an algorithmic method for showing existence of nontrivial non classical symmetries of partial differential equations without solving determining equations abstract in this paper based on differential characteristic set theory and the associated algorithm also called wu s method an algorithmic method is presented to decide on the existence of a nontrivial non classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system the theory and algorithm give a partial answer for the open problem posed by p a clarkson and e l mansfield in 21 on non classical symmetries of partial differential equations as applications of our algorithm non classical symmetries and corresponding invariant solutions are found for several evolution equations

chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

nonclassical symmetry reductions of the three dimensional incompressible navier stokes equations the nonclassical reduction method as pioneered by bluman and cole j meth mech 18 1025 42 is used to examine symmetries of the full three dimensional unsteady incompressible navier stokes equations of fluid mechanics the procedure when applied to a system of partial differential equations yields reduced sets of equations with one fewer independent variables we find eight possibilities for reducing the navier stokes equations in the three spatial and one temporal dimensions to sets of partial differential equations in three independent variables some of these reductions are derivable using the lie group method of classical symmetries but the remainder are genuinely nonclassical further investigations of one of our eight forms shows how it is possible to derive novel exact solutions of the navier stokes equations by the nonclassical method

boundedness enforced by mildly saturated conversion in a chemotaxis may nowak model for virus infection abstract we study the system u t u u v u f u w v t v v f u w w t w w v which models the virus dynamics in an early stage of an hiv infection in a smooth bounded domain r n n n for a parameter 0 and a given function f c 1 0 satisfying f 0 f 0 0 and f s k f s for all s 1 some k f 0 and r we prove that whenever 2 n solutions to exist globally and are bounded the proof mainly relies on smoothing estimates for the neumann heat semigroup and in the case 1 on a functional inequality furthermore we provide some indication why the exponent 2 n could be essentially optimal

new classes of solutions for the schwarzian korteweg de vries equation in 2 1 dimensions we have obtained new classes of solutions for the 2 1 dimensional schwarzian korteweg de vries equation by considering several types of reductions of a system equivalent to this equation the first analysis is done by studying the nonclassical reductions of the system further reductions are attained by means of other types of symmetry reductions or by ansatz based reductions most of the new classes of solutions depend on jacobian elliptic functions and solutions of a riemann wave equation including the cnoidal waves solutions the new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena according to the suitable selection of the functions these solutions depend on

the nonclassical group analysis of the heat equation abstract the nonclassical method of reduction was devised originally by bluman and cole in 1969 to find new exact solutions of the heat equation much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance however the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved although particular solutions have been given recently arrigo goard and broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical lie symmetry reduction solutions their arguments leave open the problem of what is the general nonclassical group action and its effect on the relevant solution of the heat equation in this article both these problems are solved in the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of burgers equation

a new look at fresnel field computation using the jacobi bessel series many useful applications exist for the efficient computation of fresnel and near zone fields of large antennas even small antennas in beam waveguide systems must be evaluated in the fresnel zone far zone fields computed from measured near zone measurements can be verified by both the measurement and the computation of the fresnel zone fields the authors start with the premise that the far field has been computed by a jacobi bessel series these results are used then to determine the higher order terms of a barrar kay 1 r p expansion of the fields the leading term of the 1 r p series is the far zone field classically the higher order terms are found by repetitive differentiation a laborious and often inaccurate procedure particularly since the 1 r p series is slowly convergent as d 2 r d is the diameter of antenna source the approach of the authors via the jacobi bessel 1 r p series determines the higher order terms by simple algebraic recursion the only restriction on the method is that it be used within the range of validity of the fresnel small angle fsa approximation however since the fresnel approximation is a second order approximation in terms of d r the range of validity is quite large this is demonstrated in detail the method is applicable to reflector as well as aperture field sources

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

a new result for boundedness in the quasilinear parabolic parabolic keller segel model with logistic source abstract the current paper considers the boundedness of solutions to the following quasilinear keller segel model with logistic source ks u t d u u u v u u 2 x t 0 v t v u v x t 0 d u u u v v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x where r n n 1 is a bounded domain with smooth boundary 0 and 0 one novelty of this paper is that we find a new a priori estimate u max 1 0 max 1 0 e x t d x so that we develop new l p estimate techniques and thereby obtain the boundedness results where c g n and 0 0 are the constants which are corresponding to the gagliardo nirenberg inequality see lemma 2 2 and the maximal sobolev regularity see lemma 2 3 to our best knowledge this seems to be the first rigorous mathematical result which indicates the relationship between m and that yields the boundedness of the solutions where m is the exponent of diffusion term d u the above mentioned results have significantly improved and extended previous results of several authors

reduction operators of burgers equation abstract the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed a new proof of the theorem on the special no go case of regular reduction operators is presented and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described any lie reduction of the burgers equation proves to be equivalent via the hopf cole transformation to a parameterized family of lie reductions of the linear heat equation

