complete quenching for a quasilinear parabolic equation abstract we study the homogeneous dirichlet problem for the quasilinear parabolic equation with the singular absorption term t u p u 1 u 0 u f x u in q t 0 t here r d d 1 is a bounded domain p u div u p 2 u is the p laplace operator and 0 1 is a given parameter it is assumed that the initial datum satisfies the conditions u 0 w 0 1 p l u 0 0 a e in the right hand side f r 0 is a caratheodory function satisfying the power growth conditions 0 f x s s q 1 c with positive constants c and q 1 we establish conditions of local and global in time existence of nonnegative solutions and show that if q p and and c are sufficiently small then every global solution vanishes in a finite time a e in

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

quasi factors of zero entropy systems for minimal systems x t of zero topological entropy we demon strate the sharp difference between the behavior regarding entropy of the sys tems m x t and 2x t induced by t on the spaces m x of prob ability measures on x and 2x of closed subsets of x it is shown that the system m x t has itself zero topological entropy two proofs of this the orem are given the first uses ergodic theoretic ideas the second relies on the different behavior of the banach spaces ln and ln with respect to the existence of almost hilbertian central sections of the unit ball in contrast to this theorem we construct a minimal system x t of zero entropy with a minimal subsystem y t of 2 t whose entropy is positive school of mathematics tel aviv university ramat aviv israel e mail address glasner0math tau ac il institute of mathematics hebrew university jerusalem israel e mail address weissfmath huji ac il this content downloaded from 207 46 13 117 on sun 23 oct 2016 04 35 33 utc all use subject to http about jstor org terms

extinction properties of solutions for a class of fast diffusive p laplacian equations abstract we consider the extinction properties of solutions for the homogeneous dirichlet boundary value problem for the p laplacian equation u t div u p 2 u u q u r with 1 p 2 q 1 and r 0 for 0 it is known that r p 1 is the critical extinction exponent for the weak solution for 0 we show that r p 1 is still the critical extinction exponent when q 1 moreover the precise decay estimates of solutions before the occurrence of the extinction are derived however extinction can always occur when 0 q r 1

a note on blow up of solutions of a quasilinear heat equation with vanishing initial energy abstract in this work we consider an initial boundary value problem related to the equation u t div u m 2 u f u and prove under suitable conditions on f a blow up result for solutions with vanishing or negative initial energy

well posedness problem of an anisotropic parabolic equation abstract in this paper we are concerned with well posedness of an anisotropic parabolic equation with the convection term when some diffusion coefficients are degenerate on the boundary and the others are positive on we propose a novel partial boundary value condition to study the stability of the solutions for the anisotropic parabolic equation a new concept the general characteristic function of the domain is introduced and applied the existence and stability of the solutions is established under the given partial boundary value conditions

singular and degenerate anisotropic parabolic equations with a nonlinear source abstract the dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered we state conditions that guarantee the existence and uniqueness of a global weak solution to the problem a similar result is proved for the parabolic p laplace equation

conservation laws and symmetries of radial generalized nonlinear p laplacian evolution equations abstract a class of generalized nonlinear p laplacian evolution equations is studied these equations model radial diffusion reaction processes in n 1 dimensions where the diffusivity depends on the gradient of the flow for this class all local conservation laws of low order and all lie symmetries are derived the physical meaning of the conservation laws is discussed and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases the symmetries are used to derive exact group invariant solutions from solvable three dimensional subgroups of the full symmetry group which yields a direct reduction of the nonlinear equation to a quadrature the physical and analytical properties of these exact solutions are explored some of which describe moving interfaces and green s functions

blow up phenomena for a doubly degenerate equation with positive initial energy abstract in this paper an initial boundary value problem related to the equation u t div u m p 2 u m f u is studied under suitable conditions on f we establish a blow up result for certain solution with positive initial energy and blow up time will be also considered by using the differential inequality technique the upper estimate of the blow up solution near the single blow up point is also obtained

the blow up of solutions for m laplacian equations with variable sources under positive initial energy abstract this paper deals with homogeneous dirichlet boundary value problem to a class of m laplace equations with variable reaction u t div u m 2 u u q x x t 0 the bounded domain r n n 1 with a smooth boundary we prove that the weak solutions of the above problems blow up in finite time for all q m 1 m 2 when the initial energy is positive and initial data is suitably large this result improves the recent result by zhou and yang 2015 which asserts the blow up of solutions for n m provided that q n m m n n m

existence and uniqueness of local solutions to a class of quasilinear degenerate parabolic systems abstract in this paper the authors study an initial and boundary value problem to a system of evolution p laplacian equations coupled with general nonlinear terms the authors use the method of regularization to construct a sequence of approximation solutions with the help of monotone iteration technique and hence obtain the existence of solutions to a regularized system of equations then the existence of solutions to the system of evolution p laplacian equations is obtained with the use of a standard limiting process the uniqueness of the solution is also proven

on the k property of quantized arnold cat maps we prove that some quantized arnold cat maps are entropic k systems this result was formulated by h narnhofer j math phys 33 1502 1510 1992 but the fact that the optimal decomposition for the multi channel entropy constructed there is not strictly local was not appropriately taken care of we propose a strictly local decomposition based on a construction of voiculescu

