global uniqueness in an inverse problem for time fractional diffusion equations given m g a compact connected riemannian manifold of dimension d 2 with boundary m we consider an initial boundary value problem for a fractional diffusion equation on 0 t m t 0 with time fractional caputo derivative of order 0 1 1 2 we prove uniqueness in the inverse problem of determining the smooth manifold m g up to an isometry and various time independent smooth coefficients appearing in this equation from measurements of the solutions on a subset of m at fixed time in the flat case where m is a compact subset of rd two out the three coefficients density a conductivity and q potential appearing in the equation t u div a u qu 0 on 0 t m are recovered simultaneously

on a chemotaxis model with degenerate diffusion initial shrinking eventual smoothness and expanding abstract we investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria in particular the effect of the initial attractant distribution on bacterial clustering we prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure and show the finite speed propagating property without assuming the special structure on attractant concentration and obtain an explicit formula of the population spreading speed in terms of model parameters the presented results suggest that bacterial cluster formation can be affected by chemotactic attractants and density dependent dispersal

existence results for impulsive multivalued semilinear neutral functional differential inclusions in banach spaces in this paper the existence of mild solutions for first and second order impulsive semilinear neutral functional differential inclusions in banach spaces is investigated the results are obtained by using a fixed point theorem for condensing multivalued maps due to martelli and semigroup theory

blow up for a three dimensional keller segel model with consumption of chemoattractant abstract we investigate blow up properties for the initial boundary value problem of a keller segel model with consumption of chemoattractant when the spatial dimension is three through a kinetic reformulation of the keller segel system we first derive some higher order estimates and obtain certain blow up criteria for the local classical solutions these blow up criteria generalize the results in 4 5 from the whole space r 3 to the case of bounded smooth domain r 3 lower global blow up estimate on n l is also obtained based on our higher order estimates moreover we prove local non degeneracy for blow up points

global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions we study the quasilinear chemotaxis system 1 1 in a bounded domain n n 3 with smooth boundary where the diffusion function d u satisfies d u cdum 1 for all u 0 with some cd 0 under the condition m 3 2 1 n we show that for all reasonably regular initial data the corresponding initial boundary value problem for 1 1 possesses global boundedness of solution which converges to the spatially homogeneous equilibrium u 0 0 in an appropriate sense as t where u 0 1 u 0

boundedness in a chemotaxis model with exponentially decaying diffusivity and consumption of chemoattractant abstract this paper is devoted to the following chemotaxis model u t d u u s u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in a c o n v e x smooth bounded domain r n n 2 it is proved that if d and s are sufficiently smooth nonnegative functions on 0 satisfying k 0 e s d s k 1 e s for all s 0 with some k 0 0 k 1 0 and 0 then under the condition that s s d s k 2 s for all s 0 with some k 2 0 and 0 for the initial data u 0 v 0 are sufficiently regular satisfying u 0 0 and v 0 0 the classical solutions to the system are uniformly in time bounded

numerically pricing double barrier options in a time fractional black scholes model abstract the numerical solution of the time fractional black scholes model tfbsm of order 0 1 governing european options is studied zhang et al 2016 derived a numerical scheme of second order in space we improve their results by constructing a scheme of fourth order in space while keeping 2 in time the solvability stability and convergence of the proposed numerical scheme are proved using a fourier analysis the results are demonstrated on two examples

global existence and boundedness in a keller segel stokes system involving a tensor valued sensitivity with saturation the 3d case abstract in this paper we continue to deal with the initial boundary value problem for the coupled keller segel stokes system n t u n n n s x n c c x t 0 t c t u c c c n x t 0 t u t p u n x t 0 t u 0 x t 0 t where r d is a bounded domain with smooth boundary and the chemotactic sensitivity s is not a scalar function but rather attains values in r d d and satisfies s x n c c s 1 n with some c s 0 and 0 when d 2 our previous work j differential equations 2015 has established the existence of global bounded classical solutions under the subcritical assumption 0 which is consistent with the corresponding results of the fluid free system but the method seems to be invalid in the three dimensional setting in this paper for the case d 3 we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption 1 2 which is slightly stronger than the corresponding subcritical assumption 1 3 on the fluid free system where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions the key idea here is to establish the general l p regularity of u from a rather low l p regularity of n which will be obtained by a new combinational functional

boundedness and global solvability to a chemotaxis model with nonlinear diffusion abstract in this paper we study a chemotaxis model with nonlinear diffusion u m m 1 we consider this problem in a bounded domain r 3 with zero flux boundary condition and it is shown that for any large initial datum for any m 1 the problem admits a global weak solution which is uniformly bounded

integral inequalities and mild solutions of semilinear neutral evolution equations abstract existence of mild solutions to semilinear neutral evolution equations with nonlocal conditions is proved the result is obtained by using fractional power of operators krasnoselski schaefer type fixed point theorem and by applying a modification of the bihari type inequality to the case of singular integral inequality an application to partial differential equations is given