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

a parabolic elliptic elliptic attraction repulsion chemotaxis system with logistic source abstract this paper deals with the parabolic elliptic elliptic attraction repulsion chemotaxis system with logistic source u t d u u u v u w r u u 2 x t 0 0 v u v x t 0 0 w u w x t 0 under no flux boundary conditions in bounded domain with smooth boundary where r and are assumed to be positive when r 3 d u is assumed to satisfy d 0 0 d u c d u m 1 with m 1 and c d 0 it is proved that if 0 and 1 3 then for any given u 0 w 1 the system possesses a global and bounded classical solution for the case where d u 1 and n 3 the convergence rate of the solution is established when the random motion of the chemotactic species is neglected i e d u 0 and r n n 2 is a convex domain boundedness and the finite time blow up of the solution are investigated

nonclassical symmetry reductions of a porous medium equation with convection in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method the asymptotic behaviour of some of the new solutions is analysed

global solutions in a quasilinear parabolic parabolic chemotaxis system with decaying diffusivity and consumption of a chemoattractant abstract in this study we consider the quasilinear parabolic parabolic chemotaxis model u t d u u s u v x t 0 v t v u v x t 0 subject to homogeneous neumann boundary conditions where is a convex bounded domain of r n n 2 with smooth boundary the diffusivity sensitivity d s and chemotactic sensitivity s s satisfy k 1 e s d s k 2 e s and s s d s k 3 e s for s 0 with constants k i 0 i 1 2 3 0 and 0 n 1 if the initial data are u 0 c 0 and v 0 w 1 then the classical solutions to this model are globally bounded

classical point symmetries of a porous medium equation the lie group formalism is applied to deduce symmetries of the porous medium equation we study those spatial forms that admit the classical symmetry group the reduction obtained from the optimal system of subalgebras are derived some new exact solutions can be obtained

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

boundedness of weak solutions of a chemotaxis stokes system with slow p laplacian diffusion abstract in this paper a chemotaxis stokes system with slow p laplacian diffusion n t u n n p 2 n n c x t 0 c t u c c n c x t 0 u t u p n x t 0 u 0 x t 0 is considered under homogeneous boundary conditions of neumann type for n and c and of dirichlet type for u where is a smooth bounded domain in r 3 and w 2 is a given function it is proved that global bounded weak solutions exist whenever p 23 11

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

application of an adams type inequality to a two chemical substances chemotaxis system abstract this paper deals with positive solutions of the fully parabolic system u t u u v in 0 1 v t v v w in 0 2 w t w w u in 0 under homogeneous neumann boundary conditions or mixed boundary conditions no flux and dirichlet conditions in a smooth bounded domain r n n 4 with positive parameters 1 2 0 and nonnegative smooth initial data u 0 v 0 w 0 in the lower dimensional case n 3 it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded in the case n 4 it is shown that in the radially symmetric setting solutions to the neumann boundary value problem of the system exist globally in time and remain bounded if u 0 l 1 8 2 as to the mixed boundary value problem we will establish global existence and boundedness of solutions if u 0 l 1 8 2 without radial symmetry the key ingredients are a lyapunov functional and an adams type inequality a lyapunov functional of the above problems will be constructed and the constant 8 2 is deduced from the critical constant in the adams type inequality this result is regarded as a generalization of the well known 8 problem in the keller segel system to higher dimensions

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

boundedness blowup and critical mass phenomenon in competing chemotaxis abstract we consider the following attraction repulsion keller segel system u t u u v u w x t 0 v t v u v x t 0 0 w u w x t 0 u x 0 u 0 x v x 0 v 0 x x with homogeneous neumann boundary conditions in a bounded domain r 2 with smooth boundary the system models the chemotactic interactions between one species denoted by u and two competing chemicals denoted by v and w which has important applications in alzheimer s disease here all parameters and are positive by constructing a lyapunov functional we establish the global existence of uniformly in time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction i e if the attraction dominates i e a critical mass phenomenon is found specifically speaking we find a critical mass m 4 such that the solution exists globally with uniform in time bound if m m and blows up if m m and m 4 m m n where n denotes the set of positive integers and m u 0 d x the initial cell mass

