global existence and blow up of solutions to a nonlocal quasilinear degenerate parabolic system abstract this paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system u t div u p 2 u a v x t d x v t div v q 2 v b u x t d x with zero dirichlet boundary conditions in a smooth bounded domain r n n 1 where p q 2 1 and a b 0 are positive constants under appropriate hypotheses we first establish the local existence and uniqueness of solutions then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between and p 1 q 1 in the special case of q 1 and p 1 we conclude that the solution exists globally if p 1 d x q 1 d x 1 a b while if p 1 d x q 1 d x 1 a b then the solution blows up in finite time here x and x denote the unique solution of the following elliptic problem div p 2 1 in x 0 and div q 2 1 in x 0 respectively

closed forms of the green s function and the generalized green s function for the helmholtz operator on the n dimensional unit sphere the green s function for the helmholtz differential operator 2 n 1 on the n dimensional with n 1 hyperspherical surface s n of unit radius is investigated its closed form is shown to be g n n n n 1 sn sin c n 1 2 n n where sn is the area of s n c x is the gegenbauer function of the first

global existence and l estimates of solution for doubly nonlinear parabolic equation the existence uniqueness and l estimate of the weak solution to the initial boundary value problem for the doubly nonlinear parabolic equationut div u r u m 2 u au fx t uin 0 with zero boundary condition in a bounded domain rn are established in particular the behavior of the solution as t 0 and t is investigated

numerical calculation of the diffraction coefficients for an arbitrary shaped perfectly conducting cone a method for numerical calculation of the diffraction coefficients for electromagnetic diffraction by arbitrarily shaped perfectly conducting cones is proposed the approach makes an extensive use of the analytic formulas of smyshlyaev in combination with further developments including a use of the potential theory adapted to the laplace beltrami operator on a subdomain of unit sphere this reduces the problem to a fredholm integral equation on the closed curve of the unit sphere defining the cone s geometry which can be solved numerically this strategy permits us to implement a numerical code for calculation of the diffraction coefficients for cones of rather general cross sections results of sample calculations for the circular and elliptic cones are given

boundary integral equation technique for the singular behavior of electromagnetic fields at the common tip of metallic and bi isotropic cones with arbitrary cross section a boundary integral equation technique is developed to determine the singular field behavior at the common tip of lossless bi isotropic and perfectly electrically and or magnetically conducting cones with arbitrary cross section the kernel of the set of boundary integral equations is a green s function defined on a spherical surface this green s function is the associated legendre function of the first kind the integral equations are solved with the galerkin method of moments the theory is illustrated with a number of examples that show the effects of bi isotropy on the singular field behavior

closed form of the generalized green s function for the helmholtz operator on the two dimensional unit sphere the closed representation of the generalized known also as reduced or modified green s function for the helmholtz partial differential operator on the surface of the two dimensional unit sphere is derived in its compact form the derived formula contains a legendre polynomial and a derivative of the legendre function of the first kind with respect to its index an explicit expression for that derivative is found and used to obtain an expanded and potentially more suitable in applications form of the generalized green s function for the operator in question the related problem of constructing the closed form of the generalized green s function for the legendre ordinary differential operator on the segment 1 x 1 with the boundary conditions of finiteness at x 1 is also solved

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

localization for the evolution p laplacian equation with strongly nonlinear source term abstract in this paper we study the strict localization for the p laplacian equation with strongly nonlinear source term let u u x t be a solution of the cauchy problem u t div u p 2 u u q u x 0 u 0 x where x t r n 0 t n 1 and p 2 when q p 1 we prove that if the initial data u 0 x has a compact support then the solution u t has also compactly support moreover when 1 q p 1 we show that the solution of the cauchy problem blows up at any point of r n to arbitrary compactly supported initial data