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

large time behavior of solution to an attraction repulsion chemotaxis system with logistic source in three dimensions abstract this paper studies the attraction repulsion chemotaxis system with logistic source u t u u v u w f u v t v 1 v 1 u w t w 2 w 2 u in a smooth bounded convex domain r 3 subject to nonnegative initial data and homogeneous neumann boundary conditions where i and i i 1 2 are positive parameters and the logistic source function f fulfills f s s s 1 s 0 0 and 1 it is shown that this system possesses a unique global bounded classical solution under the conditions i 1 2 and max 41 2 1 9 2 9 1 41 2 2 furthermore whenever u 0 0 and for any n the solution of the system approaches to the steady state 1 1 1 1 1 1 1 1 2 2 in the norm of l as t

boundedness in a chemotaxis model with oxygen consumption by bacteria abstract this paper deals with the keller segel model u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 with nonnegative initial data u 0 v 0 w 1 r 2 for some r n u 0 l 1 0 and v 0 l 0 this model describes bacteria movement toward the concentration gradient of the oxygen that is consumed by the bacteria it is proved that if 0 1 6 n 1 v 0 l then the corresponding initial boundary value problem possesses a unique global solution that is uniformly bounded

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

on some new inequalities related to certain inequalities in the theory of differential equations in the present paper we establish some new integral inequalities related to a certain inequality arising in the theory of differential equations the discrete analogues of the main results are also given the analysis used in the proofs is elementary and the results established here provide new estimates for these types of inequalities

the small convection limit in a two dimensional chemotaxis navier stokes system this paper deals with an initial boundary value problem for the chemotaxis navier stokes system begin aligned left begin array lcll n t u cdot nabla n delta n nabla cdot n nabla c qquad quad x in omega t 0 c t u cdot nabla c delta c nc qquad quad x in omega t 0 u t kappa u cdot nabla u delta u nabla p n nabla phi qquad quad x in omega t 0 nabla cdot u 0 qquad quad x in omega t 0 end array right end aligned in a bounded convex domain omega subset mathbb r 2 with smooth boundary with kappa in mathbb r and a given smooth potential phi omega rightarrow mathbb r it is known that for each kappa in mathbb r and all sufficiently smooth initial data this problem possesses a unique global classical solution n kappa c kappa u kappa the present work asserts that these solutions stabilize to n 0 c 0 u 0 uniformly with respect to the time variable more precisely it is shown that there exist mu 0 and c 0 such that whenever kappa in 1 1 begin aligned big vert n kappa cdot t n 0 cdot t big vert l infty omega big vert c kappa cdot t c 0 cdot t big vert l infty omega quad big vert u kappa cdot t u 0 cdot t big vert l infty omega le c kappa e mu t end aligned for all t 0 this result thereby provides an example for a rigorous quantification of stability properties in the stokes limit process as frequently considered in the literature on chemotaxis fluid systems in application contexts involving low reynolds numbers

global existence for semilinear evolution equations with nonlocal conditions abstract in this paper we study the global existence of solutions for semilinear evolution equations with nonlocal conditions via a fixed point analysis approach using the leray schauder alternative we derive conditions under which a solution exists globally

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

global existence and asymptotic stability of the fractional chemotaxis fluid system in r3 abstract in this paper we consider a model arising from biology consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in r 3 we establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument moreover the decay rates of the solutions and their higher order spatial derivatives are established towards the equilibrium by introducing the negative sobolev space h s 0 s 3 2

existence results for partial neutral functional differential equations with unbounded delay abstract we prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the leray schauder alternative

analysis of a chemotaxis model with indirect signal absorption abstract we consider the chemotaxis model u t u u v v t v v w w t w u in smooth bounded domains r n n n where 0 is a given parameter if either n 2 or v 0 l 1 3 n we show the existence of a unique global classical solution u v w and convergence of u t v t w t towards a spatially constant equilibrium as t the proof of global existence for the case n 2 relies on a bootstrap procedure as a starting point we derive a functional inequality for a functional being sublinear in u which appears to be novel in this context

