not exist any initial data in the energy space scatters

the defocusing energy supercritical nls in four space dimensions abstract we consider a class of defocusing energy supercritical nonlinear schrodinger equations in four space dimensions following a concentration compactness approach we show that for 1 s c 3 2 any solution that remains bounded in the critical sobolev space h x s c r 4 must be global and scatter key ingredients in the proof include a long time strichartz estimate and a frequency localized interaction morawetz inequality

scattering in h1 for the intercritical nls with an inverse square potential abstract we study the nonlinear schrodinger equation with an inverse square potential in dimensions 3 d 6 we consider both focusing and defocusing nonlinearities in the mass supercritical and energy subcritical regime in the focusing case we prove a scattering blowup dichotomy below the ground state in the defocusing case we prove scattering in h 1 for arbitrary data

asymptotic spreading of a diffusive competition model with different free boundaries abstract recently the authors of 22 studied a diffusive prey predator model with two different free boundaries they first obtained the existence uniqueness regularity uniform estimates and long time behaviors of global solution and then established the conditions for spreading and vanishing especially when spreading occurs they provided accurate limits of two species as t and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries motivated by the paper 22 in this paper we discuss the diffusive competition model with two different free boundaries which had been investigated by 7 11 15 21 the main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs furthermore how the solution approaches the semi wave when spreading happens is also described

non uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment abstract in many cases the movement of species within a region depends on the availability of food and other resources necessary for their survival starvation driven diffusion sdd is a non uniform dispersal strategy that increases the motility of biological organisms in unfavorable environments i e a species moves more frequently in search of food if resources are insufficient 4 in this study the proposed model represents the dispersion of an invasive species undergoing sdd where the free boundary represents the expanding front we observed that the spreading vanishing dichotomy which holds in the linear dispersal model also holds in the model undergoing sdd in the case that spreading of species occurs it is shown that the asymptotic spreading speed of the moving front is uniquely determined in relation to the semi wave speed moreover our results are compared with the results of the linear dispersal model to investigate the advantages of this strategic dispersal with respect to survival in new environments i e the conditions that allowed the species undergoing sdd to spread and the random diffusers to vanish are monitored

dynamics for a diffusive prey predator model with different free boundaries abstract to understand the spreading and interaction of prey and predator in this paper we study the dynamics of the diffusive lotka volterra type prey predator model with different free boundaries these two free boundaries which may intersect each other as time evolves are used to describe the spreading of prey and predator we investigate the existence and uniqueness regularity and uniform estimates and long time behaviors of global solution some sufficient conditions for spreading and vanishing are established when spreading occurs we provide the more accurate limits of u v as t and give some estimates of asymptotic spreading speeds of u v and asymptotic speeds of g h some realistic and significant spreading phenomena are found

global dynamics below the ground state energy for the zakharov system in the 3d radial case abstract we consider the global dynamics below the ground state energy for the zakharov system in the 3d radial case we obtain dichotomy between the scattering and the growup

a unified formulation for continuum mechanics applied to fluid structure interaction in flexible tubes this paper outlines the development of a new procedure for analysing continuum mechanics problems with a particular focus on fluid structure interaction in flexible tubes a review of current methods of fluid structure coupling highlights common limitations of high computational cost and solution instability it is proposed that these limitations can be overcome by an alternative approach in which both fluid and solid components are solved within a single discretized continuum domain a single system of momentum and continuity equations is therefore derived that governs both fluids and solids and which are solved with a single mesh using finite volume discretization schemes the method is validated first by simulating dynamic oscillation of a clamped elastic beam it is then applied to study the case of interest wave propagation in highly flexible tubes in which a predicted wave speed of 8 58m s falls within 2 of an approximate analytical solution the method shows further good agreement with analytical solutions for tubes of increasing rigidity covering a range of wave speeds from those found in arteries to that in the undisturbed fluid

