equilibria of set valued maps on nonconvex domains we present new theorems on the existence of equilibria or zeros of convex as well as nonconvex set valued maps defined on compact neighborhood retracts of normed spaces the maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets among other things we show that if k is a compact neighborhood retract with nontrivial euler characteristic in a banach space e and 1 k 2e is an upper hemicontinuous set valued map with nonempty closed convex values satisfying the tangency condition x n tk x 0 for all x c k then there exists xo c k such that 0 c 4 xo here tk x denotes a new concept of retraction tangent cone to k at x suited for compact neighborhood retracts when k is locally convex at x tk x coincides with the usual tangent cone of convex analysis special attention is given to neighborhood retracts having lipschitzian behavior called l retracts below this class of sets is very broad it contains compact homeomorphically convex subsets of banach spaces epi lipschitz subsets of banach spaces as well as proximate retracts our results thus generalize classical theorems for convex domains as well as recent results for nonconvex sets

equilibria of nonconvex valued maps under constraints abstract in the paper the notion of n tangency for set valued maps defined on a subset of a banach space is considered the existence of equilibria of upper semicontinuous map being n tangent to a sleek retract with the nontrivial euler characteristic is established

fixed point theorems and morse s lemma for lipschitzian functions abstract we prove a fixed point theorem for set valued mappings defined on a nonempty compact subset x of r n which can be represented by inequality constraints i e x x r n f x 0 f locally lipschitzian and satisfying a nondegeneracy assumption outside of x this class of sets extends significantly the class of convex compact sets with a nonempty interior topological properties of such sets x are proved continuous deformation retract of a ball acyclicity as a consequence of a generalization of morse s lemma for lipschitzian real valued function defined on r n a result also of interest for itself

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

power law distributions for the free path length in lorentz gases it is well known that in the boltzmann grad limit the distribution of the free path length in the lorentz gas with disordered scatterer configuration has an exponential density if on the other hand the scatterers are located at the vertices of a euclidean lattice the density has a power law tail proportional to xi 3 in the present paper we construct scatterer configurations whose free path lengths have a distribution with tail xi n 2 for any positive integer n we also discuss the properties of the random flight process that describes the lorentz gas in the boltzmann grad limit the convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces which in turn is a consequence of ratner s measure classification theorem

the distribution of the free path lengths in the periodic two dimensional lorentz gas in the small scatterer limit we study the free path length and the geometric free path length in the model of the periodic two dimensional lorentz gas sinai billiard we give a complete and rigorous proof for the existence of their distributions in the small scatterer limit and explicitly compute them as a corollary one gets a complete proof for the existence of the constant term c 2 3 ln 2 frac 27 zeta 3 2 pi 2 in the asymptotic formula h t 2 ln epsilon c o 1 of the ks entropy of the billiard map in this model as conjectured by p dahlqvist

free path lengths in quasicrystals previous studies of kinetic transport in the lorentz gas have been limited to cases where the scatterers are distributed at random e g at the points of a spatial poisson process or at the vertices of a euclidean lattice in the present paper we investigate quasicrystalline scatterer configurations which are non periodic yet strongly correlated a famous example is the vertex set of a penrose tiling our main result proves the existence of a limit distribution for the free path length which answers a question of wennberg the limit distribution is characterised by a certain random variable on the space of higher dimensional lattices and is distinctly different from the exponential distribution observed for random scatterer configurations the key ingredients in the proofs are equidistribution theorems on homogeneous spaces which follow from ratner s measure classification

lyapunov spectrum of the many dimensional dilute random lorentz gas for a better understanding of the chaotic behavior of systems of many moving particles it is useful to look at other systems with many degrees of freedom an interesting example is the high dimensional lorentz gas which just like a system of moving hard spheres may be interpreted as a dynamical system consisting of a point particle in a high dimensional phase space moving among fixed scatterers in this paper we calculate the full spectrum of lyapunov exponents for the dilute random lorentz gas in an arbitrary number of dimensions we find that the spectrum becomes flatter with increasing dimensionality furthermore for fixed collision frequency the separation between the largest lyapunov exponent and the second largest one increases logarithmically with dimensionality whereas the separations between lyapunov exponents of given indices not involving the largest one go to fixed limits

