bv regularity of the boltzmann equation in non convex domains we consider the boltzmann equation in a general non convex domain with the diffuse boundary condition we establish optimal bv estimates for such solutions our method consists of a new w1 1 trace estimate for the diffuse boundary condition and a delicate construction of an varepsilon tubular neighborhood of the singular set

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

existence and asymptotic behavior of positive ground state solutions for coupled nonlinear fractional kirchhoff type systems abstract this paper is concerned with the following linearly coupled fractional kirchhoff type system a b r 3 2 u 2 d x u u f u v in r 3 c d r 3 2 v 2 d x v v g v u in r 3 u v h r 3 where a c 0 b d 0 are constants 3 4 1 and 0 is a coupling parameter under the general berestycki lions conditions on the nonlinear terms f and g we prove the existence of positive vector ground state solutions of poho aev type for the above system via variational methods moreover the asymptotic behavior of these solutions as 0 is explored as well recent results from the literature are generally improved and extended

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

ground states for kirchhoff equation with hartree type nonlinearities abstract in this paper we study the kirchhoff equation with hartree type nonlinearities a b r 3 u 2 u v x u i u p u p 2 u in r 3 where a b 0 v r 3 r is a potential function and i is a riesz potential of order 0 3 under certain assumptions on v x we prove the equation has a ground state for all 3 3 p 3 by variational methods

nonlinearity estimation in hammerstein systems based on ordered observations the nonlinear subsystem of a hammerstein system is identified i e its characteristic is recovered from input output ohservations of the whole system the input and disturbance are white stochastic processes the identified characteristic satisfies a piecewise lipschitz condition only algorithms presented in the paper are calculated from ordered input output observations i e from pairs of observations arranged in a sequence in which input measurements increase in value the mean integrated square error converges to zero as the number of observations tends to infinity convergence rates are insensitive to the shape of the probability density of the input signal results of numerical simulation are also shown

the boltzmann equation with large amplitude initial data in bounded domains abstract the paper is devoted to constructing the global solutions around global maxwellians to the initial boundary value problem on the boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries we allow a class of non negative initial data which have arbitrary large amplitude and even contain vacuum the result shows that the oscillation of solutions away from global maxwellians becomes small after some positive time provided that they are initially close to each other in l 2 this yields the disappearance of any initial vacuum and the exponential convergence of large amplitude solutions to equilibrium in large time the isothermal diffuse reflection boundary condition plays a vital role in the analysis the most key ingredients in our strategy of the proof include i l x v 2 l x l v 1 l x v estimates along a bootstrap argument ii pointwise estimates on the upper bound of the gain term by the product of l norm and l 2 norm iii an iterative procedure on the nonlinear term

analysis and optimal control of time varying linear systems via block pulse functions succinct and clear sighted operational properties of block pulse functions are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index piecewise constant solutions equally distributed which are simple in form and convenient for use or implementation are consequently obtained another advantage of this method is that any positive integer can be chosen as the number of sub intervals whereas in the case of walsh function approximation the choice can only be made from 2 4 8 16 32 and so on the algorithms developed in the paper are illustrated by appropriate examples

existence and multiplicity results for perturbed kirchhoff type schrodinger systems in r3 abstract in this paper we study a class of kirchhoff type schrodinger systems with a positive parameter e by applying variational methods and the ljusternik schnirelmann category theory under some suitable conditions we obtain the existence and multiplicity results for the systems when the parameter e is sufficiently small

block pulse series analysis of linear systems incorporating observers the paper considers the analysis of linear systems incorporating observers via block pulse functions and presents recursive algorithms convenient for computers compared with walsh function approaches the results are much simpler in form an example is given to illustrate the use of the method

analysis and parameter estimation of bilinear systems via block pulse functions problems of analysis and parameter estimation of bilinear systems via block pulse functions are discussed and recursive algorithms convenient for computers are presented compared with walsh function approaches the results are much simpler in form in this method any positive integer can be chosen as the number of sub intervals consequently much computing time and storage can be saved and the recursive algorithms developed also make it possible to identify on line the parameters of bilinear systems

asymptotic stability of the boltzmann equation with maxwell boundary conditions abstract in a general c 1 domain we study the perturbative cauchy theory for the boltzmann equation with maxwell boundary conditions with an accommodation coefficient in 2 3 1 and discuss this threshold we consider polynomial or stretched exponential weights m v and prove existence uniqueness and exponential trend to equilibrium around a global maxwellian in l x v m of important note is the fact that the methods do not involve contradiction arguments