determination of the initial data in a time fractional diffusion wave problem by a final time data abstract this paper is to determine an initial value for a time fractional diffusion wave equation from the final time data the regularity of the weak solution for the direct problem is improved the existence and uniqueness of a weak solution for the adjoint problem are proved by using the fourier method the tikhonov regularization method is applied for finding a stable approximate solution in order to get the minimizer of the tikhonov regularization functional we propose a conjugate gradient algorithm based on the strict deductions of the sensitive problem and the adjoint problem four numerical examples in one dimensional and two dimensional cases are provided to show the effectiveness and stability of the proposed algorithm

simultaneous identification of three parameters in a time fractional diffusion wave equation by a part of boundary cauchy data abstract this paper is devoted to determine the fractional order the initial flux speed and the boundary neumann data simultaneously in a one dimensional time fractional diffusion wave equation from part boundary cauchy observation data we prove the uniqueness result for this inverse problem by using a new result for the mittag leffler function and laplace transform combining with analytic continuation then we use the iterative regularizing ensemble kalman method in bayesian framework to solve the inverse problem numerically and four numerical examples are provided to show the effectiveness and stability of the proposed algorithm

the fractional tikhonov regularization methods for identifying the initial value problem for a time fractional diffusion equation abstract in this paper we identify the initial value for a time fractional diffusion equation on a columnar axis symmetric domain this problem is ill posed i e the solution if it exists does not depend continuously on the domain two different kinds of fractional tikhonov methods are used to solve this problem under the a priori and the a posteriori regularization parameter choice rules the error estimates between the regularization solutions and the exact solution are obtained respectively different numerical examples are presented to illustrate the validity and effectiveness of our methods

controllability of neutral functional integrodifferential systems in banach spaces abstract sufficient conditions for controllability of neutral functional integrodifferential systems in a banach space are established the results are obtained by using the schaefer fixed point theorem an example is provided to illustrate the theory

global solutions for chemotaxis navier stokes system with robin boundary conditions abstract we consider a chemotaxis navier stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account this phenomenon results in an inhomogeneous robin type boundary condition moreover the system is studied without the logistic growth of the bacteria population we prove that in one or two dimensions the system has a unique global classical solution while the existence of a global weak solution is shown in three dimensions in the latter case we show that the energy is bounded uniformly in time a key idea is to introduce and utilise a boundary energy to derive suitable a priori estimates moreover we are able to remove the convexity assumption on the domain

on a class of non linear delay distributed order fractional diffusion equations in this paper we consider a numerical scheme for a class of non linear time delay fractional diffusion equations with distributed order in time this study covers the unique solvability convergence and stability of the resulted numerical solution by means of the discrete energy method the derivation of a linearized difference scheme with convergence order o 4 h 4 in l norm is the main purpose of this study numerical experiments are carried out to support the obtained theoretical results

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

boundedness and asymptotic stability of solutions to a two species chemotaxis system with consumption of chemoattractant this paper deals with a two competing species chemotaxis system with consumption of chemoattractant ut d1 u u 1 w w 1u 1 u a1v x t 0 vt d2 v v 2 w w 2v 1 a2u v x t 0 wt d3 w u v w x t 0 under homogeneous neumann boundary conditions in a bounded domain rn rn n 1n 1 with smooth boundary where the initial data u0 v0 c0 2 and w0 w1 w0 w1 are non negative and the parameters d1 d2 d3 0d1 d2 d3 0 1 2 0 1 2 0 a1 a2 0a1 a2 0 and 0 0 the chemotactic function i w i w i 1 2i 1 2 is smooth and satisfying some conditions it is proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution if one of the following cases hold for i 1 2i 1 2 i i w 0 i 0 i w 0 i 0 and w0 l did3n 1 0 i 2did3n 1 0 iarctan di d32n 1did3 ii 0 w0 l d33 n 1 i l 0 w0 l min 2didi d3 1 moreover we prove asymptotic stabilization of solutions in the sense that if a1 a2 0 1 a1 a2 0 1 and u0 0 v0u0 0 v0 then any global bounded solution exponentially converge to 1 a11 a1a2 1 a21 a1a2 0 as t t if a1 1 a2 0a1 1 a2 0 and v0 0v0 0 then any global bounded solution exponentially converge to 0 1 0 0 1 0 as t t if a1 1 a2 0a1 1 a2 0 and v0 0v0 0 then any global bounded solution algebraically converge to 0 1 0 0 1 0 as t t