hybrid reynolds averaged navier stokes large eddy simulations of supersonic turbulent mixing aturbulentmixinglayerformedfromtwosupersonicstreamsinitially separatedbyasplitterplateisinvestigated withahybridcomputationalmethodusingareynolds averagednavier stokes rans approachforwall bounded regions and a large eddy simulation les approach for the turbulent mixing region simulations of the mixing layer predict a vortex shedding originating from the splitter base and then a rapid transition to turbulence which is in agreement with experimental observations as a result the potential of the hybrid method is demonstrated for e ows in which a geometric feature such as the blunt splitter plate considered here provides the dominant unsteady feature leading to turbulence without imposing additional ine ow disturbances parametric studies of les subgrid modeling settings variations in the spanwise computational domain and wall temperature settings in the rans region were conducted the subgrid model cases using a baseline computational grid with small spanwise computational domain overpredicted the mixing layer turbulencelevels but only showed small variation in the mixing layer predictions with large changes in the model parameters the cases examining wider spanwise domains enabled more turbulent energy to be released in the spanwise direction which in turn reduced axial and vertical turbulence levels finally prescribing the wall temperatures in the rans regions instead of using the more traditional adiabatic wall boundary conditions further reduced turbulence levels and enabled reasonable agreement with experimental data

representations of epi lipschitzian sets abstract a closed subset m of a banach space e is epi lipschitzian i e can be represented locally as the epigraph of a lipschitz function if and only if it is the level set of some locally lipschitz function f e r for which clarke s generalized gradient does not contain 0 at points in the boundary of m i e such that m x f x 0 0 f x if x bd m this extends the characterization previously known in finite dimension and answers to a standing question

the free path in a high velocity random flight process associated to a lorentz gas in an external field we investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity the random flight models we study arise naturally as the boltzmann grad limit of a random lorentz gas in the presence of an external field by analyzing the time duration of the free path we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers as a consequence we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight

diffusion in a weakly random hamiltonian flow we consider the motion of a particle governed by a weakly random hamiltonian flow we identify temporal and spatial scales on which the particle trajectory converges to a spatial brownian motion the main technical issue in the proof is to obtain error estimates for the convergence of the solution of the stochastic acceleration problem to a momentum diffusion we also apply our results to the system of random geometric acoustics equations and show that the energy density of the acoustic waves undergoes a spatial diffusion

there is more to be learned from the lorentz model the classical lorentz model for charged noninteracting point particles in a perpendicular magnetic field is reconsidered in 2d we show that the standard boltzmann equation is not valid for this model even in the grad limit we construct a generalized boltzmann equation which is and solve the corresponding initial value problem exactly by an independent calculation we find the same solution by directly constructing the green function from the dynamics of the model in the grad limit from this solution an expression for the diffusion tensor valid for arbitrary short range forces is derived for hard disks we calculate the diffusion tensor explicitly away from the grad limit a percolation problem arises we determine numerically the percolation threshold and the corresponding geometric critical exponents the numerical evidence strongly suggests that this continuum percolation model is in the universality class of 2d lattice percolation although we have explicitly determined a number of limiting properties of the model several intriguing open problems remain

the boltzmann grad limit of the periodic lorentz gas we study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density boltzmanngrad limit a generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a markov process with memory two

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

periodic solutions for nonlinear evolution equations at resonance abstract we are concerned with periodic problems for nonlinear evolution equations at resonance of the form u t a u t f t u t where a densely defined linear operator a d a x on a banach space x is such that a generates a compact c 0 semigroup and f 0 x x is a nonlinear perturbation imposing appropriate landesman lazer type conditions on the nonlinear term f we prove a formula expressing the fixed point index of the associated translation along trajectories operator in the terms of a time averaging of f restricted to ker a by the formula we show that the translation operator has a nonzero fixed point index and in consequence we conclude that the equation admits a periodic solution

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

the averaging principle and periodic solutions for nonlinear evolution equations at resonance abstract we study the existence of t periodic solutions t 0 for the first order differential equations being at resonance at infinity where the right hand side is the perturbations of a sectorial operator our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball in terms of special geometrical assumptions imposed on the nonlinearity we also prove that the geometrical assumptions are generalizations of well known landesman lazer and strong resonance conditions the obtained index formula is used to derive the criteria determining the existence of t periodic solutions for the heat equation being at resonance at infinity

generalized poincare hopf theorem for compact nonsmooth regions this paper presents an extension of the poincare hopf theorem to generalized critical points of a function on a compact region with nonsmooth boundary m defined by a finite number of smooth inequality constraints given a function f m rn we define the generalized critical points of f over m define the index for the critical point and show that the sum of the indices of the critical points is equal to the euler characteristic of m we use the generalized poincare hopf theorem to present sufficient local conditions for the uniqueness of solutions to finite dimensional variational inequalities and the uniqueness of stationary points in nonconvex optimization problems

