regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

non simultaneous blow up for a multi coupled reaction diffusion system abstract this paper deals with a reaction diffusion system multi coupled via inner sources and nonlinear boundary flux the necessary and sufficient condition for the existence of non simultaneous blow up is proposed some regions of parameters are determined where both simultaneous and non simultaneous blow up can occur in addition sufficient conditions for non simultaneous blow up under any positive initial data are obtained

reactive dynamics of inertial particles in nonhyperbolic chaotic flows anomalous kinetics of infective e g autocatalytic reactions in open nonhyperbolic chaotic flows are important for many applications in biological chemical and environmental sciences we present a scaling theory for the singular enhancement of the production caused by the universal underlying fractal patterns the key dynamical invariant quantities are the effective fractal dimension and effective escape rate which are primarily determined by the hyperbolic components of the underlying dynamical invariant sets the theory is general as it includes all previously studied hyperbolic reactive dynamics as a special case we introduce a class of dissipative embedding maps for numerical verification

effective dimensions and chemical reactions in fluid flows we show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects the stirring protocol of the flow the local properties of the reaction and the global folding dynamics which also depends on the geometry of the container the essence of each of these components can be described by simple functional relations in an ordinary differential equation approach they determine a new chemical rate equation for the concentration which turns out to be coupled to the dynamics of an effective fractal dimension the theory predicts an exponential convergence to the asymptotic chemical state this holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics in the exponential case the theory applies to flows of chaotic time dependence chaotic flows as well

a nonlinear diffusion system with convection abstract this paper deals with a coupled nonlinear diffusion system with convection a characteristic matrix equation covering all the six nonlinear exponents of the system is introduced to get a simple description for the blow up criteria of the system in addition the influence of the four coefficients of the diffusion and the convection as well as the geometry of on global boundedness or not of the solutions is shown clearly especially for the situation with the critical sign of the characteristic parameters

critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources this paper deals with two classes of parabolic systems with localized nonlinear sources the critical exponents as well as the estimates for blow up rates and boundary layer profiles are determined

global existence and finite time blow up for a degenerate reaction diffusion system abstract this paper investigates the blow up and global existence of solutions of the degenerate reaction diffusion system u t u m u v p v t v n u q v x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 0 and p q 0 it is proved that if m n and pq m n every nonnegative solution is global whereas if m or n or pq m n there exist both global and blow up nonnegative solutions when m n and pq m n we show that there exists 1 which depends on the parameters p q m n such that all positive solutions are global if 1 while if 1 1 all positive solutions blow up in finite time where 1 is the first dirichlet eigenvalue for the laplacian on

holotransformations of bacterial colonies and genome cybernetics we present a study of colony transformations during growth of bacillus subtilis under adverse environmental conditions it is a continuation of our pilot study of adaptive self organization during growth of bacterial colonies physica a 187 1992 378 first we identify and describe the transformations pathway i e the excitation of the branching modes from bacillus subtilis 168 grown under diffusion limited conditions and the phase transformations between the tip splitting phase phase t and the chiral phase phase c which belong to the same mode this pathway shows the evolution of complexity as the bacteria are exposed to adverse growth conditions we present the morphology diagram of phases t and c as a function of agar concentration and pepton level as expected the growth of phase t is ramified fractal like or dla like at low pepton level about 1 g 1 and turns compact at high pepton level about 10 g 1 the growth of phase c is also ramified at low pepton level and turns denser and finally compact as the pepton level increases generally speaking the colonies develop more complex patterns and higher micro level organization for more adverse environments we use the growth velocity as a response function to describe the growth at low agar concentration and low pepton level phase c grows faster than phase t and for a high agar concentration about 2 phase t grows faster we observe colony transformations between the two phases phase transformations they are found to be consistent with the fastest growing morphology selection principle adopted from azoic systems the transformations are always from the slower phase to the faster one hence we observe t c transformations at low agar concentrations and c t transformations at high agar concentrations we have observed both localized and extended transformations usually the transformations are localized for more adverse growth conditions and extended for growth conditions close to the boundaries between morphologies we have observed also transformations between different branching modes as well as transformations via virtual states motivated by the contemporary knowledge about phages and plasmids we postulate a theoretical framework to comply with our experimental findings we explain our observations using these assumptions as well as our proposal of co mutations and auto catalytic mutations as presented in the above mentioned pilot paper this theoretical framework is a part of the new evolving picture of genome cybernetics we also discuss the concept of adaptive genome changes which are based on pre existing knowledge as well as the concept of genetic learning i e changes in response to a new problem which develop the potential for adaptive genome changes these concepts follow naturally if the picture of genome cybernetics is accepted we conclude with a discussion of the implications and with further predictions to be tested experimentally derived from our assumptions