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

well posedness and gradient blow up estimate near the boundary for a hamilton jacobi equation with degenerate diffusion abstract this paper is concerned with weak solutions of the degenerate diffusive hamilton jacobi equation t u p u u q with dirichlet boundary conditions in a bounded domain r n where p 2 and q p 1 with the goal of studying the gradient blow up phenomenon for this problem we first establish local well posedness with blow up alternative in w 1 norm we then obtain a precise gradient estimate involving the distance to the boundary it shows in particular that the gradient blow up can take place only on the boundary a regularizing effect for t u is also obtained

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

quantized arnold cat maps can be entropic k systems automorphisms on the irrational rotation algebra with respect to their ergodic properties are studied especially it is shown that for a dense set of the rotation parameter cat maps are entropic k systems

turbulence amalgamation and generic automorphisms of homogeneous structures we study topological properties of conjugacy classes in polish groups with emphasis on automorphism groups of homogeneous countable structures we first consider the existence of dense conjugacy classes the topological rokhlin property we then characterize when an automorphism group admits a comeager conjugacy class answering a question of truss and apply this to show that the homeomorphism group of the cantor space has a comeager conjugacy class answering a question of akin hurley and kennedy finally we study polish groups that admit comeager conjugacy classes in any dimension in which case the groups are said to admit ample generics we show that polish groups with ample generics have the small index property generalizing results of hodges hodkinson lascar and shelah and arbitrary homomorphisms from such groups into separable groups are automatically continuous moreover in the case of oligomorphic permutation groups they have uncountable cofinality and the bergman property these results in particular apply to automorphism groups of many stable 0 categorical structures and of the random graph in this connection we also show that the infinite symmetric group s has a unique non trivial separable group topology for several interesting groups we also establish serre s properties fh and fa

stability of the solutions of a convection diffusion equation abstract a class of non newtonian fluid equations with a convection term is considered if the diffusion coefficient is degenerate on the boundary the interaction between the convection and diffusion is investigated and the stability of solutions is established without any boundary condition

a new condition for the concavity method of blow up solutions to p laplacian parabolic equations abstract in this paper we consider an initial boundary value problem of the p laplacian parabolic equation u t x t div u x t p 2 u x t f u x t x t 0 u x t 0 x t 0 u x 0 u 0 0 x where p 2 and is a bounded domain of r n n 1 with smooth boundary the main contribution of this work is to introduce a new condition c p 0 u f s d s u f u u p u 0 for some 0 with 0 p 1 p p where 1 p is the first eigenvalue of p laplacian p and we use the concavity method to obtain the blow up solutions to the above equations in fact it will be seen that the condition c p improves the conditions ever known so far

bowen topological entropy of subsets for amenable group actions abstract we extend the definition of bowen topological entropy of subsets to continuous action of amenable groups on a compact metrizable space we investigate how bowen topological entropy behaves for restricted actions of finite index subgroups and verify the validity of the subgroup formula we also show that the bowen topological entropy of subsets for amenable group actions can be determined via the local entropies of measures as applications we compute the bowen topological entropy of a subset for bernoulli shift over amenable groups

local entropy theory for a countable discrete amenable group action in the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action for such an action we introduce entropy tuples in both topological and measure theoretic settings and build the variational relation be tween these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover moreover based the idea of topological entropy pairs we introduce and study two special classes of such an action uniformly positive entropy and completely positive entropy note that in the building of the local variational principle following romagnoli s ideas two kinds of measure theoretic entropy are introduced for finite borel covers these two kinds of entropy turn out to be the same where danilenko s orbital approach becomes an inevitable tool

single point gradient blow up on the boundary for a hamilton jacobi equation with p laplacian diffusion we study the initial boundary value problem for the hamilton jacobi equation with nonlinear diffusion u t delta p u nabla u q in a two dimensional domain for q p 2 it is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded we show that for suitably localized and monotone initial data the gradient blow up occurs at a single point of the boundary such a result was known up to now only in the case of linear diffusion p 2 the analysis in the case p 2 is considerably more delicate