smoothing effects of the initial boundary value problem for logarithmic type quasilinear parabolic equations abstract we give existence theorems of global solutions in l l o c 0 w 0 1 to the initial boundary value problem for quasilinear degenerate parabolic equations of the form u t div u 2 u 0 where the class of v 2 includes the logarithmic case u 2 log 1 u 2 for a typical example we assume that the initial data belong to w 0 1 p 0 p 0 2 or l r r 1 and we derive precise estimates for u t near t 0

global existence and blow up solutions for doubly degenerate parabolic system with nonlocal source abstract this paper deals with the following nonlocal doubly degenerate parabolic system u t div u m p 2 u m a u 1 x t v 1 x t d x v t div v n q 2 v n b u 2 x t v 2 x t d x with null dirichlet boundary conditions in a smooth bounded domain r n where m n 1 p q 2 i i 0 i 1 2 and a b 0 are positive constants we first get the non existence result for a related elliptic systems of non increasing positive solutions secondly by using this non existence result blow up estimates for above non newton polytropic filtration systems with the homogeneous dirichlet boundary value conditions are obtained then under appropriate hypotheses we establish local theory of the solutions and prove that the solution either exists globally or blows up in finite time in the special case 1 n q 1 2 2 m p 1 1 we also give a criterion for the solution to exist globally or blow up in finite time which depends on a b and x x as defined in the main results

a universal bound for radial solutions of the quasilinear parabolic equation with p laplace operator abstract in this paper we prove a universal bound for nonnegative radial solutions of the p laplace equation with nonlinear source u t div u p 2 u u q where p 2 and q p 1 this bound implies initial and final blowup rate estimates as well as a priori estimate or decay rate for global solutions our bound is proved as a consequence of liouville type theorems for entire solutions and doubling and rescaling arguments in this connection we use a known liouville type theorem for radial solutions along with a new liouville type theorem that is here established for nontrivial solutions in r

on the cauchy problem for the evolution p laplacian equations with gradient term and source abstract in this paper the authors study the equation u t div d u p 2 d u u q 1 u d u l in r n with p 2 we first prove that for 1 l p 1 the solution exists at least for a short time then for p 2 l p 1 the existence and nonexistence of global in time solutions are studied in various situations

the computation of conical diffraction coefficients in high frequency acoustic wave scattering when a high frequency acoustic or electromagnetic wave is scattered by a surface with a conical point the component of the asymptotics of the scattered wave corresponding to diffraction by the conical point can be represented as an asymptotic expansion valid as the wave number k rightarrow infty the diffraction coefficient is the coefficient of the principal term in this expansion and is of fundamental interest in high frequency scattering it can be computed by solving a family of homogeneous boundary value problems for the laplace beltrami helmholtz equation parametrized by a complex wave number like parameter nu on a portion of the unit sphere bounded by a simple closed contour ell and then integrating the resulting solutions with respect to nu in this paper we give the numerical analysis of a method for carrying out this computation in the case of acoustic waves via the boundary integral method applied on ell emphasizing the practically important case when the conical scat

periodic p laplacian with nonlocal terms abstract in this paper we study the existence of non trivial periodic solutions for a periodic p laplacian with nonlocal terms based on the theory of leray schauder degree the key step is dealing with the degeneracy of the p laplacian and the logistic type terms arising in the right hand side of the equation

global existence to the initial boundary value problem for a system of nonlinear diffusion and wave equations abstract we prove the global existence of weak solution pair to the initial boundary value problem for a system of m laplacian type diffusion equation and nonlinear wave equation the interaction of two equations is given through nonlinear source terms f u v and g u v

on periodic solutions of superlinear parabolic problems on etudie l existence des solutions periodiques non triviales positives des problemes paraboliques semilineaires on demontre des resultats de bifurcation

existence and uniqueness of periodic solution for a class of semilinear evolution equations this paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a hilbert space h u t au t f t u t 0 t u 0 u where a d a h h is a positive definite self adjoint operator 0 and f 0 h h satisfy caratheodory condition we present some spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions are the generalization for nonresonance condition of the self adjoint elliptic boundary value problems