existence of solutions of nonlinear neutral integrodifferential equations in banach spaces abstract in this paper we prove the existence of mild solutions of a nonlinear neutral integrodifferential equation in a banach space the results are obtained by using the schaefer fixed point theorem as an application the controllability problem for the neutral system is discussed

global existence and convergence rates to a chemotaxis fluids system with mixed boundary conditions abstract in this paper we investigate the large time behavior of strong solutions to a chemotaxis fluids system in an unbounded domain with mixed boundary conditions based on the anisotropic l p technique the elliptic estimates and stokes estimates we first establish the global existence of strong solution around the equilibrium state 0 c satn 0 with the help of the continuity arguments where c satn is the saturation value of oxygen inside the fluid then we use de giorgi s technique and energy method to show that such a solution will converge to 0 c satn 0 with an explicit convergence rate in the chemotaxis free case our assumptions and results are consistent with the experimental descriptions and the numerical analysis the novelty here consists of deriving some new elliptic estimates and stokes estimates and choosing a suitable weight in de giorgi s technique to deal with the mixed boundary conditions

the backward problem for a time fractional diffusion wave equation in a bounded domain abstract this paper is devoted to solve the backward problem for a time fractional diffusion wave equation in a bounded domain based on the series expression of the solution for the direct problem the backward problem for searching the initial data is converted into solving the fredholm integral equation of the first kind the existence uniqueness and conditional stability for the backward problem are investigated we use the tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem furthermore the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule numerical results for five examples in one dimensional case and two dimensional case show that the proposed method is efficient and stable

global stability of prey taxis systems abstract in this paper we prove the global boundedness and stability of the predator prey system with prey taxis in a two dimensional bounded domain with neumann boundary conditions by deriving an entropy like equality and a boundedness criterion we show that the intrinsic interaction between predators and preys is sufficient to prevent the population overcrowding even the prey taxis is included and strong furthermore by constructing appropriate lyapunov functionals we show that prey only steady state is globally asymptotically stable if the predation is weak and the co existence steady state is globally asymptotically stable under some conditions like the prey taxis is weak or the prey diffuses fast if the predation is strong the convergence rates of solutions to the steady states are derived in the paper

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

boundedness and stabilization in a two species chemotaxis system with signal absorption abstract this paper is concerned with the neumann initial boundary value problem for the two species chemotaxis system with consumption of chemoattractant u t u 1 u w x t 0 v t v 2 v w x t 0 w t w u v w x t 0 in a smooth bounded domain r n n 2 where the parameters 1 2 and are positive it is proved that if max 1 2 w x 0 l 2 n the problem possesses a unique global classical solution that is uniformly bounded moreover we prove that u x t 1 u x 0 v x t 1 v x 0 and w x t 0 as t uniformly with respect x

asymptotic convergence of solutions of keller segel equations in network shaped domains abstract the objective of this study is to examine the classical keller segel equations in network shaped domains first the existence of unique strict global solutions of the equations is demonstrated by using the theory of strongly elliptic differential operators in network shaped domains and the theory of abstract parabolic semilinear evolution equations furthermore by considering an angle condition as well as the ojasiewicz simon inequality it is proved that every global solution converges to a stationary solution and the convergence is dominated by the values of a lyapunov function along the global solution

existence and regularity of final value problems for time fractional wave equations abstract we study a class of final value problems for time fractional wave equations involvingcaputo s fractional derivative of order 1 2 and a symmetric uniformly elliptic operator in a bounded domain r n n 1 2 3 firstly we analyze the difficulties involved in the final value problem under consideration secondly we establish the existence and uniqueness of the mild solution finally we investigate the regularity of the mild solution in some special spaces

controllability of nonlinear neutral evolution integrodifferential systems sufficient conditions for controllability of nonlinear neutral evolution integrodifferential systems in a banach space are established the results are obtained by using the resolvent operators and the schaefer fixed point theorem an application to partial integrodifferential equation is given