fixed points and equilibria in nonconvex sets tcentre de recherches mathtmatiques universite de montreal montreal quebec h3c 357 canada psteklov institute of mathematics moscow 117966 russia and ildepartment of mathematics and statistics concordia university montreal quebec h4b lr6 canada received 30 september 1993 received forpublication 18 august 1994 key words and phrases multivalued mapping tangent cones zeros fixed points differential inclusion weak invariance nonsmooth analysis 1 introduction

on the topological regularity of the solution set of differential inclusions with constraints abstract in this paper we establish the topological regularity of the solution set of differential inclusions with constraints defined in r n the result is then extended to systems defined in banach spaces our proof makes use of an approximation result by lipschitz functions of caratheodory functions which we also prove in this paper

the galton board limit theorems and recurrence we study a particle moving in r2 under a constant external force and bouncing off a periodic array of convex domains scatterers the latter must satisfy a standard finite horizon condition to prevent ballistic collision free motion this model is known to physicists as galton board it is also identical to a periodic lorentz gas previous heuristic and experimental studies have suggested that the particle s speed v t should grow as t1 3 and its coordinate x t as t2 3 we prove these conjectures rigorously we also find limit distributions for the rescaled velocity t 1 3v t and position t 2 3x t in addition quite unexpectedly we discover that the particle s motion is recurrent that means that a ball dropped on an idealized galton board will roll down but from time to time it should bounce all the way back up with probability one

the constrained degree and fixed point index theory for set valued maps abstract in the first part of the paper we give a construction of a topological degree theory for set valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds in the second part we propose a fixed point index for inward maps on arbitrary closed convex sets

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

topological theory of non autonomous parabolic evolution inclusions on a noncompact interval and applications in this paper we consider the topological structure of the solution set of non autonomous parabolic evolution inclusions with time delay defined on non compact intervals the result restricted to compact intervals is then extended to non autonomous parabolic control problems with time delay moreover as the applications of the information about the structure we establish the existence result of global integral solutions for non autonomous cauchy problems subject to nonlocal condition and prove the invariance of a reachability set for non autonomous control problems under single valued nonlinear perturbations finally some illustrating examples are supplied

existence of equilibria of set valued maps on bounded epi lipschitz domains in hilbert spaces without invariance conditions abstract in this paper we provide a new result of the existence of equilibria for set valued maps on bounded closed subsets k of hilbert spaces we do not impose either convexity or compactness assumptions on k but we assume that k has epi lipschitz sections i e its intersection with suitable finite dimensional spaces is locally the epigraph of lipschitz functions in finite dimensional spaces the famous brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set k to itself our result could be viewed as a kind of generalization of this classical result in the context of hilbert spaces and when the function or the set valued map does not necessarily map k into itself k is not invariant under the map our approach is based firstly on degree theory for compact and for condensing set valued maps and secondly on flows generated by trajectories of differential inclusions

fixed point index for weakly inward mappings abstract the fixed point index for weakly inward mappings which are not cone mappings is defined in this paper its properties are also investigated consequently we study the derivatives of weakly inward mappings and nonzero fixed points of such mappings

computations of flow past a circular cylinder using a continuous turbulence model modeling is calculated with reynolds averaged navier stokes equations using a function that is uniquely determined by the ratio of the resolved small scale length to the integral length scale the unified treatment of reynolds averaged navier stokes large eddy simulation could be achieved with this continuous model by controlling the coarseness of the mesh the continuous reynolds averaged navier stokes large eddy simulation model is initially tested with flow past a circular cylinder at the reynolds number re 3900 both constant integral length scales and varied integral length scales based on the mixing length are used in the simulations the mesh dependence is also studied with both coarse and fine meshes the comparisons are made and analyzed between the results with different integral length scales and different mesh sizes these results are also compared with the experimental results and reynolds averaged navier stokes results the results demonstrate that with the continuous modeling the large eddy simulation like simulation can be achieved by solving reynolds averaged navier stokes equations alone the spatially varied integral length scale is necessary to capture the more accurate turbulence quantities for anisotropic wall bounded turbulent flow

constrained topological degree and positive solutions of fully nonlinear boundary value problems abstract in the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form a f where a d a e is a densely defined m accretive operator in a banach space e and f u e is a tangent field defined on an open subset u of a neighborhood retract m being invariant with respect to the resolvents of a the construction is performed under the assumption that resolvents of a are completely continuous in the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction diffusion equations and equations with one dimensional p laplacian with possibly non positive perturbations as well as some controlled neumann like problems

