spreading speeds and traveling waves for periodic evolution systems abstract the theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case then these abstract results are applied to a periodic system modeling man environment man epidemics a periodic time delayed and diffusive equation and a periodic reaction diffusion equation on a cylinder

on some higher order hardy rellich type inequalities with boundary terms abstract we determine boundary remainder terms for some higher order hardy rellich inequalities involving the polyharmonic operator m the results are proved by studying suitable auxiliary boundary eigenvalue problems the optimal constants found may not be the classical hardy rellich ones

positivity preserving property for a class of biharmonic elliptic problems the lack of a general maximum principle for biharmonic equations suggests to study under which boundary conditions the positivity preserving property holds we show that this property holds in general domains for suitable linear combinations of dirichlet and navier boundary conditions the spectrum of this operator exhibits some unexpected features radial data may generate nonradial solutions these boundary conditions are also of some interest in semilinear equations since they enable us to give explicit radial singular solutions to fourth order gelfand type problems

global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity abstract this paper is concerned with the global stability of wavefronts with minimal speeds for degenerate nonlocal dispersal equations by the method of super and subsolutions and a squeezing technique the wavefront with minimal speed is proved to be globally asymptotically stable with phase shift

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

travelling wave solutions for a nonlocal dispersal hiv infection dynamical model abstract this paper is devoted to developing a nonlocal dispersal hiv infection dynamical model the existence of travelling wave solutions is investigated by employing schauder s fixed point theorem that is we study the existence of travelling wave solutions for r 0 1 and each wave speed c c in addition the boundary asymptotic behaviour of travelling wave solutions at is obtained by constructing suitable lyapunov functions and employing lebesgue dominated convergence theorem by employing a limiting argument we investigate the existence of travelling wave solutions for r 0 1 and c c the main difficulties are that the semiflow generated by the model does not have the order preserving property and the solutions lack of regularity

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

on birman s sequence of hardy rellich type inequalities abstract in 1961 birman proved a sequence of inequalities i n for n n valid for functions in c 0 n 0 l 2 0 in particular i 1 is the classical integral hardy inequality and i 2 is the well known rellich inequality in this paper we give a proof of this sequence of inequalities valid on a certain hilbert space h n 0 of functions defined on 0 moreover f h n 0 implies f h n 1 0 as a consequence of this inclusion we see that the classical hardy inequality implies each of the inequalities in birman s sequence we also show that for any finite b 0 these inequalities hold on the standard sobolev space h 0 n 0 b furthermore in all cases the birman constants 2 n 1 2 2 2 n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in l 2 0 resp l 2 0 b we also show that these birman constants are related to the norm of a generalized continuous cesaro averaging operator whose spectral properties we determine in detail

on match lengths zero entropy and large deviations with application to sliding window lempel ziv algorithm the sliding window lempel ziv swlz algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature in this paper we undertake a finer study of these quantities under two different scenarios 1 zero entropy sources that are characterized by strong long term memory and 2 the processes with weak memory as described through various mixing conditions for zero entropy sources a general statement on match length is obtained it is used in the proof of almost sure optimality of fixed shift variant of lempel ziv fslz and swlz algorithms given in literature through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size n w a compression ratio given by o log n w n w a where a depends on n w and approaches 1 as n w is obtained under the application of fslz and swlz algorithms in addition we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy next we extend the study of ornstein and weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a class of mixing processes in addition an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

optimal hardy rellich inequalities maximum principle and related eigenvalue problem in this paper we deal with three types of problems concerning the hardy rellich s embedding for a bilaplacian operator first we obtain the hardy rellich inequalities in the critical dimension n 4 then we derive a maximum principle for fourth order operators with singular terms then we study the existence non existence simplicity and asymptotic behavior of the first eigenvalue of the hardy rellich operator 2 n 2 n 4 2 16 q x x 4 under various assumptions on the perturbation q

entropy dimension of shift spaces on monoids we consider the entropy dimension of g shifts of finite type for the case where g is a non abelian monoid entropy dimension tells us whether a shift space has zero topological entropy suppose the cayley graph cg of g has a finite representation that is cgg g g is a finite set up to graph isomorphism and relations among generators of g are determined by a matrix a we reveal an association between the characteristic polynomial of a and the finite representation of the cayley graph after introducing an algorithm for the computation of the entropy dimension the set of entropy dimensions is related to a collection of matrices in which the sum of each row of every matrix is bounded by the number of leaves of the graph furthermore the algorithm extends to g having finitely many free followers

traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity abstract this paper is concerned with traveling wave solutions of the equation u t j u u f u on r 0 where the dispersion kernel j is nonnegative and the nonlinearity f is monostable type we show that there exists c r such that for any c c there is a nonincreasing traveling wave solution with 1 and lim e 1 where 1 c is the smallest positive solution to c r j z e z d z 1 f 0 furthermore the existence of traveling wave solutions with c c is also established for c 0 we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense

convergence to diffusion waves of solutions for viscous conservation laws we study the large time behavior of solutions of viscous conservation laws it is shown that solutions tend to diffusion waves which are constructed based on the heat equation and burgers equation the convergence is in thelp 1 p sense and is obtained as a consequence of thel2 decay of the difference between the solution and its asymptotic state of diffusion waves

scaled entropy for dynamical systems in order to characterize the complexity of a system with zero entropy we introduce the notions of scaled topological and metric entropies we allow asymptotic rates of the general form e alpha a n determined by an arbitrary monotonically increasing scaling sequence a n this covers the standard case of exponential scale corresponding to a n n as well as the cases of zero and infinite entropy we describe some basic properties of the scaled entropy including the inverse variational principle for the scaled metric entropy furthermore we present some examples from symbolic and smooth dynamics that illustrate that systems with zero entropy may still exhibit various levels of complexity

on a nonlocal diffusion model with neumann boundary conditions abstract we study a nonlocal diffusion model analogous to heat equation with neumann boundary conditions we prove the existence and uniqueness of solutions and a comparison principle furthermore we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable

dynamics of layered interfaces arising from phase boundaries the dynamics of a material in two phases is studied in the context of phase field models based on a landau ginzburg free energy functional they consist of a system of two nonlinear diffusion equat

approximate the fokker planck equation by a class of nonlocal dispersal problems abstract this paper is concerned with an inhomogeneous nonlocal dispersal equation we study the limit of the re scaled problem of this nonlocal operator and prove that the solutions of the re scaled equation converge to a solution of the fokker planck equation uniformly we then analyze the nonlocal dispersal equation of an inhomogeneous diffusion kernel and find that the heterogeneity in the classical diffusion term coincides with the inhomogeneous kernel when the scaling parameter goes to zero

spectral analysis and multidimensional stability of traveling waves for nonlocal allen cahn equation in this paper the spectrum of linearized operator about a traveling wave for the nonlocal allen cahn equation is estimated and the result is applied to study multidimensional stability of planar waves

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

entire solutions in nonlocal dispersal equations with bistable nonlinearity abstract we consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one dimensional spatial domain a two dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed furthermore we show that such an entire solution is unique up to space time translations and liapunov stable a key idea is to characterize the asymptotic behaviors of the solutions as t in terms of appropriate subsolutions and supersolutions we have to emphasize that a lack of regularizing effect occurs

the hyperbolic allen cahn equation exact solutions using the first integral method a general set of analytical solutions is obtained for the hyperbolic allen cahn equation the solutions are presented by i the class of continual solutions described by profiles for traveling waves of the order parameter and ii the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system it is shown that the solutions include the previous analytical results for the parabolic allen cahn equation as a limited class of functions in which the inertial effects are omitted

zero entropy systems this paper introduces the notion of entropy dimension to measure the complexity of zero entropy dynamical systems including the probabilistic and the topological versions these notions are isomorphism invariants for measure preserving transformation and continuity we discuss basic propositions for entropy dimension and construct some examples to show that the topological entropy dimension attains any value between 0 and 1 this paper also gives a symbolic subspace to achieve zero topological entropy but with full entropy dimension

asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation abstract this paper is concerned with the asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation the asymptotic behavior of traveling fronts is obtained using ikehara s theorem by the sub super solution method the existence of entire solutions is obtained they behave as two opposite wave fronts approaching each other from either side of the x axis and then annihilating in a finite time by an entire solution we mean one which is defined over the whole space for all time t r

existence and uniqueness of solutions for a non uniformly parabolic equation abstract in this paper we study a generalized burgers equation ut u2 x tuxx which is a non uniformly parabolic equation for t 0 we show the existence and uniqueness of classical solutions to the initial value problem of the generalized burgers equation with rough initial data belonging to l r

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

systems with nonlocal vs local diffusions and free boundaries abstract we study a class of free boundary problems of ecological models with nonlocal and local diffusions which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal with the free boundary representing the spreading front of the species we first prove the existence uniqueness and regularity of global solution for the classical competition prey predator and mutualist models we show that a spreading vanishing dichotomy holds and establish the criteria of spreading and vanishing and long time behavior of the solution