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

infinitely many radial solutions for kirchhoff type problems in rn abstract in this paper we concern with a class of kirchhoff type problems in r n by using the fountain theorem we obtain three existence results of infinitely many radial solutions for the problem

identification of a class of nonlinear continuous time systems using hartley modulating functions most of the existing approaches to identification of nonlinear dynamic systems involve matching a given input output behaviour with empirical discrete time approximations such as artificial neural networks kolmogorov gabor polynomials radial basis function networks etc techniques for dealing with physically based continuous time models are either applicable to only a restricted class of systems or are computationally very demanding in this paper a new methodology is presented that is applicable to a large class of nonlinear continuous time systems by defining a set of hartley modulating functions for characterizing the continuous process signals the advantages of this new class of modulating functions are that a set of algebraic equations with real coefficients results the formulations are free from boundary conditions and the computations can be made using fast algorithms for the discrete hartley transformation the resulting estimation scheme is applied to different categories of nonlinear syst

stochastic approximation in nonparametric identification of hammerstein systems derived from the idea of stochastic approximation recursive algorithms to identify a hammerstein system are presented two of them recover the characteristic of the nonlinear memoryless subsystem while the third one estimates the impulse response of the linear dynamic part the a priori information about both subsystems is nonparametric consistency in quadratic mean is shown and the convergence rate is examined results of numerical simulation are also presented

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

the boltzmann equation with weakly inhomogeneous data in bounded domain this paper is concerned with the boltzmann equation with specular reflection boundary condition we construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum the result extends the case of cauchy problem considered by arkeryd esposito pulvirenti 4 to the specular reflection boundary value problem in bounded domain

kirchhoff equations with hardy littlewood sobolev critical nonlinearity abstract we consider the following kirchhoff choquard equation m u l 2 2 u f x u q 2 u u y 2 x y d y u 2 2 u in u 0 on where is a bounded domain in r n n 3 with c 2 boundary 2 2 n n 2 1 q 2 and f is a continuous real valued sign changing function when 1 q 2 using the method of nehari manifold and concentration compactness lemma we prove the existence and multiplicity of positive solutions of the above problem we also prove the existence of a positive solution when q 2 using the mountain pass lemma

fractional kirchhoff type equation with hardy littlewood sobolev critical exponent abstract in this paper we consider the following fractional kirchhoff type equation a b r n r n u x u y 2 x y n 2 s d x d y s u i u 2 h u 2 h 2 u in r n where n 3 s 0 1 a 0 b 0 0 n and 2 h n n 2 s is the hardy littlewood sobolev critical exponent and i is the riesz potential we establish the existence nonexistence and multiplicity of nontrivial solutions to above equation

existence and concentration of positive solutions for kirchhoff type problems with a steep well potential abstract the existence and concentration of positive solutions for a class of kirchhoff type problems with a nonnegative steep well potential in r 3 are obtained via variational methods

existence and multiplicity of solutions for nonlinear schrodinger equations with magnetic field and hartree type nonlinearities abstract we study the existence and multiplicity of solutions of the following nonlinear schrodinger equation in the presence of magnetic field i e a x 2 u v x u k x u p u p 2 u u h 1 r n where v r n r is the external potential k r n r is the response function and a a 1 a n r n r n is the magnetic vector potential associated to an external magnetic field b that is b c u r l a under suitable assumptions on the functions v x k x and a x if the parameter e is small enough we can prove the multiplicity of bound states for the equation by variational methods