global solutions for some nonlinear parabolic equations with nonmonotonic peturbations publisher summary this chapter discusses the existence uniqueness and decay property of the solutions of two problems it also presents the sobolev s lemma and gagliardo nirenberg inequality

periodic solutions of non newtonian polytropic filtration equations with nonlinear sources in this paper the authors first consider the dirichlet boundary value problem to the non newtonian polytropic filtration equation of the form u t div u m p 2 u m h x t u a in wxrwith strong nonlinear sources the existence of nontrivial periodic solutions is established based on topological degree theory the authors also studied the dirichlet boundary value problem to the equation in the form u t div u m 1u p 2 u m 1u b x t u f x t in wxrwith weak nonlinear sources the existence is treated with leray schauder fixed point theory

extinction decay and blow up for keller segel systems of fast diffusion type abstract we consider the quasi linear keller segel system of singular type where the principal part u m represents a fast diffusion like 0 m 1 we first construct a global weak solution with small initial data in the scaling invariant norm l n q m 2 for all dimensions n 2 and all exponents q 2 as for the large initial data we show that there exists a blow up solution in the case of n 2 in the second part the decay property in l r with 1 r for 1 2 n m 1 with the mass conservation is shown on the other hand in the case of 0 m 1 2 n the extinction phenomenon of solution is proved it is clarified that the case of m 1 2 n exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1 2 n for the borderline case of m 1 2 n our solution decays in l r exponentially as t

on the initial boundary value problem for some quasilinear parabolic equations of divergence form abstract in this paper we give an existence theorem of global classical solution to the initial boundary value problem for the quasilinear parabolic equations of divergence form u t div u 2 u f u u x t where u 2 may not be bounded as u as an application the logarithmic type nonlinearity u 2 log 1 u 2 which is growing as u and degenerate at u 0 is considered under f 0

blow up results and global existence of positive solutions for the inhomogeneous evolution p laplacian equations abstract this paper deals with the cauchy problem of inhomogeneous evolution p laplacian equations t u div u p 2 u u q w x with nonnegative initial data where p 1 q max 1 p 1 and w x 0 is a nonnegative continuous functions in r n we prove that q c p 1 n n p is its critical exponent provided that 2 n n 1 p n i e if q q c then every positive solution blows up in finite time whereas for q q c the equation possesses a global positive solution for some w x and some initial data meanwhile we also prove that its positive solutions blow up in finite time provided that n p

on global attractor for nonlinear parabolic equations of m laplacian type abstract existence and some regularity results of global attractor in l q q 1 for m laplacian type quasilinear parabolic equation with a perturbation like a x u u u u f x with 0 a x 0 are proved for the proofs moser s technique is used extensively

on the derivative of the legendre function of the first kind with respect to its degree we study the derivative of the legendre function of the first kind p z with respect to its degree at first we provide two contour integral representations for p z then we proceed to investigate the case of p z n with n being an integer this case is met in some physical and engineering problems since it holds that we focus on the sub case of n being a non negative integer we show that where rn z is a polynomial in z of degree n we present alternative derivations of several known explicit expressions for rn z and also add some new a generating function for rn z is also constructed properties of the polynomials vn z rn z 1 nrn z 2 and wn 1 z rn z 1 nrn z 2 are also investigated it is found that wn 1 z is the christoffel polynomial well known from the theory of the legendre function of the second kind qn z as examples of applications of the results obtained we present non standard derivations of some representations of qn z sum to closed forms some legendre series evaluate some definite integrals involving legendre polynomials and also derive an explicit representation of the indefinite integral of the legendre polynomial squared

on the solvability of a quasilinear parabolic partial differential equation at resonance abstract in this article we apply galerkin type techniques and brouwer s fixed point theorem to obtain existence theorems of time periodic solutions for quasilinear parabolic partial differential resonance equations of the dirichlet boundary condition in which the landesman lazer condition may be excluded

periodic solutions of the evolutionary p laplacian with nonlinear sources abstract this paper is concerned with the evolutionary p laplacian with nonlinear and periodic sources we will give a rather complete characterization in terms of the parameter p and the exponent q of the source of whether or not the positive periodic solutions exist