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

approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces various sufficient conditions for approximate controllability of linear evolution systems in abstract spaces have been obtained but approximate controllability of semilinear control systems usually requires some complicated and limited assumptions in this paper we show the approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable the results are obtained using new properties of symmetric operators which are proved in this paper compact semigroups the schauder fixed point theorem and or the contraction mapping principle

stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant we consider chemotaxis system with consumption of chemoattractant vt v uv ut u u v under homogeneous neumann boundary conditions it is proved that if either n 2 or 0 16 n 1 v x 0 l n 3 the global classical solution u v of this problem converges to u0 0 exponentially as t where u0 1 u x 0 dx

well posedness for chemotaxis dynamics with nonlinear cell diffusion abstract this paper investigates the well posedness of a reaction diffusion system of chemotaxis type with a nonlinear diffusion coefficient and a dynamics growth death of the cell population b and a stationary equation for the chemoattractant c with respect to other works in which a nonlinear diffusion for cells has been considered we treat here two distinct cases for this diffusion coefficient the first in which it is a positive bounded function on r and the other in which it may display a singularity at a finite value of the cell density essentially the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient the chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities for homogeneous neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the l 2 norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data in a case when the diffusion coefficient is singular we focus on a model expressed by a variational inequality describing the saturation of the cell population at the blowing up diffusion value here the proof requires the study of an intermediate problem with robin boundary conditions which may be interesting by itself in all situations uniqueness follows on a time interval included not necessarily strictly in that of the solution existence under sufficient conditions

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

large time behavior of solutions to a chemotaxis model with porous medium diffusion abstract in this paper we study the large time behavior of a chemotaxis model with nonlinear diffusion and consumption u t u m u v u 1 u v t v v u where m 1 in a previous paper 5 we have proved the existence and uniform boundedness of global weak solutions for any nonnegative initial data and any m 1 in this work we show that the weak solutions strongly converge to 1 0 in the large time limit

convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source abstract we study the global attractors to the chemotaxis system with logistic source u t u u v a u b u 2 v t v v u in r subject to the homogeneous neumann boundary conditions where smooth bounded domain r n with b 0 a r and 0 1 for the parabolic elliptic case with 0 and n 3 we obtain that the positive constant equilibrium a b a b is a global attractor if a 0 and b max n 2 n a 4 under the assumption n 3 it is proved that for either the parabolic elliptic case with 0 a 0 b max 3 a 4 or the parabolic parabolic case with 1 a 0 b a 4 large enough the system admits the positive constant equilibrium a b a b as a global attractor while the trivial equilibrium 0 0 is a global attractor if a 0 and b 0 it is pointed out that here the convergence rates are established for all of them the results of the paper mainly rely on parabolic regularity theory and lyapunov functionals carefully constructed

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

integral inequalities and global solutions of semilinear evolution equations a criterion for the nonexplosion of solutions to semilinear evolution equations on banach spaces is proved the result is obtained by applying a modification of the bihari type inequality to the case of a weakly singular nonlinear integral inequality

eventual smoothness and stabilization of large data solutions in a three dimensional chemotaxis system with consumption of chemoattractant abstract this paper deals with positive solutions of u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in bounded convex domains r 3 with smooth boundary it is shown that for arbitrarily large initial data this problem admits at least one global weak solution for which there exists t 0 such that u v is bounded and smooth in t moreover it is asserted that such solutions approach spatially constant equilibria in the large time limit

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

global bounded solutions and their asymptotic properties under small initial data condition in a two dimensional chemotaxis system for two species abstract in this paper a fully parabolic chemotaxis system for two species u t u u w x t 0 v t v v w x t 0 w t w u w v w x t 0 is considered under homogeneous neumann boundary conditions in a smooth bounded domain r 2 we obtain the global boundedness and asymptotic behavior with small initial data condition in critical space more precisely it is proved that one can find a small e 0 0 such that for any initial data u 0 v 0 w 0 satisfying u 0 l 1 e 0 and w 0 l 2 e 0 the solution of the problem above is global in time and bounded in addition u v w converges to the steady state m 1 m 2 m 1 1 m 2 as t where m 1 1 u 0 and m 2 1 v 0

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

global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread abstract in this paper we consider the following system u t u u v t d v v v m t d m m u m t 1 u m in two dimensional space with zero flux boundary conditions this model was proposed by franssen et al 7 to characterize the invasion and metastatic spread of cancer cells we first establish the global existence of uniformly bounded global strong solutions then using the decay of ecm and the positivity of mde we further improve the regularity of obtained solutions and achieve the uniform boundedness of solutions in the classical sense subsequently we also prove the uniqueness of solutions after that we turn our attention to the large time behavior of solutions and show that the global classical solution strongly converges to a semi trivial steady state in the large time limit

nonlocal cauchy problems for neutral functional differential and integrodifferential inclusions in banach spaces in this paper we investigate the existence of mild solutions on a compact interval to initial value problems for neutral functional differential and integrodifferential inclusions in banach spaces with nonlocal conditions the results are obtained by using a fixed point theorem for condensing maps due to martelli