the continuous dependence on nonlinearities of solutions of the neumann problem of a singular parabolic equation abstract in this paper we consider the singular diffusion problem u t u m 1 u x x 0 x 1 t 0 1 m u m x x 0 1 0 t 0 u t 0 u 0 x 0 x 1 with 0 m 1 we will show that there exists a unique global solution u x t m such that 0 0 1 u x t m u x t m 0 2 d x d t c m m 0 where 0 m m 0 1 and c is a explicit constant

identification of parameters in distributed parameter systems by regularization identification of spatially varying parameters in distributed parameter systems from noisy data is an ill posed problem the concept of regularization widely used in solving linear fredholm integral equations is developed for the identification of parameters in distributed parameter systems a general regularization identification theory is first presented and then applied to the identification of parabolic systems the performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation

stability analysis for genetic regulatory networks with delays the continuous time case and the discrete time case convergent dynamics are investigated in this paper for the genetic regulatory networks grns with delays in both the continuous time case and the discrete time form first by utilizing the lyapunov functional method and some inequality techniques sufficient criterion is derived to ensure the global exponential stability of the continuous time grns second by using the semi discretization technique and the email protected method as sources of reference two discrete time analogues of the continuous time grn model are formulated respectively moreover it is shown that under the same sufficient conditions derived earlier these two discrete time grn systems are also globally exponentially stable and the explicit exponential convergent rates are also given finally a numerical example is illustrated to show the validity of the obtained theoretical results and comparisons of the convergent rates for these two different discrete time analogues are also demonstrated

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

quasilinear parabolic systems with nonlinear boundary conditions ui x 0 ui0 x 0 x 0 i 1 n where 0 r is a bounded domain with smooth boundary 0 is the unit outward normal vector and the constants mi 0 i 0 mij 0 i j 1 n ui0 x i 1 n are positive c functions and satisfy the compatibility conditions 1 the present address academy of mathematics and systems sciences academia sinica beijing 100080 people s republic of china doi 10 1006 jdeq 2000 3784 available online at http www idealibrary com on

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

remarks on parameter identification ii convergence of the algorithm on demontre un theoreme de convergence pour l algorithme d identification du coefficient a x de l equation parabolique u t a x u x x f x t et on donne des resultats de stabilite

diffusivity versus absorption through the boundary for all 0 1 then we do expect to find that any solution becomes unbounded in finite time see 13 15 63 it is our goal to prove an analogous result for m f 1 for a 1 one can find that if v is a positive solution of

delay dependent stability for uncertain stochastic neural networks with time varying delay abstract this paper is concerned with the robust stability analysis problem for uncertain stochastic neural networks with time varying delay the parameter uncertainties are assumed to be norm bounded by defining a new lyapunov krasovskii functional the restrictions such as the time varying delay was required to be differentiable and its derivative was strictly smaller than one are removed based on the linear matrix inequality approach delay dependent stability criteria are obtained such that for all admissible uncertainties the stochastic neural network is globally asymptotically stable in the mean square two slack variables are introduced into the obtained stability criteria to reduce the conservatism finally a numerical example is given to illustrate the effectiveness of the developed method

global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme t u j a jk k u a 0 u f u dans 0 a jk j k u bu 1 u g u sur 0 u 0 u 0 sur ou est un domaine borne de r n n 1 de classe c 2

pth moment stability analysis of stochastic recurrent neural networks with time varying delays this paper addresses in great detail the issue of pth moment exponential stability of stochastic recurrent neural networks with time varying delays with the help of the dini derivative of the expectation of v t x t along the solution x t of the model and the technique of halanay type inequality some novel sufficient conditions on pth moment exponential stability of the trivial solution has been established results of the development as presented in this paper are more general than those reported in some previously published papers an example is also given to illustrate that our results are correct and effectiveness

novel delay dependent stability criterion for time varying delay systems with parameter uncertainties and nonlinear perturbations this study proposes a novel approach to obtain less conservative conditions for the robust stability analysis for time varying delay systems with parameter uncertainties and nonlinear perturbations specifically first we divide the time varying delay into non uniformly subintervals and decompose the corresponding integral intervals to estimate the bounds of integral terms more exactly second novel delay derivation dependent stability criteria are derived by introducing a parameter c in the lyapunov krasovskii functional third we define some new variables to deal with uncertain parameters finally two numerical examples are presented to demonstrate the effectiveness and advantages of the theoretical results