classical symmetries travelling wave solutions and conservation laws of a generalized fornberg whitham equation in this paper we consider a generalized fornberg whitham equation we make an analysis of the symmetries of this equation using the classical lie symmetry method symmetry reductions are derived from an optimal system of subalgebras and lead to ordinary differential equations we obtain travelling wave solutions in addition using the general multiplier method new conservation laws of this equation are determined

approximate potential symmetries for partial differential equations the method of approximate potential symmetries for partial differential equations with a small parameter is introduced by writing a given perturbed partial differential equation r in a conserved form an associated system s with potential variables as additional variables is obtained approximate lie point symmetries admitted by s induce approximate potential symmetries of r as applications of the theory approximate potential symmetries for a perturbed wave equation with variable wave speed and a nonlinear diffusion equation with perturbed convection terms are obtained the corresponding approximate group invariant solutions are also derived

repulsion effects on boundedness in the higher dimensional fully parabolic attraction repulsion chemotaxis system abstract this paper deals with an attraction repulsion chemotaxis system u t d u u u v u w x t 0 1 v t v u v x t 0 2 w t w u w x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 where parameters i i 1 2 and are positive and diffusion coefficient d u c 2 0 satisfies d u 0 for u 0 d u d u m 1 with d 0 and m 1 for all u 0 it is proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m 2 2 n in particular in the case 1 2 and the solution is globally bounded if m 2 2 n n 2 n 2 n 2 therefore due to the inhibition of repulsion to the attraction the range of m 2 2 n of boundedness is enlarged and the results of 21 is thus extended to the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion

global generalized solutions to a keller segel system with nonlinear diffusion and singular sensitivity abstract we consider the chemotaxis system with nonlinear diffusion and singular sensitivity u t u m u v v x t 0 v t v u v x t 0 u v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x in a smooth bounded domain r n n 2 in this work it is shown that for all reasonably regular initial data u 0 0 and v 0 0 the corresponding neumann initial boundary value problem possesses a global generalized solution provided that m 1 n 2 2 n

new similarity reductions of the steady state boundary layer equations new similarity reductions and exact solutions of the steady state boundary layer equations which are of considerable importance in physics are presented besides similarity reductions obtained by applying the non classical group method due to bluman and cole some similarity solutions which figure neither among classical nor among non classical reductions are determined using an extension of the non classical method

potential method applied to boussinesq equation in this paper a solution of the nonlinear boussinesq equation is presented using the potential similarity transformation method the equation is first written in a conserved form a potential function is then assumed reducing it to a system of equations which is further solved through the group transformation method new transformations are found

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

complex polynomial phase integration a previously published integration algorithm applicable to the numerical computation of integrals with rapidly oscillating integrands is generalized the previous algorithm involved quadratic approximation of the phase function which was assumed to be real the present generalization concerns approximation of a complex phase function by a polynomial of arbitrary degree as before the integrand is then written without approximation as a slowly varying function multiplied by the polynomial phase exponential and the slowly varying factor is approximated by a finite sum of chebyshev polynomials the integral is thus expressed as a sum of constituent integrals which are computed recursively via lu decomposition applied to a system of linear equations with a banded coefficients matrix examples are presented comparing various degree phase approximants

algorithms for the nonclassical method of symmetry reductions in this article the authors first present an algorithm for calculating the determining equations associated with so called nonclassical method of symmetry reductions a la bluman and cole for systems of partial differential equations this algorithm requires significantly less computation time than that standardly used and avoids many of the difficulties commonly encountered the proof of correctness of the algorithm is a simple application of the theory of grobner bases in the second part they demonstrate some algorithms which may be used to analyse and often to solve the resulting systems of overdetermined nonlinear pdes the authors take as their principal example a generalised boussinesq equation which arises in shallow water theory although the equation appears to be nonintegrable the authors obtain an exact two soliton solution from a nonclassical reduction

symmetries periodic plane waves and blow up of l ω systems abstract nonclassical symmetries of one dimensional reaction diffusion equations of the type have been studied the functional forms of and for which the system admits nonclassical symmetries have been determined and the corresponding reduced systems have been obtained some of these reduced systems admit symmetries which lead to further reductions among the several classes of exact solutions that have been obtained asymptotically periodic plane waves appear as similarity solutions of systems we also have obtained a family of solutions that exhibit a blow up process