fast chemical reaction and multiple scale concentration fields in singular vortices two species involved in a simple fast reaction tend to become segregated in patches composed of a single of these reactants these patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs fed by diffusion stirred by advection this boundary and the concentration fields within the patches may tend to present multiple scale characteristics based on this segregated state this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it when concentrations presenting initial self similar fluctuations are advected by a singular vortex first the two sources of singularity i e the self similar initial conditions and the singular vortex are considered isolatedly on the one hand self similar initial conditions are imposed to a diffusion reaction system for one and two dimensional cases on the other hand an imposed singular vortex advects initially on off concentration fields in combination with diffusion and reaction this problem is addressed analytically by characterizing the boundary by a box counting dimension and the concentration fields by a h older exponent and numerically by direct numerical simulations of the advection diffusion reaction equations second the way the two sources hang together shows that depending on the self similar properties of the initial concentration fields the vortex promotes the chemical activity close to its inner smoothed out core or close to the outer region where the boundary starts to spiral for all the considered situations the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws

a positive numerical scheme for a mixed type partial differential equation model for fungal growth we describe a numerical scheme for the solution of a mixed type pde system which arises in the modelling of fungal growth given the application conservation of mass and preservation of positivity are of paramount importance the scheme employs a method of lines approach in which the system is split into hyperbolic and parabolic parts positivity and conservation of mass are ensured by the use of generalised flux functions and in particular flux limiters the spatial discretisation results in stiff and non stiff components which are solved using implicit and explicit methods respectively properties of the scheme are investigated via comparison with experimental data and performance is compared with another method of solution

an explicit positivity preserving finite difference scheme for the classical fisher kolmogorov petrovsky piscounov equation abstract in this work we introduce a family of explicit finite difference schemes to approximate positive solutions of the classical fisher kolmogorov petrovsky piscounov equation from population dynamics a neumann stability analysis reveals that our methods are indeed stable under certain choices of the model and numerical parameters and numerical comparisons with analytical solutions are carried out in some detail

blow up for a degenerate reaction diffusion system with nonlinear nonlocal sources this paper investigates the global existence and blow up of nonnegative solution of the systemu email protected m u p email protected wv q 1dx v email protected n v p email protected wu q 2dx x t email protected 0 t with homogeneous dirichlet boundary conditions where email protected r n is a bounded domain with smooth boundary email protected m n 1 p 1 p 2 q 1 q 2 0 the results depend crucially on the number p i q i m n the domain w and the initial data u 0 x v 0 x moreover we obtain the blow up rate of the blow up solution under some appropriate hypotheses

critical fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux we establish the critical fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux and then determine the blow up rates and the blow up sets for the nonglobal solutions

a positive finite difference advection scheme abstract this paper examines a class of explicit finite difference advection schemes derived along the method of lines an important application field is large scale atmospheric transport the paper therefore focuses on the demand of positivity for the spatial discretization attention is confined to conservative schemes using five points per direction the fourth order central scheme and the family of schemes comprising the second order central the second order upwind and the third order upwind biased are studied positivity is enforced through flux limiting it is concluded that the limited third order upwind discretization is the best candidate from the four examined for the time integration attention is confined to a number of explicit runge kutta methods of orders two up to four with regard to the demand of positivity these integration methods turn out to behave almost equally and no best method could be identified

fractal scaling of microbial colonies affects growth the growth dynamics of filamentary microbial colonies is investigated fractality of the fungal or actinomycetes colonies is shown both theoretically and in numerical experiments to play an important role the growth observed in real colonies is described by the assumption of time dependent fractality related to the different ages of various parts of the colony the theoretical results are compared to a simulation based on branching random walks

blow up phenomena for a nonlinear reaction diffusion system with time dependent coefficients abstract we investigate the blow up phenomena for the solution to a nonlinear reaction diffusion system with time dependent coefficients subject to null dirichlet boundary conditions by virtue of kaplan s method method of subsolutions and supersolutions and modified differential inequality technique we establish the blow up criteria for the solution moreover lower and upper bounds for the blow up time are derived