existence and nonexistence of global positive solutions for the evolution p laplacian equations in exterior domains abstract this paper deals with the existence and nonexistence of global positive solutions for two evolution p laplacian equations in exterior domains with inhomogeneous boundary conditions we demonstrate that q c n p 1 n p is its critical exponent provided 2 n n 1 p n furthermore we prove that if max 1 p 1 q q c then every positive solution of the equations blows up in finite time whereas for q q c the equations admit the global positive solutions for some boundary value f x and some initial data u 0 x we also demonstrate that every positive solution of the equations blows up in finite time provided n p

liouville theorems universal estimates and periodic solutions for cooperative parabolic lotka volterra systems we consider positive solutions of cooperative parabolic lotka volterra systems with equal diffusion coefficients in bounded and unbounded domains the systems are complemented by the dirichlet or neumann boundary conditions under suitable assumptions on the coefficients of the reaction terms these problems possess both global solutions and solutions which blow up in finite time we show that any solution u v defined on the time interval 0 t satisfies a universal estimate of the form u x t v x t c 1 t 1 t t 1 where c does not depend on x t u v t in particular this bound guarantees global existence and boundedness for threshold solutions lying on the borderline between blow up and global existence moreover this bound yields optimal blow up rate estimates for solutions which blow up in finite time our estimates are based on new liouville type theorems for the corresponding scaling invariant parabolic system and require an optimal restriction on the space dimension n n 5 as an application we also prove the existence of time periodic positive solutions if the coefficients are time periodic our approach can also be used for more general parabolic systems

the critical exponents for the quasi linear parabolic equations with inhomogeneous terms abstract this paper deals with the critical exponents for the quasi linear parabolic equations in r n and with an inhomogeneous source or in exterior domains and with inhomogeneous boundary conditions for n 3 2 n and p max 1 1 we obtain that p c n 1 n 2 is the critical exponent of these equations furthermore we prove that if max 1 1 p p c then every positive solution of these equations blows up in finite time whereas these equations admit the global positive solutions for some f x and some initial data u 0 x if p p c meantime we also demonstrate that every positive solution of these equations blows up in finite time provided n 1 2 1 and p max 1 1

existence of solutions to the slow diffusion equation with a nonlinear source abstract in this paper we consider the existence of solutions to cauchy problem for the slow diffusion equation with a source term t u u m u p x r n t 0 u x 0 x 0 where m 1 and p m our purpose is to obtain lower estimates of the existence time of solutions in the uniformly local lebesgue spaces furthermore we obtain the estimate of blow up time blow up rate of solutions and blow up of critical norm

diffusion on a sphere with localized traps mean first passage time eigenvalue asymptotics and fekete points a common scenario in cellular signal transduction is that a diffusing surface bound molecule must arrive at a localized signaling region on the cell membrane before the signaling cascade can be completed the question then arises of how quickly such signaling molecules can arrive at newly formed signaling regions here we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere in the presence of n partially absorbing traps of small radii the rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the laplace operator on the sphere with small localized traps the asymptotic analysis relies on the method of matched asymptotic expansions together with detailed properties of the green s function for the laplacian and the helmholtz operators on the surface of the unit sphere the asymptotic results compare favorably with

hopf algebras cyclic cohomology and the transverse index theorem in this paper we solve a longstanding internal problem of noncommutative geometry namely the computation of the index of transversally elliptic operators on foliations we show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific hopf algebra associated to each integer codimension this hopf algebra reduces transverse geometry to a universal geometry of affine nature the structure of this hopf algebra its relation with the lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the hochschild cohomology of a hopf algebra carries a highly nontrivial cyclic structure

existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms abstract in this paper the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of leray schauder s degree

geometric invariants of the quantum hall effect we study the two dimensional hall effect with a random potential the hall conductivity is identified as a geometric invariant associated with an algebra of observables using the pairing betweenk theory and cyclic cohomology theory we identify this geometric invariant with a topological index thereby giving the hall conductivity a new interpretation