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

symmetry reduction and explicit solutions of the 2 1 dimensional boiti leon pempinelli system abstract through the standard truncated painleve expansion of the boiti leon pempinelli equation the residual symmetry is localized in the properly prolonged system with the lie point symmetry vector some different transformation invariances are derived through the obtained symmetries at the same time the symmetry of the equation is also derived utilizing the clarkson kruskal direct method from which through solving the characteristic equations several types of the explicit reduction solutions that related the hyperbolic tangent function are obtained finally some kink type soliton excitations are depicted from one of them

on certain symmetries of the nonlinear schrodinger equation in this paper we reduce the problem of symmetries for the ginzburg landau equation to a specific problem of non classical symmetries for the linear free schrodinger equation with the help of an immersion result this new problem is related to the local u 1 invariance of the model and allows us to construct projective representations for arbitrary lie groups acting on the space time manifold of the related sigma model to achieve this result we propose an enlargement of the usual notion of non classical symmetry

applications of nonclassical symmetries in this paper we discuss new applications of the nonclassical method of symmetry reduction the equations treated are well known in mathematical physics and are used to describe multisoliton solutions of the burger equation they are also useful in the study of kac moody algebras and in the examination of nonlinear waves in the theory of elasticity with the help of the nonclassical symmetry method the partial differential equations are reduced to ordinary differential equations which are solvable explicitly

self similar blow up for a reaction diffusion system abstract this work is concerned with the following system which is a model to describe several phenomena in which aggregation plays a crucial role as for instance motion of bacteria by chemotaxis and equilibrium of self attracting clusters when the space dimension n is equal to three we show here that s has radial solutions with finite mass that blow up in finite time in a self similar manner when n 2 however no radial solution with finite mass may give rise to self similar blow up

quadratic phase integration using a chebyshev expansion an integration algorithm is described which is particularly effective in the numerical treatment of integrands having rapidly varying phase and slowly varying amplitude the algorithm involves approximating the phase function by a quadratic polynomial and rewriting the integrand without approximation as a slowly varying function multiplied by this quadratic phase exponential the slowly varying function is then approximated by chebyshev expansion and the desired integral is thus expressed as a sum of constituent integrals with integrands containing a chebyshev polynomial multiplied by the quadratic phase factor these constituent integrals are computed by means of lu decomposition applied to a system of linear equations with a banded coefficients matrix example results are presented indicating that a substantial reduction in computation time may be realized by means of this approach

correction to a new integration algorithm and its application to the analysis of symmetrical cassegrain microwave antennas an efficient numerical algorithm for physical optics po analysis of reflectors is presented cylindrical symmetry of the reflector is not essential but is assumed for purposes of illustration and this leads to the use of fourier representation in the azimuthal variable the algorithm is based on quadratic approximation of the phase and polynomial approximation of the remaining portion of the integrand the necessary integrals are generated recursively and they involve the fresnel integral additional efficiency is achieved by means of an interpolation scheme therein one interpolates all but the rapidly varying edge diffraction contribution to the integral and then adds the explicitly calculated edge contribution at each far field point a few illustrative examples are included

the coupled modified korteweg de vries equations similarity reduction lie backlund symmetries and integrability the lie point symmetries of n coupled modified korteweg de vries equations n cmkdv is derived and shown that the similarity reduction associated with symmetries passes the painleve property indicating its integrability the nonclassical symmetry analysis of bluman and cole and the direct method of clarkson and kruskal to n cmkdv equations show that there exists no new similarity reductions a sequence of lie backlund symmetries for n cmkdv equations is derived explicitly establishing its complete integrability the question of constructing the recursion operator of n cmkdv a characteristic of integrable systems is also discussed

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

global existence and boundedness in a keller segel navier stokes system with signal dependent sensitivity abstract in this paper we consider the following keller segel navier stokes system n t u n n n c c x t 0 c t u c c c n x t 0 u t u u u p n x t 0 u 0 x t 0 where r n n 2 3 is a bounded domain with smooth boundary r and c is assumed to generalize the prototype c 0 1 c 2 c 0 it is proved that i for 0 and n 2 or 0 and n 2 3 the corresponding initial boundary problem admits a unique global classical solution which is bounded ii for 0 and n 3 the corresponding initial boundary problem possesses at least one global weak solution

boundedness of solutions to a quasilinear parabolic elliptic keller segel system with logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u 1 m 1 u u u 1 q 1 v a u b u r x t 0 0 v v u x t 0 in a bounded domain r n n 1 with smooth boundary with parameters m 1 r 1 a 0 b q 0 it is shown that when q 1 max r m 2 n or b b 0 n r m 2 r m n 2 r 2 if q 1 r then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded the results improve the results of wang et al 2014 37 and cao and zheng 2014 6

