well posedness for the schrodinger korteweg de vries system we study well posedness of the cauchy problem associated to the schrodinger korteweg de vries system we obtain local well posedness for weak initial data where the best result obtained is for data in the sobolev space l 2 r x h 3 4 this result implies in particular the global well posedness in the energy space h 1 r x h 1 r both results considerably improve the previous ones by bekiranov ogawa and ponce 1997 guo and miao 1999 and tsutsumi 1993

on the periodic schrodinger boussinesq system abstract we study the local and global well posedness of the periodic boundary value problem for the nonlinear schrodinger boussinesq system the existence of periodic traveling wave solutions as well as the orbital stability of such solutions are also considered

perturbation theory and the rayleigh quotient the characteristic frequencies of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the rayleigh quotient expressed in terms of the corresponding exact eigenfunctions in problems that can be analytically expanded in a small parameter it is shown that when an approximate eigenfunction is known with an error o n the rayleigh quotient gives the frequency with an error o 2n a gain of n orders this result generalizes a well known theorem for n 1 a non trivial example is presented for n 4 whereby knowledge of the 3rd order eigenfunction error being 4th order gives the eigenvalue with an error that is 8th order the 6th order term thus determined provides an unambiguous derivation of the shear coefficient in timoshenko beam theory

longitudinal vibrations in circular rods a systematic approach a systematic method is developed for expressing the frequency squared 2 and the corresponding displacement fields of harmonic waves in a long thin rod as an even power series in qa where q is the wavenumber along the rod and a is a representative transverse dimension for longitudinal waves in a circular rod the evaluation is reduced to algebraic recursion giving coefficients analytically in terms of poisson s ratio v to many orders the second nontrivial coefficient corresponding to rayleigh love theory in the present longitudinal case and timoshenko theory in the flexural case is thus put on a firm footing without reliance on ad hoc physical assumptions the results are compared to available exact predictions and shown to be accurate for moderate values of qa 5 accuracy for qa 1 5 with just two terms improvements based on the rayleigh quotient guarantee positivity and the correct asymptotic power and the variational principle further ensures that the accuracy improves monotonically with the order of approximation with these features accurate results are obtained for larger qa 5 accuracy for qa 3 so that results are valid for rods that are by no means thin application of these methods to the flexural case has been presented separately

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

symmetry breaking bifurcation of analytic solutions to free boundary problems an application to a model of tumor growth in this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e the technique is worked out primarily for a free boundary problem describing a model of a stationary tumor we prove the existence of infinitely many branches of symmetry breaking solutions which bifurcate from any given radially symmetric steady state these asymmetric solutions are analytic jointly in the spatial variables and in e 1 the model and main result in this paper we present a general technique for establishing analyticity of solutions of systems of partial differential equations which depend analytically on a parameter e the method works not only for boundary value problems but also for free boundary problems in this latter context it can be used to establish long time existence of transient solutions and also to study the existence of spatially asymmetric steady solutions since free boundary problems are typically more challenging than their boundary value counterparts we shall concentrate here on a free boundary problem from developmental biology namely a model of tumor growth to further exemplify the generality of our approach an instance of a boundary value problem in a fixed domain is presented in the last section of the paper a variety of other problems are amenable to the same analysis including in particular the hele shaw model of fluid flow 11 within the last several decades a number of mathematical models have been developed that aimed at describing the evolution of carcinomas see e g 1 5 6 8 12 13 and the references cited there the main objective of these models has been to qualitatively describe under various simplifying assumptions the growth and stability of tumor tissue analysis and simulations of such models are helping to assess the relative importance of various mechanisms affecting tumor growth as well as the efficacy of certain cancer treatments on the other hand the description of the stationary dormant configurations that arise from the models has only been addressed in the case of spherical tumors but otherwise it remains largely unexplored in this paper we develop a method for establishing analyticity of received by the editors august 17 1999 1991 mathematics subject classification primary 35b32 35r35 secondary 35b30 35b60 35c10 35j85 35q80 92c15 95c15

local well posedness for the sixth order boussinesq equation abstract in this work we study the local well posedness of the initial value problem for the nonlinear sixth order boussinesq equation u t t u x x u x x x x u x x x x x x u 2 x x where 1 we prove the local well posedness with initial data in non homogeneous sobolev spaces h s r for negative indices of s r

