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

global well posedness for the 2d incompressible four component chemotaxis navier stokes equations abstract the four component chemotaxis navier stokes system n t u n n n c n m c t u c c c m m t u m m n m u t u u p u n m u 0 is considered in r 2 by using fourier localization technique and the structure of equations we obtain the existence and uniqueness of weak solutions for the above system for a large class of initial data

an rqp algorithm using a differentiable exact penalty function for inequality constrained problems in this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function the algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an armijo type line search procedure that avoids the need to evaluate second order derivatives of the problem functions we prove that the algorithm possesses global and superlinear convergence properties numerical results are reported

global existence and asymptotic behavior of classical solutions to a fractional logistic keller segel system abstract in this paper we consider a fractional parabolic elliptic keller segel system with a logistic source on r n first we establish the regularity of weak solutions of the fractional parabolic equation using blow up arguments combined with liouville type theorems next by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions moreover the global existence and boundedness of classical solutions for given initial data are obtained under some conditions finally we show the asymptotic behavior of the global solutions with strictly positive initial data

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic type source abstract we consider the fully parabolic keller segel system with singular sensitivity and logistic type source u t u u v v r u u k v t v v u under the non flux boundary conditions in a smooth bounded convex domain r n r 0 k 1 a global very weak solution for the system with n 2 is obtained under one of the following conditions i r 2 4 for 0 2 or r max 2 4 1 p 0 2 1 for 2 with p 0 4 k 1 4 2 k k 2 if k 2 1 n 2 ii 2 min 2 r r 2 k 4 k k 1 k 2 if k 2 furthermore this global very weak solution should be globally bounded in fact provided r and the initial data u 0 l 2 v l 4 suitably small for n 2 3 in addition if k 3 n 2 n 4 replaces k 2 in the condition ii the system admits globally bounded classical solutions all these describe the influence of the exponent k 1 in the logistic type source r u u k to the behavior of solutions for the considered fully parabolic keller segel system with singular sensitivity

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

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

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

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

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 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 existence and asymptotic behavior of solutions to a two species chemotaxis system with two chemicals in this paper we investigate the competitive parabolic elliptic parabolic elliptic two species chemotaxis system with two chemicals in a smooth bounded domain rn n 1 the existence of global bounded classical solution is proved under mild assumptions on the initial data and appropriate conditions on the strength of the damping death effects moreover for the case when both competition parameters a1 and a2 lie in 0 1 it is shown that such solution stabilizes to spatially homogeneous equilibria in the large time limit

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

existence of weak and regular solutions for keller segel system with degradation coupled to fluid equations abstract we establish the global well posedness for the following chemotaxis fluid system t n u n n n c n q t c u c c c n t u u u p u n u 0 in r d d 2 3 where 0 q 2 1 d and 0 1 for either q 2 d 1 2 or q 2 d 0 3 we prove the global existence of regular solutions in case that q 2 1 d and 0 very weak solutions are constructed as well

a new approach toward stabilization in a two species chemotaxis model with logistic source abstract this paper aims at providing an alternative approach to study global dynamic properties for a two species chemotaxis model with the main novelty being that both populations mutually compete with the other on account of the lotka volterra dynamics more precisely we consider the following neumann initial boundary value problem u t d 1 u 1 u w 1 u 1 u a 1 v x t 0 v t d 2 v 2 v w 2 v 1 a 2 u v x t 0 0 d 3 w w u v x t 0 in a bounded domain r n n 1 with smooth boundary where d 1 d 2 d 3 1 2 1 2 a 1 a 2 are positive constants when a 1 0 1 and a 2 0 1 it is shown that under some explicit largeness assumptions on the logistic growth coefficients 1 and 2 the corresponding neumann initial boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state u v w of above system in the large time limit the respective decay rate of this convergence is shown to be exponential when a 1 1 and a 2 0 1 if 2 is suitable large for all sufficiently regular nonnegative initial data u 0 and v 0 with u 0 0 and v 0 0 the globally bounded solution of above system will stabilize toward 0 1 1 as t in algebraic