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a patlak keller segel model this paper is concerned with a chemotaxis aggregation model for cells more precisely with a parabolic elliptic semilinear patlak keller segel system in a ball of rn exhibiting a critical mass phenomenon for n 2 the main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case we stress that this covers in particular the classical keller segel system with n 2 well known for its critical mass 8 and that the result improves on the known results even for this most studied problem a key tool is an associated one dimensional degenerate parabolic problem the proof exploits its formal gradient flow structure ut f u t on an infinite dimensional riemannian manifold in particular we show a new hardy type inequality equivalent to the strict convexity of f at any steady state of subcritical mass which heuristically explains the exponential speed of convergence

reflector antenna fields an exact aperture like approach a new computational approach is presented which allows fast analysis of radiation from large reflector antennas for an aperture a fourier transform ft relationship does exist between far field and aperture distribution accordingly the far field can be exactly reconstructed from the knowledge of approximately one sample per lobe shannon whittaker theorem applied at nyquist rate the finite reflector curvature introduces an extra factor in the radiation integral so that the radiation integral is no longer a ft in order to overcome this difficulty a new pseudosampling expansion which explicitly takes into account the extra factor is developed for parabolic reflector the sampling functions are related to the fresnel integrals and the far field can be exactly reconstructed in terms of aperture far field samples which can be computed using the fast fourier transform fft numerical computations and error analysis show the excellent performance of the method which can be generalized to deal with arbitrary reflector surfaces and near field evaluation

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem

global existence and stabilization in a degenerate chemotaxis stokes system with mildly strong diffusion enhancement abstract a class of chemotaxis stokes systems generalizing the prototype n t u n n m 1 n n c c t u c c n c u t p u n u 0 is considered in bounded convex three dimensional domains where w 2 is given the paper develops an analytical approach which consists in a combination of energy based arguments and maximal sobolev regularity theory and which allows for the construction of global bounded weak solutions to an associated initial boundary value problem under the assumption that 0 1 m 9 8 moreover the obtained solutions are shown to approach the spatially homogeneous steady state 1 n 0 0 0 in the large time limit this extends previous results which either relied on different and apparently less significant energy type structures or on completely alternative approaches and thereby exclusively achieved comparable results under hypotheses stronger than 0 1

analytic n solitary wave solution of a variable coefficient gardner equation from fluid dynamics and plasma physics in this paper the investigation is focused on a variable coefficient gardner equation with quadric and cubic nonlinearities from fluid dynamics and plasma physics using the hirota bilinear method the one two and three solitary wave solutions of the variable coefficient gardner equation are derived and the analytic n solitary wave solution is presented for the first time in this paper with the aid of symbolic computation figures are plotted to illustrate the solutions obtained in this paper

eventual smoothness of generalized solutions to a singular chemotaxis stokes system in 2d abstract we study the chemotaxis fluid system n t u n n n c c x t 0 c t u c c n c x t 0 u t p u n x t 0 u 0 x t 0 under homogeneous neumann boundary conditions for n and c and homogeneous dirichlet boundary conditions for u where r 2 is a bounded domain with smooth boundary and c 2 from recent results it is known that for suitable regular initial data the corresponding initial boundary value problem possesses a global generalized solution we will show that for small initial mass n 0 these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties moreover from the analysis of certain energy type inequalities arising during the investigation of the eventual regularity we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of n 0 in l 1 and in l log l u 0 in l 4 and of c 0 in l 2

simplified technique for evaluating the radiation integrals a simplified numerical method is described for computing the radiation integral of a reflector antenna which is associated with either the aperture field method afm or the induced current method icm the method involves linear approximation of the phase function and absorption of the approximation error into amplitude function then the integrand is rewritten as a slowly varying function multiplied by the phase exponential the slowly varying function is expanded in terms of a polynomial for the radial direction and trigonometric functions for the circumferential direction expansion coefficients are determined by matching at points on the aperture the matching points are chosen in a manner similar to that of a two dimensional numerical integration scheme

some traveling wave solutions for the dissipative zabolotskaya khokhlov equation by using the theory of nonclassical symmetry reductions some traveling wave solutions of the dissipative zabolotskaya khokhlov equation are obtained in particular we derive some singular rational traveling wave solutions describing coherent structures traveling with constant velocity