the rohlin property for shifts on uhf algebras and automorphisms of cuntz algebras abstract we show that the shift on the uhf algebra m n has the rohlin property for any odd n hence by brotteli et al the crossed product of a uhf algebra by a shift ergodic theory dynam systems 13 1993 615 626 for any n 2 3 and that any automorphism of the cuntz algebra o n n m is outer for any m 0

independence in topological and c dynamics we develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods one of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties we also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to c dynamics

study of weak solutions for parabolic equations with nonstandard growth conditions abstract the authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for u t div u d 0 u p x t 2 u f x t localization property of weak solutions is also discussed

global and blow up solutions to a p laplacian equation with nonlocal source this paper deals with a p laplacian equation ut div u p 2 u uq x t dx with null dirichlet boundary conditions in a bounded domain rn where p 2 q 1 under appropriate hypotheses we establish local theory of the solution and obtain that the solution either exists globally or blows up in finite time

extinction and positivity for the evolution p laplacian equation with l1 initial value abstract the aims of this paper are to discuss extinction and positivity for the evolution p laplacian equation u t x u x p 2 u x with initial value u 0 x l 1 and zero boundary value conditions where p 1 in particular the necessary and sufficient condition for extinction is obtained

extinction and positivity for the evolution p laplacian equation in rn abstract the aim of this paper is to discuss extinction and positivity for the solution of the initial boundary value problem of evolution p laplacian equation it is proved that the weak solution will be extincted for 1 p 2 and will be positive for p 2 for large t

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

combinatorial lemmas and applications to dynamics abstract the well known combinatorial lemma of karpovsky milman and alon and a very recent one of kerr and li are extended the obtained lemmas are applied to study the maximal pattern entropy introduced in the paper it turns out that the maximal pattern entropy is equal to the supremum of sequence entropies over all sequences both in topological and measure theoretical settings moreover it is shown the maximal pattern entropy of any topological system is log k for some k n with k the maximal length of intrinsic sequence entropy tuples and a zero dimensional system has zero sequence entropy for any sequence if and only if the maximal pattern with respect to any open cover is of polynomial order

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

crossed products and entropy of automorphisms abstract let a be an exact c algebra let g be a locally compact group and let a g be a c dynamical system each automorphism g induces a spatial automorphism ad g on the reduced crossed product a g in this paper we examine the question first raised by e stormer of when the topological entropies of g and ad g coincide this had been answered by n brown for the particular case of discrete abelian groups using different methods we extend his result to preservation of entropy for g when the subgroup of aut g generated by the corresponding inner automorphism ad g has compact closure this property is satisfied by all elements of a wide class of groups called locally fia this class includes all abelian groups both discrete and continuous as well as all compact groups

the problem of dirichlet for evolution one dimensional p laplacian with nonlinear source in the present paper we consider the dirichlet problem for one dimensional p laplacian with nonlinear source we obtain new a priori estimates of a solution and of the gradient of a solution and formulate conditions guaranteeing the global solvability of this problem our consideration includes singular case as well

global existence and blow up of solutions to an evolution p laplace system coupled via nonlocal sources abstract the aim of this paper is to investigate the behavior of positive solutions to the following system of evolution p laplace equations coupled via nonlocal sources u t u x p 1 1 u x x 0 a v m 1 t d x t in 0 a 0 t v t v x p 2 1 v x x 0 a u m 2 t d x t in 0 a 0 t with nonlinear boundary conditions u x x 0 0 u x x a u q 11 v q 12 x a v x x 0 0 v x x a u q 21 v q 22 x a and the initial data u 0 v 0 where p 1 p 2 1 m 1 m 2 q 11 q 12 q 21 q 22 0 under appropriate hypotheses the authors first prove a local existence result by a regularization method then the authors discuss the global existence and blow up of positive weak solutions by using a comparison principle

dynamical entropy in banach spaces we introduce a version of voiculescu brown approximation entropy for isometric automorphisms of banach spaces and develop within this framework the connection between dynamics and the local theory of banach spaces as discovered by glasner and weiss our fundamental result concerning this contractive approximation entropy or ca entropy characterizes the occurrence of positive values both geometrically and topologically this leads to various applications for example we obtain a geometric description of the topological pinsker factor and show that a c algebra is type i if and only if every multiplier inner automorphism has zero ca entropy we also examine the behaviour of ca entropy under various product constructions and determine its value in many examples including isometric automorphisms of l p for 1 p and noncommutative tensor product shifts