blow up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions abstract in this paper we consider the following reaction diffusion systems with gradient nonlinearity under nonlinear boundary condition u t u u p v q u x t 0 t v t v v r u s v x t 0 t u g u v h v x t 0 t u x 0 u 0 x v x 0 v 0 x x where r n n 1 is a bounded region with smooth boundary p q r s 0 1 t is a possible blow up time when blow up occurs by constructing an appropriate auxiliary functions and by means of payne weinberger or scott s method a lower bound on blow up time when blow up occurs is derived

positive solutions of quasilinear parabolic systems with dirichlet boundary condition abstract coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems including systems of parabolic and ordinary differential equations are investigated the aim of this paper is to show the existence uniqueness and asymptotic behavior of time dependent solutions also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i 1 n and the boundary condition is u i 0 using the method of upper and lower solutions we show that a unique global classical time dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a scalar polynomial growth problem a coupled system of polynomial growth problem and a two component competition model in ecology

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

a skew symmetry preserving computational technique for obtaining the positive and the bounded solutions of a time delayed advection diffusion reaction equation abstract in this work we consider a one dimensional time delayed advective version of the well known fisher kolmogorov petrovsky piscounov equation from population dynamics which extends several models from mathematical physics including the classical wave equation the nonlinear klein gordon equation a fitzhugh nagumo equation from electrodynamics and the burgers huxley equation and the newell whitehead segel equation from fluid mechanics we propose a skew symmetry preserving finite difference scheme for approximating the solutions of the model under investigation and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded respectively the derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of m matrices in fact the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type we provide simulations in order to show that the technique is indeed conditionally positivity preserving and boundedness preserving

effective dynamics in hamiltonian systems with mixed phase space an adequate characterization of the dynamics of hamiltonian systems at physically relevant scales has been largely lacking here we investigate this fundamental problem and we show that the finite scale hamiltonian dynamics is governed by effective dynamical invariants which are significantly different from the dynamical invariants that describe the asymptotic hamiltonian dynamics the effective invariants depend both on the scale of resolution and the region of the phase space under consideration and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the hamiltonian system is modeled as a chain of hyperbolic systems

models for branching networks in two dimensions in this paper mathematical models for branching networks growing in one and two dimensions are described continuum equations are formulated to represent evolving spatial distributions of density given a variety of assumptions about branching and crosslinking kinetics in accommodating the influenceof angular branch distributions a set of integropartial differential equations are obtained it is found that crosslinking which eliminates apical growth acts as a density regulating mechanism if the branching angle varphi is small this mechanism further leads to a phenomenon of orientation selection it is shown analytically that for small varphi a spatially homogeneous network with an initial uniform distribution of branch orientations will align along a single axis as a result of instability of the uniform steady state to small perturbations that are nonuniform in the angular variable because instability first occurs to a mode e i theta for which l 2 such networks eventually contain

optimal classification for blow up phenomena in heat equations coupled via exponential sources abstract this paper deals with simultaneous and non simultaneous blow up solutions for heat equations coupled via exponential sources subject to null dirichlet boundary conditions the main results complete the previously known results on the optimal classification for simultaneous and non simultaneous blow up solutions by covering the whole ranges of exponents moreover all kinds of simultaneous and non simultaneous blow up rates are obtained

a bacterial colony is not self similar abstract the geometry of complex bacterial colonies has been studied different values of the exponent in scaling law have been recorded on different length scales thus a colony is not a self similar object and has a multiple fractal structure

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

blow up for a semilinear reaction diffusion system coupled in both equations and boundary conditions we study the blow up behavior for a semilinear reaction diffusion system coupled in both equations and boundary conditions the main purpose is to understand how the reaction terms and the absorption terms affect the blow up properties we obtain a necessary and sufficient condition for blow up derive the upper bound and lower bound for the blow up rate and find the blow up set under certain assumptions

global and nonglobal weak solutions to a degenerate parabolic system this paper deals with a degenerate parabolic system coupled via general reaction terms of power type global weak solutions are obtained by means of energy estimates and the de giorgi s technique in particular the criterion for global nonexistence of weak solutions is proved by introducing suitable weak sub solutions together with a weak comparison principle in summary the critical exponent for weak solutions of the degenerate parabolic system is determined