nonlinear evolution inclusions topological characterizations of solution sets and applications this paper deals with a nonlinear delay differential inclusion of evolution type involving m dissipative operator and source term of multi valued type in a banach space under rather mild conditions the r structure of c0 solution set is studied on compact intervals which is then used to obtain the r property on non compact intervals secondly the result about the structure is furthermore employed to show the existence of c0 solutions for the inclusion mentioned above subject to nonlocal condition defined on right half line no nonexpansive condition on nonlocal function is needed as samples of applications we consider a partial differential inclusion with time delay and then with nonlocal condition at the end of the paper

a mechanical model of brownian motion we consider a dynamical system consisting of one large massive particle and an infinite number of light point particles we prove that the motion of the massive particle is in a suitable limit described by the ornstein uhlenbeck process this extends to three dimensions previous results by holley in one dimension

positive stationary solutions for p laplacian problems with nonpositive perturbation the paper is devoted to the existence of positive solutions of nonlinear elliptic equations with p laplacian we provide a general topological degree that detects solutions of the problem a u f u u m where a x d a x is a maximal monotone operator in a banach space x and f m x is a continuous mapping defined on a closed convex cone m x next we apply this general framework to a class of partial differential equations with p laplacian under dirichlet boundary conditions in the paper we employ general ideas from cwiszewski and kryszewski 2009 5 where a setting suitable for the one dimensional p laplacian was introduced

semi markov approach to continuous time random walk limit processes continuous time random walks ctrws are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences their scaling limits are typically non markovian and the computation of their finite dimensional distributions is an important open problem this paper develops a general semi markov theory for ctrw limit processes in mathbb r d with infinitely many particle jumps renewals in finite time intervals the particle jumps and waiting times can be coupled and vary with space and time by augmenting the state space to include the scaling limits of renewal times a ctrw limit process can be embedded in a markov process explicit analytic expressions for the transition kernels of these markov processes are then derived which allow the computation of all finite dimensional distributions for ctrw limits two examples illustrate the proposed method

on open loop and feedback attainability of a closed set for nonlinear control systems abstract in this note we investigate the existence of controls which allow to reach a given closed set k through trajectories of a nonlinear control system in the case where the set is sufficiently regular we give a condition allowing to find a feedback control law which ensures the existence of trajectories to reach the set we also consider the case where all the trajectories reach k when k is not necessarily attainable but only viable we build a set valued feedback for which the set is invariant our approach concerns continuous dynamics possibly not c1 so our methods do not come from geometric control theory furthermore we do not require any regularity of the set k in order to obtain our results except when we want to establish the existence of a feedback control law to achieve our goals

fractal mobile immobile solute transport 1 a fractal mobile immobile model for solute transport assumes power law waiting times in the immobile zone leading to a fractional time derivative in the model equations the equations are equivalent to previous models of mobile immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times the solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase in this regard the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase the solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers including power law breakthrough curves at late time and power law decline in the measured mobile mass the made site mobile tritium mass decline is consistent with a fractional time derivative of order 0 33 while haggerty et al s 2002 stream tracer test is well modeled by a fractional time derivative of order 0 3

heterogeneous memorized continuous time random walks in an external force fields in this paper we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated memorized waiting times which involves reimann liouville fractional derivative or reimann liouville fractional integral we show that the mean squared displacement of the test particle x x which is dependent on its location x of the form el wakil and zahran chaos solitons fractals 12 1929 1935 2001 begin aligned langle mathbb x x 2 rangle t langle delta x x t 2 rangle 0 sim x theta t gamma quad 0 gamma 1 quad theta d w 2 end aligned 1 where d w 2 is the anomalous exponent the diffusion exponent gamma is dependent on the model parameters we obtain the fokker planck type dynamic equations and their stationary solutions are of the boltzmann gibbs form these processes obey a generalized einstein stokes smoluchowski relation and the second einstein relation we observe that the asymptotic behavior of waiting times and subordinations are of stretched gaussian distributions we also discuss the time averaged in the case of an harmonic potential and show that the process exhibits aging and ergodicity breaking