on evaluation of the diffraction coefficients for arbitrary nonsingular directions of a smooth convex cone diffraction coefficients are fundamental objects determining principal amplitudes in asymptotic expansions for a general high frequency diffraction problem and are to be found from an associated canonical problem for a class of canonical problems of plane wave diffraction by an arbitrarily shaped smooth convex cone formulated as a boundary value problem for the helmholtz equation with dirichlet or neumann boundary conditions we develop a method for evaluating the diffraction coefficients in arbitrary nonsingular directions i e in all the directions where they are well defined the approach builds further on the analytic representations of v p smyshlyaev wave motion 12 1990 pp 329 339 and the constructions of v m babich d b dement ev and b a samokish wave motion 21 1995 pp 203 207 who have previously developed a method for calculating the diffraction coefficients in certain restricted directions for which an integral over a complex contour determining the diffraction

an optimal liouville type theorem of the quasilinear parabolic equation with a p laplace operator abstract in this paper we consider nonnegative solutions of the quasilinear parabolic equation with p laplace operator u t div u p 2 u u q 1 u where p 2 and q p 1 our main result is that there is no nontrivial positive bounded radial entire solution the proof is based on intersection comparison arguments which can be viewed as a sophisticated form of the maximum principle and has been used to deal with the semilinear heat equation by polacik and quittner peter polacik pavol quittner a liouville type theorem and the decay of radial solutions of a semilinear heat equation nonlinear analysis tma 64 2006 1679 1689 and the porous medium equation by souplet ph souplet an optimal liouville type theorem for radial entire solutions of the porous medium equation with source j differential equations 246 2009 3980 4005

the supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations abstract in this paper we provide supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations u t u m a x t u p with m 0 p 0 by using a new method to deal with the blow up sequence we obtain the l estimate with optimal exponent 1 p m p s where ps is the sobolev critical exponent then we apply topological degree method to prove the existence of positive periodic solutions if p m p s n 3 and is star shaped we establish rellich pohozaev type identity to prove the nonexistence of positive periodic solutions we improve the results of mizoguchi 10 therem 1 by filling the gap m 1 p m 1 2 m n in addition we also give quite complete results about the existence of periodic solutions for the fast diffusion case 0

global existence and blow up solutions and blow up estimates for a non local quasilinear degenerate parabolic system this paper deals with p laplacian systems ut div u p 2 u a u 1 x t v 1 x t dx x t 0 vt div v q 2 v b u 2 x t v 2 x t dx x t 0 with null dirichlet boundary conditions in a smooth bounded domain rn where p q 1 i i 0 i 1 2 and a b 0 are positive constants we first get the non existence result for a related elliptic systems of non increasing positive solutions secondly by using this non existence result blow up estimates for above p laplacian systems with the homogeneous dirichlet boundary value conditions are obtained under br x rn x 0 then under appropriate hypotheses we establish local theory of the solutions and obtain that the solutions either exists globally or blow up in finite time

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

extinction and non extinction for a p laplacian equation with nonlinear source abstract in this paper we deal with the extinction of solutions of the initial boundary value problem of the p laplacian equation u t div u p 2 u u q in a bounded domain of r n with n 2 for 1 p 2 we show that q p 1 is the critical exponent of extinction for the weak solution furthermore for 1 p 2 and q p 1 we prove that the extinction and non extinction of the solution depends strongly on the first eigenvalue of the problem div p 2 p 2 in 0

local and global existence for quasilinear parabolic systems with a strongly nonlinear source abstract we study the cauchy problem for quasilinear parabolic systems with a strongly nonlinear source on the assumptions that the initial values are taken in the sense of l l o c 1 r n we obtain the local existence global existence and nonexistence of solutions particularly in the case of global solvability some exact estimates of a solution are obtained