gradient divergence of fluid dynamic quantities in rarefied gases on smooth boundaries the behavior of fluid dynamic or macroscopic quantities of rarefied gases is studied with a special interest in its non analytic feature near boundaries it is shown that their gradients normal to the boundary diverge even if the boundary is smooth irrespective of the value of the nonzero knudsen number the boundary geometry determines the diverging rate on a planar or concave boundary the logarithmic divergence ln s should be observed where s is the normal distance from the boundary in other cases the diverging rate is enhanced to be the inverse power s 1 n where n ge 2 is the degree of the dominant terms of the polynomial which locally represents the boundary some numerical demonstrations are given as well

n kirchhoff choquard equations with exponential nonlinearity abstract this article deals with the study of the following kirchhoff equation with exponential nonlinearity of choquard type see k c below we use the variational method in the light of moser trudinger inequality to show the existence of weak solutions to k c moreover analyzing the fibering maps and minimizing the energy functional over suitable subsets of the nehari manifold we prove existence and multiplicity of weak solutions to convex concave problem p m below

brief parameter identification of a class of hammerstein plants we consider parameter identification of plants that can be described by a piecewise affine nonlinear element in series with linear dynamics the identification scheme is based on a suitable plant parametrization obtained by an orthogonal expansion of the nonlinearity then a persistently exciting pe input sequence is generated to ensure the convergence of the estimated model to the true one

existence and multiplicity of non trivial solutions for schrodinger kirchhoff type equations with radial potential abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u q x f u in r n u x 0 as x is studied four existence theorems of nontrivial solutions and a sequence of high energy solutions for problem p are obtained by the mountain pass theorem and symmetric mountain pass theorem

a pmgop method of parameter identification for a class of continuous nonlinear systems a series of piecewise multiple general orthogonal polynomials pmgops is introduced and applied to the parameter identication problem for a class of continuous nonlinear systems after introducing pmgops and discussing their main properties an effective procedure for the parameter identification of a large class of continuous systems called parameter separable systems is proposed some relevant algorithms are presented the procedure given in the paper has the following advantages compared with other methods the identification algorithm ia obtained is computationally fast and accurate the ia can be implemented in a recursive fashion the ia is effective for a small number of data points the ia does not require a priori knowledge of the estimated parameters the ia is tolerant to choices of expanding orders when pmgops are applied some numerical examples together with an experimental example taken from a biochemical fermentation process are given for illustrative purposes

the global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball iii the 3 d boltzmann equation abstract this paper is a continuation of the works in 35 and 37 where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases respectively in this paper we are concerned with the global existence and large time behavior of compressible boltzmann gases in an infinitely expanding ball such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time due to the conservation of mass the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball which is easily observed in finite time in the present paper we will confirm this physical phenomenon for the boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function

system identification based on hammerstein model we are considering non linear system identification based on the hammerstein model i e a non linear static gain in series with linear dynamics the static gain characteristic is any non linear function f an identification scheme is designed to get estimates of both the plant dynamics model and a set of n different points x f x where n is arbitrarily chosen by the user such a scheme involves least squares and prediction error algorithms as well as algebraic transformations such as singular values decomposition svd interestingly the proposed scheme ensures persistent excitation allowing thus exact model identification in the case of no external disturbances

regularity of the boltzmann equation in convex domains a basic question about regularity of boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non local mixing of the collision operator consider the boltzmann equation in a strictly convex domain with the specular bounce back and diffuse boundary condition with the aid of a distance function toward the grazing set we construct weighted classical c 1 solutions away from the grazing set for all boundary conditions for the diffuse boundary condition we construct w 1 p solutions for 1 p 2 and weighted w 1 p solutions for 2 le p le infty as well

berestycki lions conditions on ground state solutions for kirchhoff type problems with variable potentials based on the strategy introduced by chen and tang adv nonlinear anal 9 496 515 2020 and some new tricks we prove that the nonlinear problem of kirchhoff type a b r3 u 2dx u v x u f u x r3 in h1 r3 admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a simple minimax characterization of the ground state energy our results improve and extend recent theorems in several directions