bilinear space time estimates for linearised kp type equations on the three dimensional torus with applications abstract a bilinear estimate in terms of bourgain spaces associated with a linearised kadomtsev petviashvili type equation on the three dimensional torus is shown as a consequence time localized linear and bilinear space time estimates for this equation are obtained applications to the local and global well posedness of dispersion generalised kp ii equations are discussed especially it is proved that the periodic boundary value problem for the original kp ii equation is locally well posed for data in the anisotropic sobolev spaces h x s h y e t 3 if s 1 2 and e 0

on a check on the accuracy of timoshenko s beam theory in a recent article rneton 1 presented a comparison between the standing wave natural frequency predictions of timoshenko beam theory tbt for a long beam of thin rectangular cross section when the flexural mode is sinusoidal in the axial co ordinate x and those of a plane stress elastodynamic solution the latter may be regarded as the exact benchmark renton s main concern was the accuracy of the lower of the two frequency predictions of tbt tbt1 although the results presented in figure 1 of reference 1 also show the frequency prediction of the so called second spectrum tbt2 and re presents evidence that this mode should be disregarded

the cauchy problem for higher order modified camassa holm equations on the circle abstract in this paper we investigate the cauchy problem for the shallow water type equation u t x 2 n 1 u 1 2 x u 2 x 1 x 2 1 u 2 1 2 u x 2 0 with low regularity data in the periodic settings firstly we proved that the bilinear estimate related to the nonlinear term of the equation in space w s defined in page 5 is invalid with s n 2 1 then the locally well posed of the cauchy problem for the periodic shallow water type equation is obtained in h s t with s n 3 2 n 2 for arbitrary initial data thus our result improves the result of himonas and misiolek commun partial differ equ 23 1998 123 139 where they have proved that the problem is locally well posed for small initial data in h s t with s n 2 1 n n with the aid of the standard fourier restriction norm method

global existence for 2d nonlinear schrodinger equations via high low frequency decomposition method abstract we study global existence of solutions for the cauchy problem of the nonlinear schrodinger equation i u t u u 2 m u in the 2 dimension case where m is a positive integer m 2 using the high low frequency decomposition method we prove that if 10 m 6 10 m 5 s 1 then for any initial value h s r 2 the cauchy problem has a global solution in c r h s r 2 and it can be split into u t e i t y t with y c r h 1 r 2 satisfying y t h 1 1 t 2 1 s 10 m 5 s 10 m 6 where is an arbitrary sufficiently small positive number

an improved local well posedness result for the one dimensional zakharov system abstract the 1d cauchy problem for the zakharov system is shown to be locally well posed for low regularity schrodinger data u 0 h k p and wave data n 0 n 1 h l p h l 1 p under certain assumptions on the parameters k l and 1 p 2 where u 0 h k p k u 0 l p generalizing the results for p 2 by ginibre tsutsumi and velo especially we are able to improve the results from the scaling point of view and also allow suitable k 0 l 1 2 i e data u 0 l 2 and n 0 n 1 h 1 2 h 3 2 which was excluded in the case p 2

a new method to determine the shear coefficient of timoshenko beam theory abstract the frequencies of flexural vibrations in a uniform beam of arbitrary cross section and length l are analysed by expanding the exact elastodynamics equations in powers of the wavenumber q m l where m is the mode number 2 a 4 q 4 a 6 q 6 the coefficients a 4 and a 6 are obtained without further assumptions the former captures euler bernoulli theory while the latter when compared with timoshenko beam theory rendered into the same form unambiguously yields the shear coefficient for any cross section the result agrees with the consensus best values in the literature and provides a derivation of that does not rely on physical assumptions

a free boundary problem for a singular system of differential equations an application to a model of tumor growth first published in transactions of the american mathematical society in volume 355 issue 9 published by the american mathematical society

asymptotic smoothing effect for a weakly damped nonlinear schrodinger equation in t2 abstract we prove that the global attractor for a weakly damped two dimensional nonlinear schrodinger equation in the usual energy space is in fact included and compact in a more regular energy space the method relies on a suitable approximation when time goes to infinity of the high frequency modes of the solutions

