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

persistence property in a two species chemotaxis system with two signals this paper deals with a two species chemotaxis system with two different signals under homogeneous neumann boundary conditions in a bounded convex domain with the non negative initial data this system is a generalization of the classical keller segel chemotaxis models to the case of two species which are attracted by two different chemical signals under suitable conditions it is proved that for any non negative global classical solutions the masses of two species do not extinct at any time

solving the fifth order caudrey dodd gibbon cdg equation using the exp function method abstract in this work the exp function method is applied to search for a generalized solitary solution of the fifth order caudrey dodd gibbon cdg equation the obtained result includes naturally those in open literature as special cases the work highlights the significant features of the exp function method

analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method abstract the tanh method is used to derive explicit travelling wave solutions for two fifth order integrable nonlinear evolution equations the fifth order kdv equation fkdv and the fifth order caudrey dodd gibbon cdg equation are used as vehicles for our analysis new families of solitary wave solutions and periodic solutions are formally obtained for these equations the presented framework reveals quite a number of useful features of the method used here

the variational iteration method for a reliable treatment of the linear and the nonlinear goursat problem abstract the linear and the nonlinear goursat problems are investigated the he s variational iteration method vim is used to construct exact solutions to these problems the linear and the nonlinear structures are handled in a like manner without any need to restrictive assumptions the accuracy level of the obtained results reveal the power of this method over existing numerical methods

a new result for global existence and boundedness of solutions to a parabolic parabolic keller segel system with logistic source abstract we consider the following fully parabolic keller segel system with logistic source ks u t u u v a u u 2 x t 0 v t v v u x t 0 over a bounded domain r n n 1 with smooth boundary the parameters a r 0 0 it is proved that if 0 then ks admits a global weak solution while if n 2 n c n 2 1 1 n 2 1 then ks possesses a global classical solution which is bounded where c n 2 1 1 n 2 1 is a positive constant which is corresponding to the maximal sobolev regularity apart from this we also show that if a 0 and n 2 n c n 2 1 1 n 2 1 then both u t and v t decay to zero with respect to the norm in l as t

generalization of exp function and other standard function methods exp function and other standard function methods for solving nonlinear differential equations are generalized in this paper an analytical criterion determining if a solution can be expressed in a form comprising standard functions is derived new computational algorithm for automatic identification of the structure of the solution is constructed the algorithm provides information if the solution can be expressed as a sum of standard functions a ratio of sums of standard functions or even a more complex algebraic form involving standard functions several examples are used to illustrate the proposed concept

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

boundedness and decay behavior in a higher dimensional quasilinear chemotaxis system with nonlinear logistic source this paper is concerned with a class of quasilinear chemotaxis systems generalizing the prototype 0 1 ut um uv uur x t 0 vt vv u x t 0 u v 0 x t 0 u x 0 u0 x v x 0 v0 x x in a smooth bounded domain rn n2 with parameters m r1 and 0 the pde system in 0 1 is used in mathematical biology to model the mechanism of chemotaxis that is the movement of cells in response to the presence of a chemical signal substance which is in homogeneously distributed in space it is shown that if m 22nif11 n 22r n 2ifn 22rn 2n 1ifr n 22 and the nonnegative initial data u0 v0 c w1 0 then 0 1 possesses at least one global bounded weak solution apart from this it is proved that if 0 then both u t and v t decay to zero with respect to the norm in l as t

blow up in a higher dimensional chemotaxis system despite logistic growth restriction abstract we study radially symmetric solutions of a class of chemotaxis systems generalizing the prototype u t u u v u u x t 0 0 v m t u x t 0 in a ball r n with parameters 0 0 0 and 1 and m t 1 u x t d x it is shown that when n 5 and 3 2 1 2 n 2 then there exist initial data such that the smooth local in time solution of blows up in finite time this indicates that even superlinear growth restrictions may be insufficient to rule out a chemotactic collapse as is known to occur in the corresponding system without any proliferation