topological degree methods for perturbations of operators generating compact c 0 semigroups abstract the main aim of this paper is to construct a topological degree for maps a f m d a e where a densely defined closed operator a d a e of a banach space e is such that a is the generator of a compact c 0 semigroup and f m e is a locally lipschitz map defined on a neighborhood retract m e if m is a closed convex cone then a degree formula allowing an effective computation of the degree is proved this formula provides an infinite dimensional counterpart of the well known krasnosel skii theorem by the use of the introduced topological degree and an abstract result concerning branching of fixed points the bifurcation of periodic points of the parameterized boundary value problem u au f t u 0 u t m u 0 u t is studied examples of applications to partial differential equations are discussed

diffusion approximation for a class of transport processes with physical reflection boundary conditions consider a stochastic process consisting of the pair of a position and a velocity in a piecewise el d dimensional domain in the interior of the domain the dynamics are assigned by a potential and by random changes of the velocity occurring at exponentially distributed times according to a probability distribution which may depend on the current position and velocity on the boundary the process reflects physically the angle of reflection equals the angle of incidence first it is shown that the process is well defined for all times then when the coefficients depend on a diverging parameter n in particular such that the speed and the jump rate of the velocity go to xo with order rn and at least n respectively a diffusion approximation is sought the position process is represented as a solution of a skorohod reflection equation a skewing effect on the boundary results from the interaction between the dynamics and the reflection law so that the direction of reflection is in general oblique the assumption that the mean change of the velocity in the interior is linear in the current velocity up to order at least 1 2 in 1 n ensures that the cone of directions of reflection is independent of n the continuity properties of the skorohod oblique reflection problem enable one to show tightness of the position processes without having to estimate explicitly local times boundary layers and together with a suitable law of large numbers for the velocity allow one to identify the limit stochastic differential equation with oblique reflection the theory is illustrated by several applications in particular one to a mechanical model of brownian motion

the bolzano mean value theorem and partial differential equations abstract we study the existence of solutions to abstract equations of the form 0 a u f u u k e where a is an abstract differential operator acting in a banach space e k is a closed convex set of constraints being invariant with respect to resolvents of a and a perturbation f satisfies a certain tangency condition such problems are closely related to the so called poincare miranda theorem being the multi dimensional counterpart of the celebrated bolzano intermediate value theorem in fact our main results should be regarded as infinite dimensional variants of bolzano and miranda poincare theorems along with single valued problems we deal with set valued ones yielding the existence of the so called constrained equilibria of set valued maps the abstract results are applied to show existence of strong steady state solutions to some weakly coupled systems of drift reaction diffusion equations or differential inclusions of this type in particular we get the existence of strong solutions to the dirichlet neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type certain aspects of the bernstein theory for bvp for second order ode are studied too no assumptions concerning structural coupling monotonicity cooperativity are undertaken

on the solution sets of differential inclusions and the periodic problem in banach spaces abstract in this paper the topological structure of the solution set of a constrained semilinear differential inclusion in a banach space e is studied it is shown that the set of all mild solutions with values in a closed and in general thin subset d e is an r set provided natural boundary conditions and appropriate geometrical assumptions on d which hold e g when d is convex are satisfied applications to the periodic problem and to the existence of equilibria are given

homotopy invariants for tangent vector fields on closed sets abstract in the paper we construct a topological degree theory for single and convex valued tangent vector fields defined on locally compact closed subsets of a banach space the obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds the degree allows to study vector fields on sets which are neither smooth nor convex and may be applied to study continuation and bifurcation of equilibria in parameterized families of closed sets

fixed points of weakly inward multis soit x un espace de banach reel k x convexe ferme borne f k 2 x semicontinue superieurement et condensant f x convexe ferme x k et f faiblement interieur sur k alors f a un point fixe

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

coincidences for admissible and φ maps and minimax inequalities abstract a coincidence theory for and admissible maps is presented in this paper one can also deduce from this theory new minimax inequalities of von neumann sion type

existence for reaction diffusion systems a compactness method approach the authors study the existence of weak solutions for the following system ut u f u v vt v g u v in 0 t u v 0 on 0 t u 0 x u0 x v 0 x v0 x in the region rn with smooth boundary the functions r r are assumed to be continuous and nondecreasing with 0 0 0 u0 v0 l f g r2 2r with f an upper semicontinuous mapping u s c the following local existence results are shown 1 for the diffusive case i e when both and are strictly increasing with u s c g 2 for the semi diffusive case only one function is strictly increasing with g being either with separated variables i e having the form of the product or of the sum of two functions g u and h v or globally lipschitz with respect to its second variable i e g u v g u v l v v for each u b r v v r and some l l b additional conditions of linear form on the growth of f and g are indicated to guarantee global existence results