sharp local well posedness for the good boussinesq equation abstract in the present article we prove the sharp local well posedness and ill posedness results for the good boussinesq equation on 1d torus the initial value problem is locally well posed in h 1 2 t and ill posed in h s t for s 1 2 well posedness result is obtained from reduction of the problem into a quadratic nonlinear schrodinger equation and the contraction argument in suitably modified x s b spaces the proof of the crucial bilinear estimates in these spaces especially in the lowest regularity rely on some bilinear estimates for periodic functions in x s b spaces which are generalization of the bilinear refinement of the l 4 strichartz estimate on r our result improves the known local well posedness in h s t with s 3 8 given by oh stefanov 2012 similar ideas also establish the sharp local well posedness in h 1 2 r and ill posedness below h 1 2 for the nonperiodic case which improves the result of tsugawa and the author 2010 in h s r with s 1 2

well posedness for the cauchy problem associated to a periodic boussinesq equation abstract we consider the initial value problem associated to the nonlinear periodic sixth order boussinesq equation for given initial data in sobolev spaces h s t the local well posedness of this equation is established for s 1 2

low regularity solutions for a 2d quadratic nonlinear schrodinger equation we establish that the initial value problem for the quadratic nonlinear schrodinger equation iu t u u 2 where u r 2 x r c is locally well posed in h s r 2 when s 1 the critical exponent for this problem is s c 1 and previous work by colliander delort kenig and staffilani 2001 established local well posedness for s 3 4

low regularity solutions of non homogeneous boundary value problems of a higher order boussinesq equation in a quarter plane abstract in this article we study an initial boundary value problem of the sixth order boussinesq equation on a half line with nonhomogeneous boundary conditions u t t u x x u x x x x u x x x x x x u 2 x x 0 x 0 t 0 u x 0 x u t x 0 x x 0 u 0 t h 1 t u x x 0 t h 2 t u x x x x 0 t h 3 t t 0 where 1 it is shown that the problem is locally well posed in c r l o c h s r with initial condition h s r h s 1 r and boundary condition h 1 h 2 h 3 in the product space h s 1 3 r h s 1 3 r h s 3 3 r for 1 2 s 0

well posedness for a quadratic derivative nonlinear schrodinger system at the critical regularity abstract we consider the cauchy problem for a quadratic derivative nonlinear schrodinger system in sobolev space h s in this paper under the null condition and the mass resonance relation we prove large data local well posedness if the space dimension d 3 and s s c d 2 and s s c or d 1 and s 0 where s c is the scaling critical regularity moreover under the same assumptions we also prove small data global well posedness and scattering in h s if d 3 and s s c our proof is based on a contraction argument using the u p and v p type function spaces

on the existence of a maximizer for the strichartz inequality it is shown that a maximizing function u l 2 does exist for the strichartz inequality e it x 2 u l 6 t l 6 x s u l 2 with s 0 being the sharp constant

lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth abstract in this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth we first establish a general result for differential equations in banach spaces possessing a lie group action which maps a solution into new solutions we prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either d a or d a type by using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem we prove that if the surface tension coefficient is larger than a threshold value then the unique stationary solution is asymptotically stable modulo translations provided the constant c is sufficiently small whereas if then this stationary solution is unstable

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

bilinear estimates and applications to 2d nls the three bilinearities uv uv uv for functions u v r2 0 t 7 c are sharply estimated in function spaces xs b associated to the schrodinger operator i t these bilinear estimates imply local wellposedness results for schrodinger equations with quadratic nonlinearity improved bounds on the growth of spatial sobolev norms of finite energy global in time and blow up solutions of the cubic nonlinear schrodinger equation and certain generalizations are also obtained

the cauchy problem for nonlinear quadratic interactions of the schrodinger type in one dimensional space in this work we study the well posedness of the cauchy problem associated with the coupled schrodinger equations with quadratic nonlinearities which appears modeling problems in nonlinear optics we obtain the local well posedness for data in sobolev spaces with low regularity to obtain the local theory we prove new bilinear estimates for the coupling terms of the system in the continuous case concerning global results in the continuous case we establish the global well posedness in hs r hs r for some negatives indexes s the proof of our global result uses the i method introduced by colliander et al

on schrodinger maps we study the question of well posedness of the cauchy problem for schrodinger maps from 2 to the sphere 2 or to 2 the hyperbolic space the idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear schrodinger system of equations and then study this modified schrodinger map system msm we then prove local well posedness of the cauchy problem for the msm with minimal regularity assumptions on the data and outline a method to derive well posedness of the schrodinger map itself from it in proving well posedness of the msm the heart of the matter is resolved by considering truly quatrilinear forms of weighted l2 functions 2002 wiley periodicals inc