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

finite time blowup and global in time unbounded solutions to a parabolic parabolic quasilinear keller segel system in higher dimensions abstract in this paper we consider quasilinear keller segel type systems of two kinds in higher dimensions in the case of a nonlinear diffusion system we prove an optimal with respect to possible nonlinear diffusions generating explosion in finite time of solutions finite time blowup result in the case of a cross diffusion system we give results which are optimal provided one assumes some proper non decay of a nonlinear chemical sensitivity moreover we show that once we do not assume the above mentioned non decay our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross diffusion terms to this end we provide an example interesting by itself of global in time unbounded solutions to the nonlinear cross diffusion keller segel system with chemical sensitivity decaying fast enough in a range of parameters in which there is a finite time blowup result in a corresponding case without nonlinear cross diffusion

the variational iteration method a reliable analytic tool for solving linear and nonlinear wave equations in this work he s variational iteration method vim is used for analytic treatment of the linear and the nonlinear wave equations in bounded and unbounded domains wave like equations are also investigated the method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or adomian polynomials the study highlights the power of the vim method

solitons and discrete eigenfunctions of the recursion operator of non linear evolution equations i the caudrey dodd gibbon sawada kotera equation the solution of the third order isospectral equation of the caudrey dodd gibbon sawada kotera equation cdgske for soliton potential is obtained recursively from the riccati equation derived by iterating once the auto backlund transformation it is then shown that the discrete eigenfunctions of the sixth order recursion operator for this equation can be written in terms of the solutions of the isospectral equation the behaviour of the 1 soliton solution which has certain novel features is studied a sine gordon like equation resembling the double sine gordon equation is derived from the cdgske

the exp function method and generalized solitary solutions the exp function method is applied to a system of nonlinear partial differential equations and generalized solitary solutions with free parameters are obtained the solution procedure is simple with the help of symbolic computation

boundedness in a quasilinear parabolic parabolic keller segel system with logistic source abstract we consider a quasilinear parabolic parabolic keller segel system involving a source term of logistic type 0 1 u t u u u v g u x t 0 t v t v v u x t 0 t with nonnegative initial data under neumann boundary condition in a smooth bounded domain r n n 1 here and are supposed to be smooth positive functions satisfying c 1 s p and c 1 s q s c 2 s q when s s 0 with some s 0 1 and we assume that g is smooth on 0 fulfilling g 0 0 and g s a s s 2 for all s 0 with constants a 0 and 0 within this framework it is proved that whenever q 1 for any sufficiently smooth initial data there exists a unique classical solution which is global in time and bounded our result is independent of p

boundedness of solutions to a quasilinear parabolic parabolic chemotaxis model with nonlinear signal production abstract this work is concerned with a quasilinear parabolic parabolic chemotaxis model with nonlinear signal production u t 1 u u u 1 u 1 v f u v t v v u with nonnegative initial data under homogeneous neumann boundary conditions in a smooth bounded domain where r and 0 the logistic type source term f u satisfies that either f u 0 or f u r u u k with r r 0 and k 1 the global in time existence and uniform in time boundedness of solutions are established under specific parameters conditions which improves the known results

boundedness of solutions to a quasilinear chemotaxis haptotaxis model we study global solutions of a class of chemotaxis haptotaxis systems generalizing the prototype u t u 1 m 1 u u u 1 q 1 v u u 1 p 1 w h u w 0 v v u w t v w in a bounded domain r n n 1 with smooth boundary h u w u 1 u r 1 w with parameters m 1 r 1 and positive constants p q it is shown that either max q 1 p 2 p m max m 2 n r or max q 1 p 2 p m r and b 0 is large enough then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded the results of this paper improve the results of tao and winkler 2014 46 51

global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the parabolic chemotaxis system u t u u v u u 2 x t 0 v t v v u x t 0 in bounded convex domains r n n 1 with smooth boundary it is shown that if the ratio is sufficiently large then the unique nontrivial spatially homogeneous equilibrium given by u v 1 is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data u 0 v 0 such that u 0 0 the above problem possesses a uniquely determined global classical solution u v with u v t 0 u 0 v 0 which satisfies u t 1 l 0 and v t 1 l 0 as t

boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear logistic source abstract this paper deals with a fully parabolic chemotaxis system with nonlinear logistic source 0 1 u t u u v u 1 u r 1 v t v v u under homogeneous neumann boundary conditions in a smooth bounded convex domain r n with parameters 0 r 2 it is shown that if r 2 or r 2 and n 4 then for all sufficiently smooth initial data the associated initial boundary value problem 0 1 possesses a unique global in time classical solution that is bounded in 0 which satisfies lim sup t u t l max t 0 n 4 t 2 r t t r min r 1 2 moreover with the assumption u 0 0 and appropriate growth assumptions the globally asymptotical stability of 1 1 r 1 1 1 r 1 is established

boundedness in a three dimensional chemotaxis fluid system involving tensor valued sensitivity with saturation abstract we consider the chemotaxis fluid system cf n t u n d n n n s x n c c a n b n 2 x t 0 c t u c c c n x t 0 u t p u n g x t x t 0 u 0 x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r 3 where w 1 a 0 and b 0 here g c 1 0 l 0 d n u m 1 s x n c 1 n and the parameter 0 if m max 6 5 1 3 then for all reasonably regular initial data a corresponding initial boundary value problem for cf possesses a globally defined weak solution through the moser type iteration

global weak solutions in a three dimensional keller segel navier stokes system with nonlinear diffusion abstract the coupled quasilinear keller segel navier stokes system ksnf n t u n n m n c x t 0 c t u c c c n x t 0 u t u u p u n x t 0 u 0 x t 0 is considered under neumann boundary conditions for n and c and no slip boundary conditions for u in three dimensional bounded domains r 3 with smooth boundary where m 0 r are given constants w 1 if m 2 then for all reasonably regular initial data a corresponding initial boundary value problem for ksnf possesses a globally defined weak solution

an optimal result for global existence and boundedness in a three dimensional keller segel stokes system with nonlinear diffusion abstract this paper investigates the following keller segel stokes system with nonlinear diffusion k s f n t u n n m n c x t 0 c t u c c c n x t 0 u t p u n x t 0 u 0 x t 0 under homogeneous boundary conditions of neumann type for n and c and of dirichlet type for u in a three dimensional bounded domains r 3 with smooth boundary where w 2 m 0 it is proved that if m 4 3 then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system k s f which in view of the known results for the fluid free system mentioned below see introduction is an optimal restriction on m

nonlocal nonlinear reaction preventing blow up in supercritical case of chemotaxis system abstract this paper is devoted to the analysis of non negative solutions for the chemotaxis model with nonlocal nonlinear source in bounded domain the qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction when the growth factor is stronger than the dampening effect with the help of the nonlocal nonlinear term in the reaction for appropriately chosen exponents and arbitrary initial data the model admits a classical solution which is uniformly bounded moreover when the growth factor has the same order with the dampening effect the nonlocal nonlinear exponents can prevent the chemotactic collapse

the variational iteration method for solving linear and nonlinear systems of pdes in this work the variational iteration method vim is used for analytic treatment of the linear and nonlinear systems of partial differential equations the method reduces the calculation size and overcomes the difficulty of handling nonlinear terms numerical examples are examined to highlight the significant features of the vim method the method shows improvements over existing numerical techniques

on a quasilinear parabolic elliptic chemotaxis system with logistic source abstract this paper deals with a quasilinear parabolic elliptic chemotaxis system with logistic source under homogeneous neumann boundary conditions in a smooth bounded domain for the case of positive diffusion function it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded moreover if the diffusion function is zero at some point or a positive diffusion function and the logistic damping effect is rather mild we proved that the weak solutions are global existence finally it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

coexistence periodic solutions of a doubly nonlinear parabolic system with neumann boundary conditions abstract this paper is concerned with a competitive and cooperative mathematical model for two biological populations which dislike crowding diffuse slowly and live in a common territory under different kind of intra and inter specific interferences the model consists of a system of two doubly nonlinear parabolic equations with nonlocal terms and neumann boundary conditions based on the theory of the leray schauder degree we obtain the coexistence periodic solutions namely the existence of two non trivial non negative periodic solutions representing the densities of the two interacting populations under different intra and inter specific interferences on their natural growth rates