entropy and mixing for amenable group actions for i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space written asft g 2i acting on x f we sayft g 2i has completely positive entropy or simply cpe for short if for any flnite and nontrivial partition p of x the entropy h t p is not zero our goal is to demonstrate what is well known for actions of and even d that actions of completely positive entropy have very strong mixing properties let si be a list of flnite subsets of i we say the si spread if any particular 6 id belongs to at most flnitely many of the sets sis i1 i theorem 0 1 for ft g 2i an action of i of completely positive entropy and p any flnite partition for any sequence of flnite sets si i which spread we have

the extinction versus the blow up global and non global existence of solutions of source types of degenerate parabolic equations with a singular absorption abstract we consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term t u u x p 2 u x x u u 0 f u x t in i 0 t with the homogeneous zero boundary condition on i x 1 x 2 an open bounded interval in r through this paper we assume that p 2 and 0 1 to show the local existence result we prove first a sharp pointwise estimate for u x one of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t 0 the so called global solutions or by the contrary a finite time blow up 0 0 arises such that lim t 0 u t l i moreover we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough finally we show that the condition f 0 x t 0 x t i 0 is a necessary and sufficient condition for the existence of solution of equations of this type

on stability of linear time varying infinite dimensional discrete time systems the asymptotic behaviour of linear time varying infinite dimensional discrete lime systems is considered the introduced notions are weak power equistability power equistability uniform power equistability l p equistability uniform l p equistability and l p x equistability it is shown that they are identical a generalization of the concept of spectral radius of a single operator is also proposed it is proven that any time varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one

blow up versus extinction in a nonlocal p laplace equation with neumann boundary conditions abstract this paper studies a fast diffusive p laplace equation with the nonlocal source u q u q d x in a bounded domain subject to homogeneous neumann boundary value condition a critical criterion is determined that the changing sign solutions blow up in finite time with q 1 and non positive initial energy associated and must be global for any initial energy if q 1 in particular the conditions are obtained under which the changing sign solutions vanish in finite time

stability of linear infinite dimensional systems revisited the paper is devoted to a study of stability questions for linear infinite dimensional discrete time and continuous time systems the concepts of power stability and l p instability for a linear discrete time system x k 1 ax k where x k e x x is a banach space a is linear and bounded are introduced and studied relationships between these concepts and the inequality r a 1 where r a denotes the spectral radius of a are also given the discrete time results are used for a simple derivation of some well known properties of exponentially stable and lp stable linear continuous time systems described by xdot t ax t a generates here a strongly continuous semigroup of linear and bounded operators on x some remarks on norms related to stable systems are also included

generically there is but one self homeomorphism of the cantor set we describe a self homeomorphism r of the cantor set x and then show that its conjugacy class in the polish group h x of all homeomorphisms of x forms a dense g subset of h x we also provide an example of a locally compact second countable topological group which has a dense conjugacy class

combinatorial independence in measurable dynamics abstract we develop a fine scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence the concepts of measure ie tuples and measure in tuples are introduced and studied in analogy with their counterparts in topological dynamics local characterizations of the pinsker von neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence l 1 geometry and voiculescu s completely positive approximation entropy among the novel features of our local study is the treatment of general discrete acting groups with the structural assumption of amenability in the case of entropy

global existence and finite time blow up of the solution for a thin film equation with high initial energy abstract this paper is devoted to the study of a thin film equation which was considered extensively in recent years we obtain the conditions for global existence and finite time blow up of solution for the high initial energy case i e j u 0 d while the previous works only considered the case j u 0 d here d is the mountain pass level moreover for the blow up solution with j u 0 d we obtain the lower bound estimate of the blow up time and blow up rate

stability of linear continuous time systems with stochastically switching delays necessary and sufficient conditions for the stability of linear continuous time systems with stochastically switching delays are presented in this paper it is assumed that the delay random paths are piece wise constant functions of time where a finite number of values may be taken by the delay the stability is assessed in terms of the second moment of the state vector of the system the solution operators of individual linear systems with constant delays chosen from the set of all possible delay values are extended to form new augmented operators then for proper formulation of the second moment in continuous time tensor products of the augmented solution operators are used finally the finite dimensional versions of the stability conditions that can be obtained using various time discretization techniques are presented some examples are provided that demonstrate how the stability conditions can be used to assess the stability of linear systems with stochastic delay

finite uniform generators for ergodic finite entropy free actions of amenable groups let x be a lebesgue space we prove in the first part of this paper that any ergodic 2 action on x with finite entropy hlogk has generating partition p that is uniform and has k atoms in the second part we prove a similar result for any ergodic free g action with finite entropy hlog k 2 for any discrete amenable group g