analysis of a mathematical model of the growth of necrotic tumors abstract in this paper we study a model of necrotic tumor growth the tumor comprises necrotic cells which occupy a radially symmetric core and life proliferating cells which occupy a radially symmetric shell adjacent to the core the proliferating cells receive nutrients through diffusion from the outer boundary as well as by means of blood flow through a network of capillary vessels the mathematical model describes the evolution of the nutrient concentration between the boundary of the necrotic core r t and the outer boundary of the tumor r r t within the core itself the concentration is a constant nec a level under which life cells cannot be sustained both of the surfaces r t and r r t are free boundaries which are unknown in advance under some assumptions on the parameters we prove that i there exists a stationary solution with radii r s r r s ii for any initial data near the stationary solution the time dependent model has a unique solution r t with free boundaries r t r r t and iii t s and r t r s as t

some non existence results for the semilinear schrodinger equation without gauge invariance we consider the cauchy problem for the semilinear schrodinger equation nls i t u u p t x 0 t rd u 0 x f x x rd where u u t x is a c valued unknown function c 0 p 1 0 f f x is a c valued given function and t is a maximal existence time of the solution our first aim in the present paper is to prove a large data blow up result for nls in hs critical or hs subcritical case p ps 1 4 d 2s for some s 0 more precisely we show that in the case 1 0 such that for any 0 the following estimates t c and limt t 0 u t hs if1 p ps u ltr 0 t b 2s ifp ps hold where c 0 are constants independent of and r is an admissible pair see theorem 2 3 our second aim is to prove non existence local weak solution for nls in the hs supercritical case p ps which means that if p ps then there exists an hs function f such that if there exist t 0 and a weak solution u to nls on 0 t then 0 see theorem 2 5

low regularity global well posedness for the zakharov and klein gordon schrodinger systems we prove low regularity global well posedness for the id zakharov system and the 3d klein gordon schrodinger system which are systems in two variables u r d x x r t c and n r d x x r t r the zakharov system is known to be locally well posed in u n l 2 x h and the klein gordon schrodinger system is known to be locally well posed in u n l 2 x l 2 here we show that the zakharov and klein gordon schrodinger systems are globally well posed in these spaces respectively by using an available conservation law for the l 2 norm of u and controlling the growth of n via the estimates in the local theory

improved local well posedness for the periodic good boussinesq equation abstract we prove that the good boussinesq model is locally well posed in the space h h 2 3 8 in the proof we employ the method of normal forms which allows us to explicitly extract the rougher part of the solution while we show that the remainder is in the smoother space c 0 t h t min 1 3 1 2 thus we establish a smoothing effect of order min 1 2 1 2 for the nonlinear evolution this is new even in the previously considered cases 0 1 4

on eigenvalues of the linearization of a free boundary problem modeling two phase tumor growth abstract in this paper we study eigenvalues of the linearization of a free boundary problem modeling the growth of a tumor containing two species of cells proliferating cells and quiescent cells such eigenvalues are potential bifurcation points from which nonradial solutions of the free boundary problem might bifurcate from the radial solution a special feature of this problem is that it contains a singular ordinary differential equation which causes the main difficulty of this problem by using the spherical harmonic expansion method combined with some techniques for solving singular differential integral equations developed in some previous literature eigenvalues of the linearized problem are completely determined invertibility of some linear operators related to the linearized problem in suitable function spaces is also studied which might be useful in the analysis of the original free boundary problem

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 the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

critical nonlinear schrodinger equations with data in homogeneous weighted l2 spaces abstract we consider the initial value problem for nonlinear schrodinger equations with the critical nonlinearities 1 u 2 n u where im 1 0 when the space dimension n 1 2 we prove the global existence of small solutions in homogeneous weighted l 2 r n spaces it is shown that the small solutions decay uniformly like t n 2 for t 1 if im 1 0 the higher uniform time decay rates t n 2 log t n 2 for t 1 are obtained if im 1 0

well posedness and nonlinear smoothing for the good boussinesq equation on the half line abstract in this paper we study the regularity properties of the good boussinesq equation on the half line we obtain local existence uniqueness and continuous dependence on initial data in low regularity spaces moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data obtaining half derivative smoothing of the nonlinear term in some cases our paper improves the result in 17 being the first result that constructs solutions for the initial and boundary value problem of the good boussinesq equation below the l 2 space our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in 20 and 13