regularizing feynman path integrals using the generalized kontsevich vishik trace a fully regulated definition of feynman s path integral is presented here the proposed re formulation of the path integral coincides with the familiar formulation whenever the path integral is well defined in particular it is consistent with respect to lattice formulations and wick rotations i e it can be used in euclidean and minkowski space time the path integral regularization is introduced through the generalized kontsevich vishik trace that is the extension of the classical trace to fourier integral operators physically we are replacing the time evolution semi group by a holomorphic family of operators such that the corresponding path integrals are well defined in some half space of c the regularized path integral is thus defined through analytic continuation this regularization can be performed by means of stationary phase approximation or computed analytically depending only on the hamiltonian and the observable i e known a priori in either case the computational effort to eval

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

global boundedness in a quasilinear attraction repulsion chemotaxis model with nonlinear sensitivity abstract this paper considers the attraction repulsion chemotaxis model with homogeneous neumann boundary conditions in a smooth bounded convex domain in this model when the scaling constant is zero and the chemotactic sensitivity functions are nonlinear we prove that this system possesses a unique global classical solution that is uniformly bounded under some assumptions when the scaling constant is one and one of the chemotactic sensitivity functions is nonlinear we also obtain a unique bounded global classical solution under some assumptions

large time behavior of solution to a fully parabolic chemotaxis haptotaxis model in higher dimensions abstract this paper deals with the chemotaxis haptotaxis model of cancer invasion u t u u v u w u 1 u w x t 0 v t v v u x t 0 w t v w x t 0 in a bounded smooth domain r n with zero flux boundary conditions where and are positive parameters it is shown that if is suitably large then for all sufficiently smooth initial data the associated initial boundary value problem possesses a unique global in time classical solution that is bounded in 0 and if the initial data w 0 is small w becomes asymptotically negligible moreover we prove that when domain is convex 1 1 0 is globally asymptotically stable provided that u 0 0 and thereby extends the result of hillen et al 2013 18 to the higher space dimensions

a new result for global solvability and boundedness in the n dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant abstract we consider the following chemotaxis model 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 on a bounded domain r n n 1 with smooth boundary and are positive constants here d is supposed to be smooth positive function satisfying d u c d u 1 m 1 for all u 0 with some c d m 0 besides appropriate smoothness assumptions in this paper it is only required that m 1 2 1 max 1 s 2 0 s v 0 l 2 3 if n 2 1 if n 3 then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded where 0 is a positive constant which is corresponding to the maximal sobolev regularity the results of this paper extends the results of jin j differential equations 263 9 2017 5759 5772 who proved the possibility of boundness of weak solutions in the case m 1 and n 3 to the best of our knowledge this is the first result which gives the relationship between m and that yields to the boundedness of the solution

global existence and asymptotic behavior of the fractional chemotaxis system with signal dependent sensitivity abstract in this paper we study a fractional chemotaxis model with signal dependent sensitivity and the source term which involves two parabolic equations with fractional diffusions we develop a suitable mathematical framework for a unified treatment of the existence and decay estimates based on an appropriate functional space we show the existence uniqueness and temporal decay of the global classical solutions to the problem simultaneously under the smallness initial assumptions

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

solitary solutions periodic solutions and compacton like solutions using the exp function method in this paper the exp function method is used for seeking solitary solutions periodic solutions and compacton like solutions of nonlinear differential equations the combined kdv mkdv equation and the liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method

exact solutions of the drinfel d sokolov wilson equation using the exp function method abstract the generalized solitary solutions of the classical drinfel d sokolov wilson equation dswe are obtained using the exp function method then some of these solutions are easily converted into kink shaped solutions and blow up solutions by a simple transformation

multiple soliton solutions for the fifth order caudrey dodd gibbon cdg equation abstract in this work we use the hirota s direct method combined with the simplified hereman method to determine the n soliton solutions for the fifth order caudrey dodd gibbon cdg equation the one soliton solution will be handled by the tanh coth method the work highlights the significant features of the employed methods