boundedness in the higher dimensional keller segel model with signal dependent motility and logistic growth this paper concerns a higher dimensional keller segel model with signal dependent motility and logistic growth it is shown that the strong logistic damping can prevent blow up in the higher dimensions this paper concerns a higher dimensional keller segel model with signal dependent motility and logistic growth it is shown that the strong logistic damping can prevent blow up in the higher dimensions

sub logistic source can prevent blow up in the 2d minimal keller segel chemotaxis system it is well known that the neumann initial boundary value problem for the minimal keller segel chemotaxis system in a 2d bounded smooth domain has no blow ups for any presence of logistic source of cell kinetics here for a large class of cell kinetics including sub logistic sources we find an explicit condition involving the chemotactic strength the asymptotic damping rate and the initial mass of cells to ensure the uniform in time boundedness for the corresponding 2d neumann initial boundary value problem our finding in particular shows that sub logistic source can prevent blow up in 2d indicating that logistic damping is not the weakest damping to guarantee boundedness for the 2d keller segel minimal chemotaxis model

global boundedness in a three dimensional chemotaxis haptotaxis model abstract this paper studies 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 3 with zero flux boundary conditions where the parameters and are positive it is shown that the corresponding initial boundary problem possesses a unique classical solution which is global in time and bounded under the explicit condition 11 2 2 w 0 l 2 2 37 2 4 2 w 0 l 2 and suitable regularity assumptions on the initial data u 0 v 0 w 0

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

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

long time behaviour of solutions to a chemotaxis model with volume filling effect the lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume filling effect for some typical case it is proved that the limit set of any trajectory consists of regular stationary solutions some lower and upper bounds on the stationary solutions are found for a given range of parameters there are stationary solutions which are inhomogeneous in space

type ii blowup of solutions to a simplified keller segel system in two dimensional domains abstract we consider a parabolic elliptic system which is introduced as a simplified version of the so called keller segel system in particular we consider the system in a bounded domain of two dimensional euclidean space in that situation we can find solutions to the system blowing up in finite time then these solutions become the sum of an l 1 function and delta functions at the blowup time as regards the blowup speed there exists a radial solution whose blowup speed is faster than that of backward self similar solutions we refer to that blowup as type ii blowup in this paper we investigate whether finite time blowup solutions to the system exhibit type ii blowup or not

eventual smoothness and asymptotics in a three dimensional chemotaxis system with logistic source abstract we prove existence of global weak solutions to the chemotaxis system u t u u v u u 2 v t v v u under homogeneous neumann boundary conditions in a smooth bounded convex domain r n for arbitrarily small values of 0 additionally we show that in the three dimensional setting after some time these solutions become classical solutions provided that is not too large in this case we also consider their large time behaviour we prove decay if 0 and the existence of an absorbing set if 0 is sufficiently small

global existence of solutions to a parabolic parabolic system for chemotaxis with weak degradation abstract we study the global existence of solutions to a parabolic parabolic system for chemotaxis with a logistic source in a two dimensional domain where the degradation order of the logistic source is weaker than quadratic we introduce nonlinear production of a chemoattractant and show the global existence of solutions under certain relations between the degradation and production orders

convergence rate of a quasilinear parabolic elliptic chemotaxis system with logistic source abstract this paper deals with the quasilinear parabolic elliptic chemotaxis system u t d u u u v u 1 u x t 0 0 v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary where 0 and 0 d u is supposed to satisfy d 0 0 d u u with 0 1 when n 2 the convergence rate of the solution is investigated

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

minimum audible angle for clicks with simulated echoes effects of azimuth and standard minimum audible angle maa was measured in free field with a center position at 0 and 25 or 50 on the right stimuli were 6 ms wide band noise bursts presented as single bursts or noise pairs under precedence effect conditions in the single noise condition one burst was presented from the right or left in precedence lead discrimination a leading sound was presented from the left or right speaker and a lagging sound from a speaker at center in precedence lag discrimination the lagging sound was on the right or left and the leading sound was at center the test stimulus was either presented in isolation or was preceded by a single standard noise at center maas were lowest for the single condition slightly higher for lead discrimination and much higher for lag discrimination presence of a standard yielded a benefit in lag discrimination especially at 0 thresholds at 50 were significantly higher than at 0 and 25 especially in lag discrimination the data can be described by a model in wh