a hyperbolic free boundary problem modeling tumor growth asymptotic behavior first published in transactions of the american mathematical society in volume 357 issue 12 published by the american mathematical society

the cauchy problem for the dissipation modified kadomtsev petviashvili equation abstract considered herein is the dissipation modified kadomtsev petviashvili equation in two space dimensional case it is established that the cauchy problem associated to this equation is locally well posed in anisotropic sobolev spaces it is also shown in some sense that this result is sharp in addition the global well posedness for this equation under suitable conditions is proved

ill posedness for a system of quadratic nonlinear schrodinger equations in two dimensions abstract this paper concerns the ill posedness issue for a system of quadratic nonlinear schrodinger equations in two dimensions from previous studies for the large time behavior of the solution to the system one may expect that the critical regularity to show the well posedness depends on the dispersion coefficients to prove the ill posedness we show the failure of the uniform continuity of the data solution map for the mass resonance case and show the norm inflation for other cases

low regularity bilinear estimates for a quadratic nonlinear schrodinger equation abstract in this article we establish the bilinear estimates corresponding to the 1d and 2d nls with a quadratic nonlinearity c 1 u 2 c 2 u 2 which imply the local well posedness of the cauchy problem in h s for s 1 in the 1d case and for s 1 in the 2d case this is a continuation of our study n kishimoto local well posedness for the cauchy problem of the quadratic schrodinger equation with nonlinearity u 2 commun pure appl anal 7 2008 1123 1143 on the 1d nls with nonlinearity c u 2 previous papers by kenig ponce and vega and colliander delort kenig and staffilani established local well posedness for s 3 4 in 1d and in 2d respectively and when the nonlinearity is restricted to c u 2 papers by bejenaru and tao and bejenaru and de silva improved these results to s 1 in 1d and s 1 in 2d the bilinear estimate for 2d also yields an improvement on the growth rate of sobolev norms of finite energy global in time solutions to the 2d cubic nls

on optimal exponential decay properties of solutions to the korteweg de vries equation abstract we consider the cauchy problem associated to the korteweg de vries equation kdv and study the preservation of exponential decay of order 3 2 on the right of the x axis as time evolves more precisely for a solution of the equation which decays at t 0 as e a 0 x 3 2 for x 0 we find an optimal function a t with a 0 a 0 such that the solution decays as e a t x 3 2 for x t 0

quadratic dispersive generalized benjamin ono equation abstract this paper is concerned with quadratic dispersive generalized benjamin ono equation 1 1 1 2 we prove sharp local well posedness in h s s 1 4 with a solution map that is analytic from h s r to c 0 t h s r and the ill posed result in h s r as soon as s s in the sense that the flow map u 0 u t cannot be continuous

a free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids abstract this paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid tumor cells consist of two types of cells proliferating cells and quiescent cells which have different chemotactic responses to an extracellular nutrient supply the model is a free boundary problem for a nonlinear system of reaction diffusion advection equations where the free boundary is the outer boundary of the spheroid the free boundary condition is quite novel due to different velocity of two types of cells the global existence and uniqueness of solutions to the model is proved the proof is based on a fixed point argument together with the l p theory for parabolic equations with the third boundary condition

global solutions of the derivative schrodinger equation in a class of functions analytic in a strip lower bounds on the rate of decrease in time of a uniform radius of spatial analyticity for solutions of the derivative schrodinger equation are derived the bounds depend algebraically on time they are valid as long as the initial datum is of order one in l2 norm and satisfies suitable spatial decay requirements on the real axis

solvability of the cauchy problem of nonlinear beam equations in besov spaces abstract in this paper we study solvability of the cauchy problem of the nonlinear beam equation t 2 u 2 u u p with initial data in besov spaces we prove that for any 1 q the cauchy problem of this equation is locally well posed in the besov spaces b 2 q s p r n and b 2 q s r n where s p n 2 4 p 1 and s s p and globally well posed in these spaces if initial data are small moreover we obtain scattering results in b 2 q s p r n

quadratic forms for the 1 d semilinear schrodinger equation this paper is concerned with 1 d quadratic semilinear schr6dinger equations we study local well posedness in classical sobolev space hs of the associated initial value problem and periodic boundary value problem our main interest is to obtain the lowest value of s which guarantees the desired local well posedness result we prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies