the blow up rate for a system of heat equations with nonlinear boundary conditions this paper deals with the blow up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions the upper and lower bounds of blow up rate are obtained

on the asymptotic geometry of nonpositively curved graphmanifolds in this paper we study the tits geometry of a 3 dimensional graphmanifold of nonpositive curvature in particular we give an optimal upper bound for the length of nonstandard components of the tits metric in the special case of a 2 metric we determine the whole length spectrum of the nonstandard components

asymptotic rank of spaces with bicombings the question under what geometric assumptions on a space x an n quasiflat in x implies the existence of an n flat therein has been investigated for a long time it was settled in the affirmative for busemann spaces by kleiner and for manifolds of non positive curvature it dates back to anderson and schroeder we generalize the theorem of kleiner to spaces with bicombings this structure is a weak notion of non positive curvature not requiring the space to be uniquely geodesic beside a metric differentiation argument we employ an elegant barycenter construction due to es sahib and heinich by means of which we define a riemannian integral serving us in a sort of convolution operation

the local structure of length spaces with curvature bounded above we show that a number of different notions of dimension coincide for length spaces with curvature bounded above we then apply this result showing that if x is a locally compact cat 0 space with cocompact isometry group then the dimension of the tits boundary and the asymptotic cone s of x are determined by the maximal dimension of a flat in x

blow up solutions for reaction diffusion equations with nonlocal boundary conditions abstract we devote to investigating the following nonlinear reaction diffusion equations with nonlocal boundary conditions u t u u k 1 t f u in d 0 t u k 2 t d g u d x on d 0 t u x 0 u 0 x 0 in d where d is a bounded convex region in r n n 2 and the boundary d is smooth by constructing some auxiliary functions and using differential inequality technique we derive that the solution blows up at some finite time moreover upper and lower bounds of the blow up time are obtained

blow up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions abstract by constructing different auxiliary functions and using hopf s maximum principle the sufficient conditions for the blow up and global solutions are presented for nonlinear parabolic equation u t a u b x c t u f x u q t with different kinds of boundary conditions the upper bounds of the blow up time and the upper estimates of global solutions are provided finally some examples are presented as the application of the obtained results

properties of non simultaneous blow up in heat equations coupled via different localized sources abstract this paper deals with u t u u m x t e pv 0 t v t v u q 0 t e nv x t subject to homogeneous dirichlet boundary conditions the complete classification on non simultaneous and simultaneous blow up is obtained by four sufficient and necessary conditions it is interesting that in some exponent region large initial data u 0 v 0 leads to the blow up of u v and in some betweenness simultaneous blow up occurs for all of the nonnegative exponents we find that u v blows up only at a single point if m 1 n 0 while u v blows up everywhere for 0 m 1 n 0 moreover blow up rates are considered for both non simultaneous and simultaneous blow up solutions

blow up estimate in a reaction diffusion equation with nonlinear nonlocal flux and source abstract this paper deals with a reaction diffusion equation with nonlocal inner source and nonlocal boundary flux besides the influence of exponents we find out that the weighted functions and the measure of the domain could lead to the existence of global solutions for any initial data larger weighted functions require smaller initial data in guaranteeing the existence of global solutions to the problem blow up rates are obtained under different dominations of nonlinearities in one dimensional space the upper and lower bounds of blow up time are determined for all spatial dimensions respectively moreover the solutions blow up only on the boundary when the nonlocal flux dominates the blow up

discrete analogues in harmonic analysis i l 2 estimates for singular radon transforms this paper studies the discrete analogues of singular radon transforms we prove the 2 boundedness for those operators that are quasi translation invariant the approach used is related to the circle method of hardy and littlewood and requires multi dimensional extensions of weyl sums and gauss sums as well as variants that replace scalar sums by operator sums 1 introduction the purpose of this paper is to study discrete analogues of singular radon transforms their analysis will involve considerations originating in analytic number theory related to the circle method of hardy and littlewood and necessitates an extension of the notion of weyl sums and gauss sums this will require that we pass from the usual setting of scalar valued sums to certain operator analogues of these sums

existence and uniqueness results for parabolic equations with robin type boundary conditions in a non regular domain of r3 this paper is devoted to the analysis of the following two space dimensional linear parabolic equation tu x 1 2u x 2 2u f subject to robin type conditions x 1u bu 0 on the lateral boundary where the coefficient b satisfies suitable non degeneracy assumptions the right hand side f of the equation is taken in l 2 the problem is set in a domain of the form t x 1 r 2 0

generalizations of some reverse integral inequalities abstract we establish the inequality equation where ck n p q q 1 u x r x u2 x r x q k and r x x0r t u t dt which is sharper than that obtained by b mond j pecaric and i peric j math anal appl 203 1996 381 387

asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions abstract this paper is concerned with some dynamical property of a reaction diffusion equation with nonlocal boundary condition under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function the asymptotic behavior of the time dependent solution is characterized in relation to a finite or an infinite set of constant steady state solutions this characterization is determined solely by the initial function and it leads to the stability and instability of the various steady state solutions in the case of finite constant steady state solutions the time dependent solution blows up in finite time when the initial function in greater than the largest constant solution also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain

reverse dynamic inequalities and higher integrability theorems abstract in this paper we prove some reverse dynamic inequalities and apply them to prove some new higher integrability theorems for nonincreasing functions on time scales as a special case we will derive the classical integral inequalities due to alzer and formulate the discrete versions of these inequalities which are essentially new

a volume convergence theorem for alexandrov spaces with curvature bounded above we obtain a volume convergence theorem for alexandrov spaces with curvature bounded above with respect to the gromov hausdorff distance as one of the main tools proving this we construct an almost isometry between alexandrov spaces with curvature bounded above with weak singularities which are close to each other furthermore as an application of our researches of convergence phenomena for given positive integer n we prove that if a compact geodesically complete n dimensional cat 1 space has the volume sufficiently close to that of the unit n sphere then it is bi lipschitz homeomorphic to the unit n sphere

blow up and global solutions for nonlinear reaction diffusion equations with nonlinear boundary condition abstract this paper deals with the blow up of positive solutions for a nonlinear reaction diffusion equation subject to nonlinear boundary conditions we obtain the conditions under which the solutions may exist globally or blow up in finite time moreover an upper bound of the blow up time an upper estimate of the blow up rate and an upper estimate of the global solutions are given at last we give two examples to which the theorems obtained in the paper may be applied

roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition in this paper we investigates the blow up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition we obtain the blow up condition and its blow up rate estimate

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

blow up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition abstract in this paper we consider a semilinear heat equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove global existence theorem for max p l 1 some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given

blow up rate estimates for semilinear parabolic systems abstract this paper deals with the blow up rate estimates of positive solutions for semilinear parabolic systems with null dirichlet boundary conditions the upper and lower bounds of blow up rate were obtained

higher rank hyperbolicity the large scale geometry of hyperbolic metric spaces exhibits many distinctive features such as the stability of quasi geodesics the morse lemma the visibility property and the homeomorphism between visual boundaries induced by a quasi isometry we prove a number of closely analogous results for spaces of rank n ge 2 in an asymptotic sense under some weak assumptions reminiscent of nonpositive curvature for this purpose we replace quasi geodesic lines with quasi minimizing locally finite n cycles of r n volume growth prime examples include n cycles associated with n quasiflats solving an asymptotic plateau problem and producing unique tangent cones at infinity for such cycles we show in particular that every quasi isometry between two proper text cat 0 spaces of asymptotic rank n extends to a class of n 1 cycles in the tits boundaries

commensurators of abelian subgroups in cat 0 groups we study the structure of the commensurator of a virtually abelian subgroup h in g where g acts properly on a mathrm cat 0 space x when x is a hadamard manifold and h is semisimple we show that the commensurator of h coincides with the normalizer of a finite index subgroup of h when x is a mathrm cat 0 cube complex or a thick euclidean building and the action of g is cellular we show that the commensurator of h is an ascending union of normalizers of finite index subgroups of h we explore several special cases where the results can be strengthened and we discuss a few examples showing the necessity of various assumptions finally we present some applications to the constructions of classifying spaces with virtually abelian stabilizers

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

the tits boundary of a text cat 0 2 complex we investigate the tits boundary of cat 0 2 complexes that have only a finite number of isometry types of cells in particular we show that away from the endpoints a geodesic segment in the tits boundary is the ideal boundary of an isometrically embedded euclidean sector as applications we provide sufficient conditions for two points in the tits boundary to be the endpoints of a geodesic in the 2 complex and for a group generated by two hyperbolic isometries to contain a free group we also show that if two cat 0 2 complexes are quasi isometric then the cores of their tits boundaries are bi lipschitz

a reaction diffusion system with mixed type coupling this paper deals with asymptotic behavior of solutions to a reaction diffusion system coupled via localized and local sources u t du v p x t t v t dv u q both the initial boundary problem with null dirichlet boundary condition and the cauchy problem are considered to study the interaction between the two kinds of sources for the initial boundary problem we prove that the nonglobal solutions blow up everywhere in the bounded domain with uniform blow up profiles in addition it is interesting to observe that the cauchy problem admits an infinity fujita exponent namely the solutions blow up under any nontrivial and nonnegative initial data whenever pq 1 all these imply that the blow up behavior of solutions is governed by the localized source for the two problems with mixed type coupling

global existence and blow up for a porous medium system with nonlocal boundary conditions and nonlocal sources abstract this paper is devoted to a porous medium system subject to nonlocal boundary conditions and with nonlocal sources we investigate the global existence and blow up in finite time of a nonnegative solution by using a sub super solution method moreover under appropriate hypotheses for the conditions we obtain asymptotic properties of the solutions

discrete analogues in harmonic analysis spherical averages in this paper we prove an analogue in the discrete setting of d o fthe spherical maximal theorem for d the methods used are two fold the application of certain sampling techniques and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular the circle method the results we obtained are by necessity limited to d 5 and moreover the range of p for the l p estimates differs from its analogue in d

hyperbolization of polyhedra hyperbolization is a process for converting a simplicial complex into a metric space with nonpositive curvature in the sense of gromov several such processes are described in 19 3 4 one of the purposes of this paper is to elaborate this idea of gromov another purpose is to use it to construct the three examples described below our approach to hyperbolization is based on the following construction of williams 32 suppose that x is a space and that x is a map onto the standard z simplex suppose also that k is an dimensional simplicial complex to these data williams associates a space xak constructed by replacing each n simplex in the barycentric subdivision of k by a copy of x the pair x is a hyperbolized simplex if x is a nonpositively curved manifold with boundary and has appropriate properties it is proved in 4 that hyperbolized simplices exist if x is a hyperbolized simplex then xak is nonpositively curved it is called a hyperbolization of k in all three examples we begin with a polyhedral homology manifold having a desired feature a hyperbolization then has the added feature of nonpositive curvature the first example is a closed aspherical fourmanifold which cannot be triangulated taking the product of this example with a a2 torus we obtain an aspherical manifold of any dimension 4 which is not homotopy equivalent to a pl manifold the second example is a closed smooth manifold of dimension n 5 which carries a topological metric of nonpositive curvature while its universal cover though contractible is not homeomorphic to a euclidean n space r as we shall see such a manifold cannot carry a pl or smooth metric of nonpositive curvature the third example is a further refinement m carries a topological metric of strict negative curvature and its universal

an improvement of the constants in a reverse integral inequality abstract we present an upper bound for the ratio formula where f is a positive decreasing function satisfying 1 t t 0 f x q dx cf t q 1 1 t t 0 f x dx for all t 0 a our result sharpens an inequality of l nania

structure of manifolds of nonpositive curvature clearly 1 rank m dim m we may regard rank m as a measurement of the flatness of m since a parallel field x along yv and perpendicular to yv is a jacobi field if and only if the sectional curvatures of the planes i t a x t are zero for all t e r in particular m is flat if rank m dim m hence a compact surface of nonpositive curvature has rank one unless it is a torus or a klein bottle it also follows that rank m1 x m2 rank ml rank m2 finally it is important that the definition of rank m above coincides with the usual one if m is locally symmetric

blow up of solutions for nonlinear parabolic equation with nonlocal source and nonlocal boundary condition this paper is devoted to a nonlinear parabolic equation subject to nonlocal boundary condition and with nonlocal reaction source some criteria on this problem which determine whether the solutions blow up in a finite time or exist globally are given moreover the blow up rate estimates are also obtained