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

critical exponents and blow up rate for a nonlinear diffusion equation with logarithmic boundary flux abstract in this paper we establish the critical global existence exponent and the critical fujita exponent for the nonlinear diffusion equation u t log 1 u u x x in r 0 subject to a logarithmic boundary flux log 1 u u x 0 t 1 u log p 1 u 0 t t 0 furthermore give the blow up rate for the nonglobal solutions

remarks on parameter identification i in this paper a method for constructing a spatially varying diffusion coefficient for a parabolic partial differential equation is given this function is obtained as the limit of a sequence of functions which are obtained by solving a sequence of finite dimensional optimization problems

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

mean square exponential stability of stochastic genetic regulatory networks with time varying delays in this paper we study the mean square exponential stability of stochastic genetic regulatory networks with time varying delays two kinds of time varying delays are considered one is differentiable with bounded delay derivative the other is continuous without constraints on the delay derivative in order to investigate the mean square exponential stability in stochastic genetic regulatory networks some novel rate dependent independent mean square exponential stability criteria are derived by constructing lyapunov krasovskii functional the sufficient conditions are given in terms of linear matrix inequalities moreover illustrative examples are used to substantiate the effectiveness and less conservativeness of our results

new results on delay dependent robust stability criteria of uncertain fuzzy systems with state and input delays this paper focuses on the delay dependent stability analysis and stabilization for uncertain t s fuzzy control systems with state and input delays the key features of the approach include the introduction of uncorrelated augmented matrix items into the lyapunov functional and the use of a tighter bounding technology in fact these techniques lead to more general and less conservative stability conditions that guarantee a wide stability region our delay dependent stability conditions thus obtained are given in terms of linear matrix inequalities lmis finally some examples are included to show that the proposed criteria improve the existing results significantly

the asymptotic self similar behavior for the quasilinear heat equation with nonlinear boundary condition in this paper the quasilinear heat equation with the nonlinear boundary condition is studied the blow up rate and existence of a self similar solution are obtained it is proved that the rescaled function v y t t t 1 2 p a 2 u t t p 1 2 p a 2 y t behaves as t t like a nontrivial self similar profile

blow up and blow up rate for a reaction diffusion model with multiple nonlinearities abstract this paper deals with interactions among three kinds of nonlinear mechanisms nonlinear diffusion nonlinear reaction and nonlinear boundary flux in a parabolic model with multiple nonlinearities the necessary and sufficient blow up conditions are established together with blow up rate estimates for the positive solutions of the problem

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

the role of non linear diffusion in non simultaneous blow up we study a parabolic system of two non linear reaction diffusion equations completely coupled through source terms and with power like diffusivity under adequate hypotheses on the initial data we prove that non simultaneous blow up is sometimes possible i e one of the components blows up while the other remains bounded the conditions for non simultaneous blow up rely strongly on the diffusivity parameters and significant differences appear between the fast diffusion and the porous medium case surprisingly flat homogeneous in space solutions are not always a good guide to determine whether non simultaneous blow up is possible

convective porous medium equations with nonlinear forcing at the boundary abstract this article deals with the global solutions and blow up problems for the convective porous medium equationut um xx n un x x 0 1 t 0 with the nonlinear boundary conditions um x 0 t aup 0 t um x 1 t 0 and positive initial datau x 0 u0 x 0 x 0 1 here m n a andpare all positive constants necessary and sufficient conditions for the global existence of all positive solutions are obtained

the balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation abstract we study nonnegative solutions of u t u m xx x t 0 l 0 t u m x 0 t u p 0 t t 0 t u m x l t u q l t t 0 t u x 0 u 0 x x 0 l where m p q and l are positive parameters for different values of the parameters three situations may occur 1 all solutions of this problem exist for all t 0 2 for certain initial data functions the solution exists for all t 0 while for others the solution blows up as t t for some finite t 3 excepting the trivial solution when u0 0 all solutions blow up as t t for some finite t we identify in terms of the parameters which of them actually happens for solutions which blow up we find the blow up rate and the blow up set

on critical fujita exponents for the porous medium equation with a nonlinear boundary condition we establish the critical fujita exponents for the solution of the porous medium equation ut u m x r n t 0 subject to the nonlinear boundary condition u m x1 u p x1 0 t 0

quasilinear reaction diffusion systems with nonlinear boundary conditions this paper deals with the existence and nonexistence of global positive solutions of quasilinear reaction diffusion systems with nonlinear boundary conditions necessary and sufficient conditions on the global existence of all positive solutions are obtained

doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions in this paper we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions we first prove a local existence result by the regularization method next we construct a weak comparison principle then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions necessary and sufficient conditions on the global existence of all positive weak solutions are obtained