dynamics for the diffusive mutualist model with advection and different free boundaries abstract in this paper we study the diffusive mutualist model with advection and different free boundaries in one space dimension these two free boundaries may intersect each other as time evolves and can be used to describe the spreading of invasive and native species directly we first prove the existence and uniqueness regularity and uniform estimates of global solution then we provide the criteria governing spreading and vanishing at last we investigate long time behaviors and asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries

a free boundary problem for the diffusive intraguild predation model with intraspecific competition abstract in this paper we investigate a free boundary problem for the diffusive intraguild predation model with intraspecific competition in one dimensional space the main objective is to portray the asymptotic behavior of spread of an invasive or new predator species via a free boundary in this model we assume both the ig prey and the ig invasive new predator species with intraspecific competition and the ig predator species can only expand further into the new environment from the right end of the initial region in both cases we prove a spreading vanishing dichotomy for this model specifically the ig predator species either successfully spreads to infinity as t at the front and survives in the new environment or spreads within a bounded area and dies out in the long run the long time behavior of r n p and criteria for spreading and vanishing are also obtained

the diffusive competition problem with a free boundary in heterogeneous time periodic environment abstract in this paper we consider the diffusive competition problem with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment consisting of an invasive species with density u and a native species with density v we assume that v undergoes diffusion and growth in r n and u exists initially in a ball b h 0 0 but invades into the environment with spreading front r h t the effect of the dispersal rate d 1 the initial occupying habitat h 0 the initial density u 0 of invasive species u and the parameter see 1 3 on the dynamics of this free boundary problem are studied a spreading vanishing dichotomy is obtained and some sufficient conditions for the invasive species spreading and vanishing are provided moreover when spreading of u happens some rough estimates of the spreading speed are also given

scattering for h 1 2 bounded solutions to the cubic defocusing nls in 3 dimensions we show that if a solution of the defocusing cubic nls in 3d remains bounded in the homogeneous sobolev norm of order 1 2 in its maximal interval of existence then the interval is infinite and the solution scatters no radial assumption is made

a novel pressure velocity formulation and solution method for fluid structure interaction problems in the standard approach for simulating fluid structure interaction problems the solution of the set of equations for solids provides the three displacement components while the solution of equations for fluids provides the three velocity components and pressure in the present paper a novel reformulation of the elastodynamic equations for hookean solids is proposed so that they contain the same unknowns as the navier stokes equations namely velocities and pressure a separate equation for pressure correction is derived from the constitutive equation of the solid material the system of equations for both media is discretised using the same method finite volume on collocated grids and the same iterative technique simple algorithm is employed for the pressure velocity coupling with this approach the continuity of the velocity field at the interface is automatically satisfied a special pressure correction procedure that enforces the compatibility of stresses at the interface is also developed the new method is employed for the prediction of pressure wave propagation in an elastic tube computations were carried out with different meshes and time steps and compared with available analytic solutions as well as with numerical results obtained using the flugge equations that describe the deformation of thin shells for all cases examined the method showed very good performance

different spreading speeds in a weak competition model with two free boundaries abstract in this paper we focus on the spreading behavior of a two species weak competition system with two free boundaries in a spherically symmetric setting this model may describe how two weakly competing species spread into the right empty space when both two species spread successfully and one species cannot keep pace with the other one it can be expected that two species spread asymptotically in a linear fashion but spread with different speeds by the help of a semi wave system and a construction of supersolution and subsolution we establish the precise spreading speeds and population distribution of two weakly competing species which provides a new spreading behavior

note on a two species competition diffusion model with two free boundaries abstract in guo and wu 2015 and wu 2015 the authors studied a two species competition diffusion model with two free boundaries these two free boundaries describing the spreading fronts of two competing species respectively may intersect each other as time evolves the existence uniqueness and long time behavior of global solution have been established in this note we discuss the conditions for spreading and vanishing and more accurate limits of u v as t when spreading occurs some new results and simpler proofs will be provided