global existence and asymptotic behavior of classical solutions to a parabolic elliptic chemotaxis system with logistic source on rn in the current paper we consider the following parabolic elliptic semilinear keller segel model on rnrn ut u u v au bu2 x rn t 0 0 i v u x rn t 0 where 0 a 0 b 0 are constant real numbers and n is a positive integer we first prove the local existence and uniqueness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 with u x 0 u0 u0 x u x 0 u0 u0 x for various initial functions u0 x u0 x next under some conditions on the constants a b a b and the dimension n we prove the global existence and boundedness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 for given initial functions u0 x u0 x finally we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions under some conditions on the constants a b a b and the dimension n we show that for every strictly positive initial function u0 u0 limt supx rn u x t u0 ab v x t u0 ab 0 and that for every nonnegative initial function u0 u0 with non empty and compact support supp u0 supp u0 there are 0 clow u0 cup u0 such that limt sup x ct u x t u0 ab v x t u0 ab 0 0 c clow u0 and limt sup x ct u x t u0 v x t u0 0 c cup u0

boundedness in the higher dimensional quasilinear chemotaxis growth system with indirect attractant production abstract this paper deals with the following quasilinear chemotaxis growth system u t d u u u v u 1 u x t 0 v t v v w x t 0 w t w u x t 0 in a smoothly bounded domain r n n 3 under zero flux boundary conditions the parameters and are positive and the diffusion function d u is supposed to generalize the prototype d u d 0 u with d 0 0 and r under the assumption 1 4 n it is proved that whenever 0 0 and 0 for any given nonnegative and suitably smooth initial data u 0 v 0 w 0 satisfying u 0 0 the corresponding initial boundary problem possesses a unique global solution which is uniformly in time bounded the novelty of the paper is that we use the boundedness of the v t w 1 s with s 1 2 n n 2 to estimate the boundedness of v t l 2 q q 1 moreover the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by hu et al 2016 under the condition that d u 1 and n 3

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

boundedness in quasilinear keller segel systems of parabolic parabolic type on non convex bounded domains abstract this paper deals with the quasilinear fully parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary n n the diffusivity d u is assumed to satisfy some further technical conditions such as algebraic growth and d 0 0 which says that the diffusion is allowed to be not only non degenerate but also degenerate the global in time existence and uniform in time boundedness of solutions are established under the subcritical condition that s u d u k u e for u 0 with 2 n k 0 and e 0 when d 0 0 this paper represents an improvement of tao and winkler 17 because the domain does not necessarily need to be convex in this paper in the case r n and d 0 0 uniform in time boundedness is an open problem left in a previous paper 7 this paper also gives an answer to it in bounded domains

new abundant solutions for the burgers equation this paper applies a new modified exp function method to search for exact solitary solutions of the burgers equation using the imaginary transformation suggested by he and wu solitary solutions can be easily converted into periodic or compacton like solutions many new solutions are appearing for the first time in the open literature

boundedness of solutions to a quasilinear parabolic parabolic keller segel system with a logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u u u v a u b u r x t 0 v t v v u x t 0 in a bounded domain r n n 1 with smooth boundary u u 1 u u u 1 1 and the parameters r 1 a 0 b 0 and r it is proved that if 0 max r 1 2 n or b is big enough if r 1 then the classical solutions to the above system are uniformly in time bounded our results improve the results of wang et al 2014 28 and cao 2014 3 and also enlarge the range of the results of tao and winkler 2012 25 and ishida et al 2014 14

preventing blow up in a chemotaxis model abstract this paper deals with a chemotactic model of keller segel type the main feature of the keller segel model is the possibility of blow up of solutions in a finite time to eliminate the possibility of blow up a modified version of keller segel model is introduced the blow up control relies on the presence of a pressure function which increases faster than a logarithm for high enough cells densities for such a pressure function the solutions cannot blow up in a finite time some other conditions are introduced to ensure the global boundedness

a study on the bilinear caudrey dodd gibbon equation abstract in this paper we consider the hirota transformation of the caudrey dodd gibbon equation cdge from another point of view as a result the local equivalence between the cdge and its bilinear equation is established and a new type of backlund transformation which is defined by a second order ode along with the appropriate initial values is presented to construct new solutions for the bilinear cdge from the seed solutions of original cdge