on fractional regularity methods for a class of nonlocal problems abstract in the past years the phenomenon of fractional regularity has been addressed for a large class of linear and or quasilinear differential operators mostly in terms of certain besov spaces as it turned out for equations governed by the p laplacian in general the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness moreover despite its own interest fractional regularity methods may be used as a tool for the investigation of some partial differential equations which are not usually addressed in this manner thus the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p laplacian this is done by means of explicit a priori estimates regarding lebesgue and nikolskii spaces which are part of the present contribution as a consequence this approach allows a relaxation on some of the standard conditions employed in this class of problems

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

neural network approach for identification of hammerstein systems in this paper four different on line gradient based learning algorithms for training neural network hammerstein models are presented in a unified framework these algorithms namely the backpropagation for series parallel models the backpropagation sensitivity method and truncated backpropagation through time algorithm bptt for parallel models are derived analysed and compared for the truncated bptt it is shown that determination of the number of unfolding time steps necessary to calculate the gradient with an assumed degree of accuracy can be made on the basis of impulse response functions of sensitivity models the algorithms are shown to differ in their computational complexity gradient approximation accuracy and convergence rates numerical examples are also included to compare the performance of the algorithms

existence results for kirchhoff equations with hardy littlewood sobolev critical nonlinearity abstract in this paper we study a class of kirchhoff equations with hardy littlewood sobolev critical nonlinearity in r n more precisely we consider a b r n u 2 d x u a u 2 u i u p u p 2 u i u 2 2 u 2 2 2 u where a 0 b 0 n 3 0 4 n 3 n 4 2 n n p 2 2 n n 2 0 and i is a riesz potential we prove the existence and multiplicity of solutions for the equation by variational methods together with concentration compactness principle

infinitely many solutions for fractional kirchhoff schrodinger poisson systems in this paper we study the existence of infinitely many solutions for a fractional kirchhoff schrodinger poisson system based on variational methods especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by kajikiya for the critical case we obtain infinitely many solutions for the system under certain assumptions the novelties of this article lie in the appearance of the possibly degenerate kirchhoff function and weak assumptions on the nonlinear term which are quite mild in this paper we study the existence of infinitely many solutions for a fractional kirchhoff schrodinger poisson system based on variational methods especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by kajikiya for the critical case we obtain infinitely many solutions for the system under certain assumptions the novelties of this article lie in the appearance of the possibly degenerate kirchhoff function and weak assumptions on the nonlinear term which are quite mild

on a periodic schrodinger equation with nonlocal superlinear part we consider the choquard pekar equation open image in new window and focus on the case of periodic potential v for a large class of even functions w we show existence and multiplicity of solutions essentially the conditions are that 0 is not in the spectrum of the linear part v and that w does not change sign our results carry over to more general nonlinear terms in arbitrary space dimension n 2

sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems abstract this paper is concerned with nonlocal kirchhoff s equation a b u 2 u f x u in u 0 on via variational methods and invariant sets of descent flow we obtain existence of signed and sign changing solutions with asymptotically 3 linear bounded nonlinearity

two positive solutions for the kirchhoff type elliptic problem with critical nonlinearity in high dimension abstract we investigate the kirchhoff type elliptic problem with critical nonlinearity 1 u 2 d x u u q u 2 1 u 0 in u 0 on where n 5 r n is a bounded domain with smooth boundary 0 r 2 2 n n 2 and q 1 2 1 we prove the existence of two solutions of it via the variational method since n 5 and 0 the uniqueness assertion for the associated limiting problem may fail this causes serious difficulties in controlling concentrating palais smale sequences we overcome these by introducing new techniques for a mountain pass type solution we utilize the limit function of the fibering maps of the concentrating palais smale sequence this tool is based on our careful setting of nehari type sets on the other hand a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution this is the first work which proves the multiplicity of positive solutions of the kirchhoff type critical problem in high dimension