roles of weight functions in a nonlinear nonlocal parabolic system abstract this paper deals with a semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal dirichlet boundary conditions we establish the conditions for global and non global solutions it is interesting to observe that the weight functions for the nonlocal dirichlet boundary conditions play substantial roles in determining not only whether the solutions are global or non global but also whether for the non global solutions the blowing up occurs for any positive initial data or just for large ones

varieties fibred over abelian varieties with fibres of log general type abstract let x b be a complex projective klt pair and let f x z be a surjective morphism onto a normal projective variety with maximal albanese dimension such that k x b is relatively big over z we show that such pairs have good log minimal models

blow up rate estimates for a doubly coupled reaction diffusion system abstract this paper deals with a reaction diffusion system with coupled nonlinear inner sources and a nonlinear boundary flux blow up rates are determined for four different blow up situations the so called characteristic algebraic system is introduced to get a very simple and clear description for the desired blow up rate estimates it is pointed out that one cannot directly use super and sub solutions to establish blow up rate estimates since they do not share the same blow up time in general

a nonstandard finite difference scheme for convection diffusion equations having constant coefficients in this note we derive using the subequation method a new nonstandard finite difference scheme nsfd for a class of convection diffusion equations having constant coefficients despite the fact that this scheme has nonlinear denominator functions of the step sizes even for linear pdes it has a couple of favourable properties it is explicit and due to its construction it reproduces important properties of the solution of the parabolic pde this proposed method conserves by construction the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the black scholes equation for the valuation of standard options since it avoids negative values for the calculated prices finally we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature

on the abundance theorem in the case of numerical kodaira dimension zero we present a shorter proof of the abundance theorem in the case of numerical kodaira dimension 0 proved by nakayama and its log generalization

on rational connectedness of globally f regular threefolds in this paper we show that projective globally f regular threefolds defined over an algebraically closed field of characteristic p 11 are rationally chain connected

discrete time model for chemical or biological decay in chaotic flows reentrance phase transitions we consider a discrete time model of advection reaction and diffusion on a lattice to investigate the steady state spatial structure of chemically decaying substances the time discretization of the dynamics has a considerable impact on these structures additional smooth filamental phase transitions nonexistent in the continuous time description appear we show how these structures and their scaling properties depend on the time step of the discrete dynamics exploiting the analogies of this discrete model with the logistic map some general features are discussed

simultaneous and non simultaneous blow up for a cross coupled parabolic system this paper deals with a parabolic system cross coupled via a nonlinear source and a nonlinear boundary flux we get a necessary and sufficient condition for the existence of non simultaneous blow up in particular four different simultaneous blow up rates are obtained in different regions of parameters described by an introduced characteristic algebraic system it is observed that different initial data may result in different simultaneous blow up rates even in the same region of parameters

asymptotic analysis to a parabolic equation with a weighted localized source abstract this paper deals with a nonlinear parabolic equation with a more complicated source term which is a product of localized source u q 0 t local source u p x t and weight function a x we investigate how the three factors influence the asymptotic behavior of solutions it is shown that the blow up set consists of single point x 0 if p 1 when 0 p 1 with p q 1 the blow up take place everywhere in b moreover the blow up rate estimate is established with more precise coefficients determined

the positivity of low order explicit runge kutta schemes applied in splitting methods splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right hand side function such systems arise for instance as method of lines discretizations of evolutionary partial differential equations in many applications we consider the choice of explicit runge kutta rk schemes in implicit explicit splitting methods our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency a three stage second order explicit rk method is proposed which has optimized positivity properties this method compares well with standard s stage explicit rk schemes of order s s 2 3 it has advantages in the low accuracy range and this range is interesting for an application in splitting methods numerical results are presented

reactions in flows with nonhyperbolic dynamics we study the reaction dynamics of active particles that are advected passively by 2d incompressible open flows whose motion is nonhyperbolic this nonhyperbolicity is associated with the presence of persistent vortices near the wake wherein fluid is trapped we show that the fractal equilibrium distribution of the reactants is described by an effective dimension d mathit eff which is a finite resolution approximation to the fractal dimension furthermore d mathit eff depends on the resolution and on the reaction rate 1 ensuremath tau as ensuremath tau is increased the equilibrium distribution goes through a series of transitions where the effective dimension increases abruptly these transitions are determined by the complex structure of cantori surrounding the kolmogorov arnold moser kam islands