a complete characterization of nonlinear absorption for the evolution p laplacian equations to have positive or extinctive solutions this paper is concerned with long time behaviors of solutions to the initial boundary value problem of the evolution p laplacian equations with nonlinear absorption a long time behavior of solutions to these equations has been studied so far for particular types of nonlinear absorption term the purpose of this paper is to give a complete characterization of the nonlinear absorption term according to the parameter p in p laplacian operator and the growth of the nonlinear absorption term near the origin in order to determine whether the solution to the equations is extinctive or positive in addition we also give upper bounds for extinction times of solutions

liouville type results for a class of quasilinear parabolic problems the aim of this paper is to study the properties of positive solutions to the nonlinear diffusion equation with a nonlinear bistable source we show that there exists a universal a priori upper bound for all positive solutions moreover under some assumptions we prove the convergence of these solutions to a constant if t

remarks on the stability of linear infinite dimensional discrete time systems abstract various results are presented which complement the existing theory of stability for linear time varying infinite dimensional discrete time systems the concept of generalized spectral radius of a uniformly bounded sequence of elements of a normed algebra is studied in detail and several new formulas for computing this quantity are offered next the problem of stability of perturbed systems is considered finally a quantitative analysis of the so called perron condition is given

stability and stabilizability of infinite dimensional systems in this paper we are concerned with stability problems for infinite dimensional systems first we review the theory for linear systems where the dynamics are governed by strongly continuous semigro

on proximality with banach density one abstract let x t be a topological dynamical system a pair of points x y x 2 is called banach proximal if for any e 0 the set n z d t n x t n y e has banach density one we study the structure of the banach proximal relation a useful tool is the notion of the support of a topological dynamical system we show that a dynamical system is strongly proximal if and only if every pair in x 2 is banach proximal a subset s of x is banach scrambled if every two distinct points in s form a banach proximal pair but not asymptotic we construct a dynamical system with the whole space being a banach scrambled set even though the banach proximal relation of the full shift is of first category it has a dense mycielski invariant banach scrambled set we also show that for an interval map it is li yorke chaotic if and only if it has a cantor banach scrambled set

strictly ergodic models for dynamical systems on etudie l action d un groupe g d homeomorphismes d un espace metrique compacte quand g est commutatif l action ergodique a un modele strictement ergodique

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

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

topological pressure for sub additive potentials of amenable group actions abstract we introduce the topological pressure for any sub additive potentials of a countable discrete amenable group action and any given open cover and establish a local variational principle for it

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

generalized specification property and distributionally scrambled sets abstract suppose the dynamical system x f has generalized specification property in this paper we research the existence of the invariant mycielski distributionally scrambled sets for dynamical system x f moreover some properties about measure center are discussed and a dichotomy theorem is proved

a gradient estimate to a degenerate parabolic equation with a singular absorption term the global quenching phenomena we prove global existence of nonnegative solutions to the one dimensional degenerate parabolic problems containing a singular term we also show the global quenching phenomena for l1 initial datums moreover the free boundary problem is considered in this paper

a subadditive thermodynamic formalism for mixing repellers the thermodynamical description of fractals that has recently attracted much interest both experimentally and theoretically in the study of dynamical systems is in some ways limited being essentially an additive theory the author presents a subadditive thermodynamic formalism for which he derives a variational principle and shows how it may be used to study the dynamics of non conformal transformations in particular the author discusses an analogue of bowen s formula for the dimension of a mixing repeller

global solvability results for parabolic equations with p laplacian type diffusion abstract we give conditions that assure global existence of bounded weak solutions to the cauchy problem of general conservative 2nd order parabolic equations with p laplacian type diffusion p 2 and initial data u 0 l 1 r n l r n related results of interest are also given

conditional entropy and fiber entropy for amenable group actions abstract in this paper we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions and prove three variational principles for conditional entropy and fiber entropy moreover as an application of our variational principle we prove that the countable to one extension and the distal extension have zero conditional topological entropy in particular the topological entropy of a distal system is zero

classification of blowup solutions for a parabolic p laplacian equation with nonlinear gradient terms abstract we mainly consider the following degenerate p 2 parabolic equation u t p u u m u q with homogeneous dirichlet boundary condition in a bounded domain r n before studying the properties of the solutions of this equation we first establish the local in time existence of its weak solutions then in different ranges of reaction exponents we give the complete classification of blowup results including l blowup and gradient blowup moreover under the subcritical condition of q p 1 m p 1 we can also prove that the solution is global in time

strong asymptotic abelianness for entropic k systems we prove that in entropick systems of type ii1 the automorphism is strongly asymptotically abelian