analysis and identification of hammerstein model non linear delay systems using block pulse function expansions algorithms for the analysis and identification of non linear delay systems described by a hammerstein model which consists of a single value non linearity followed by a linear time delayed plant are presented by using the finite dimensional block pulse expansion the solution of a hammerstein model non linear dynamic system with time delay is reduced to the solution of a set of linear algebraic equations in addition through the block pulse expansions of the measured input output data parameters of both the linear time delayed plant and the polynomial approximation of the non linear gain are estimated using the least squares method examples are given to illustrate the utility of this method

multiplicity of solutions for p x polyharmonic elliptic kirchhoff equations abstract in this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p x polyharmonic kirchhoff equations including the new delicate degenerate case not yet covered in the literature the main tool is the symmetric mountain pass theorem of ambrosetti and rabinowitz

the boltzmann equation with specular boundary condition in convex domains we establish the global well posedness and stability of the boltzmann equation with the specular reflection boundary condition in general smooth convex domains when an initial datum is close to the maxwellian with or without a small external potential in particular we have completely solved the longstanding open problem after an announcement by shizuta and asano in 1977 2017 wiley periodicals inc

existence and multiplicity of solutions for kirchhoff type equations abstract in this paper the existence and multiplicity of weak solutions are obtained for a class of kirchhoff type problems with dirichlet boundary value conditions by using the mountain pass theorem the local linking theorem the fountain theorem and the symmetric mountain pass lemma in critical point theory

non isothermal boundary in the boltzmann theory and fourier law in the study of the heat transfer in the boltzmann theory the basic problem is to construct solutions to the following steady problem v cdot nabla x f frac 1 rm k rm n q f f qquad x v in omega times mathbf r 3 quad quad 0 1 f x v n x cdot v 0 f x v prime n x cdot v prime dv prime quad x in partial omega quad quad 0 2 where is a bounded domain in mathbf r d 1 leq d leq 3 kn is the knudsen number and mu theta frac 1 2 pi theta 2 x rm exp frac v 2 2 theta x is a maxwellian with non constant non isothermal wall temperature x based on new constructive coercivity estimates for both steady and dynamic cases for theta theta 0 leq delta ll 1 and any fixed value of kn we construct a unique non negative solution fs to 0 1 and 0 2 continuous away from the grazing set and exponentially asymptotically stable this solution is a genuine non equilibrium stationary solution differing from a local equilibrium maxwellian as an application of our results we establish the expansion f s mu theta 0 delta f 1 o delta 2 and we prove that if the fourier law holds the temperature contribution associated to f1 must be linear in the slab geometry

existence and concentration of sign changing solutions to kirchhoff type system with hartree type nonlinearity abstract in this paper we discuss the existence and the concentration of sign changing solutions to a class of kirchhoff type systems with hartree type nonlinearity in r 3 by the minimization argument on the sign changing nehari manifold and a quantitative deformation lemma we prove that the system has a sign changing solution moreover concentration behaviors of sign changing solutions are obtained when the coefficient of the potential function tends to infinity specially our results cover general schrodinger equations kirchhoff equations and schrodinger poisson systems

a note on kirchhoff type equations with hartree type nonlinearities abstract we consider the following kirchhoff type equation in r 3 a b r 3 u 2 d x u 1 g x u 1 x u p u p 2 u where a 0 b 0 are constants 0 3 p 2 6 0 is a parameter and g x is a nonnegative continuous potential satisfying some conditions by using the nehari manifold and the concentration compactness principle we establish the existence of ground state solutions for the equation if the parameter is large enough moreover some concentration behaviors of these solutions as are discussed

positive and sign changing solutions to a nonlinear choquard equation abstract we consider the problem u w x u 1 x u p u p 2 u u h 0 1 where is an exterior domain in r n n 3 0 n p 2 2 n n 2 w c 0 r n inf r n w 0 and w x v 0 as x under symmetry assumptions on and w which allow finite symmetries and some assumptions on the decay of w at infinity we establish the existence of a positive solution and multiple sign changing solutions to this problem having small energy