periodic doubly degenerate parabolic equation with nonlocal terms this paper is a remark on gao and wang 2007 4 under some weaker conditions than 4 we establish the existence of nontrivial nonnegative periodic solutions for a doubly degenerate parabolic equation with nonlocal terms the key is the use of de giorgi iteration technique and the theory of leray schauder degree

quasi uniformity for the abstract neumann antimaximum principle and applications with a priori estimates abstract in this paper we prove a general result giving the maximum and the antimaximum principles in a unitary way for linear operators of the form l i provided that 0 is an eigenvalue of l with associated constant eigenfunctions to this purpose we introduce a new notion of quasi uniform maximum principle named k uniform maximum principle it holds for belonging to certain neighborhoods of 0 depending on the fixed positive multiplier k 0 which selects the good class of right hand sides our approach is based on a l l p estimate for some related problems as an application we prove some generalization and new results for elliptic problems and for time periodic parabolic problems under neumann boundary conditions

global existence and decay properties for a degenerate keller segel model with a power factor in drift term abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u q 1 v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where m 1 q 2 0 or 1 and n 1 the aim of this paper is to prove the existence of a time global weak solution u v of ks with the l 0 l r n bound such a global bound is obtained in the case of i m q 2 n for large initial data and ii 1 m q 2 n for small initial data in the case of ii the decay properties of the solution u v are also discussed

global existence and gradient estimates for the quasilinear parabolic equations of m laplacian type with a nonlinear convection term abstract in this paper we derive precise estimates for u t including smoothing effects near t 0 and decay as t as well as global existence of the solutions u t to the initial boundary value problem in a bounded domain in r n for the quasilinear parabolic equation of the m laplacian type with a nonlinear convection term b u u for the initial data u 0 we only assume u 0 l q 1 q

a pairing between super lie rinehart and periodic cyclic homology we consider a pairing producing various cyclic hochschild cocycles which led alain connes to cyclic cohomology we are interested in geometrical meaning and homological properties of this pairing we define a non trivial pairing between the homology of a lie rinehart super algebra with coefficients in some partial traces and relative periodic cyclic homology this pairing generalizes the index formula for summable fredholm modules the connes kubo formula for the hall conductivity and the formula computing the k0 group of a smooth noncommutative torus it also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected lie group finally we compare it with the characteristic map for the hopf cyclic cohomology

global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where 0 or 1 we show here that the system of ks with m 1 has a time global weak solution u v with a uniform bound in time when u 0 v 0 is a nonnegative function and in l 1 l r n l 1 h 1 w 1 r n u 0 m h 1 r n the decay properties of the solution with small initial data are also discussed

instability of positive periodic solutions for semilinear pseudo parabolic equations with logarithmic nonlinearity abstract in this paper we consider the periodic problem for semilinear heat equation and pseudo parabolic equation with logarithmic source after establishing the existence of positive periodic solutions we discuss the instability of such solutions

on the cauchy problem for the evolution p laplacian equations with gradient term and source and measures as initial data abstract in this paper we study the cauchy problem for the evolution p laplacian equations with gradient term and source on the assumption that measures as initial conditions

positive periodic solutions for a system of anisotropic parabolic equations abstract we consider a system of two degenerate parabolic equations with nonlocal terms and dirichlet boundary conditions more precisely the degeneracy in each equation of the system is of the type r x laplacian where r x is a function depending on x where is a bounded smooth domain of r n the system models the diffusion and the interaction between two different biological species sharing the same territory the paper provides conditions on the parameters of the problem that guarantee the coexistence of a t periodic non negative solution u v with both non trivial u v

periodic solutions of evolutionp laplacian equations with nonlinear sources abstract in this paper we are concerned with the time periodic solutions to the evolution p laplacian equations of the form formula and formula with the dirichlet boundary value condition where p 2 0 is a bounded convex domain in r n with smooth boundary m x t is continuous on r periodic in t with period and positive in r the existence of nontrivial nonnegative periodic solutions is established provided that n 1 and p 1 p 1 p n