boundary estimates for solutions of nonlinear degenerate parabolic systems on etablit la regularite partout jusqu a la frontiere pour des solutions faibles de systemes paraboliques du type u l t div du p 2 du l f l x t u du dans t l 1 2 m u t r m m n p max 1 2n n 2 est un ouvert borne de r n n 2 0 t t 0 t

exponential stability of discrete time genetic regulatory networks with delays discrete time versions of the continuous time genetic regulatory networks grns with sum regulatory functions are formulated and studied in this letter sufficient conditions are derived to ensure the global exponential stability of the discrete time grns with delays an illustrative example is given to demonstrate the effectiveness of the obtained results

a nonlinear diffusion system coupled via nonlinear boundary flux abstract in this paper we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux by constructing various kinds of sub and super solutions and using the basic properties of m matrix we give the necessary and sufficient conditions for global existence of nonnegative solutions the critical curve of fujita type is conjectured with the aid of some new results which extend the recent results of wang et al nonlinear anal 71 2009 2134 2140 and li et al j math anal appl 340 2008 876 883 to more general equations

existence and nonexistence of global solutions of the convective porous medium equations with a nonlinear forcing at the boundary this article deals with the existence and nonexistence of global positive solutions of the following equations begin array l hskip2em l u t u m xx displaystyle mu over n u n x x in 0 1 t 0 u m x x 0 0 t 0 u m x x 1 au p t 0 u x 0 u o x 0 x in 0 1 end array wherem n p a 0 the necessary and sufficient conditions for the global existence of solutions are obtained

an lmi approach to asymptotical stability of multi delayed neural networks abstract in this paper some criteria are derived for global asymptotic stability of a class of neural networks with multiple constant or time varying delays based on the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi approach some delay independent criteria for neural networks with multiple constant delays and delay dependent criteria for neural networks with multiple time varying delays are provided to guarantee global asymptotic stability of these networks the main results are generalizations of some recent results reported in the literature

existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary this paper deals with the existence and nonexistence of global positive solutions to ut a ln 1 u in q x 0 oo a ln 1 u 1 u ln 1 u 0 on oq x 0 oo an and u x 0 uo x in q here a 0 is a parameter q c rn is a bounded smooth domain after pointing out the mistakes in global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary siam j math anal 24 1993 317 326 by n wolanski which claims that for q br the ball of rn the positive solution exists globally if and only if a 1 we reconsider the same problem in general bounded domain q and obtain that every positive solution exists globally if and only if a 1 2

an adaptive numerical method to handle blow up in a parabolic system we study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval coupled at the boundary in a nonlinear way in certain cases the system develops a blow up singularity in finite time fixed mesh methods are not well suited to approximate the problem near the singularity as an alternative to reproduce the behaviour of the continuous solution we present an adaptive in space procedure the scheme recovers the conditions for blow up and non simultaneous blow up it also gives the correct non simultaneous blow up rate and set moreover the numerical simultaneous blow up rates coincide with the continuous ones in the cases when the latter are known finally we present numerical experiments that illustrate the behaviour of the adaptive method

singular reaction diffusion equations of porous medium type abstract a class of reaction diffusions equations with nonlinear boundary conditions and porous medium type of diffusion is investigated by the method of upper and lower solutions the diffusion term in the equation includes the so called slow diffusion m 1 fast diffusion 0 m 1 and super diffusion m 0 and the reaction term may involve a singular function the aim of this paper is to show the existence of a global time dependent solution the existence of positive minimal and maximal steady state solutions including the uniqueness of the positive solution and the asymptotic behavior of the time dependent solution in relation to the positive steady state solutions of particular concern is the convergence of the time dependent solution to a unique positive steady state solution and the global attraction property of the reaction diffusion problem applications are given to several specific problems which are of considerable interest in the current literature

doubly nonlinear parabolic equation with nonlinear boundary conditions this paper deals with the existence and nonexistence of global positive solutions of the doubly nonlinear parabolic equation with nonlinear boundary conditions necessary and sufficient conditions in order that all positive solutions exist globally are obtained by using the upper and lower solutions method

stability and instability for solutions of the convective porous medium equation with a nonlinear forcing at the boundary ii abstract we continue our study of the long time behavior of nonnegative solutions for the degenerate parabolic equation u t u m xx n u n x 0 x t 0 subject to nonlinear boundary conditions here we consider solutions of this equation subject to the boundary conditions here we consider solutions of this equation subject to the boundary conditions u m x 0 t au p 0 t u 1 t 0 t 0 as in part i j r anderson j differential equations to appear we give the bifurcation diagrams for the stationary solutions due to the existence of singular stationary states some of these diagrams are three dimensional we also examine the stability properties of these states when n p m 1

global solvability of nonlinear diffusion equations with forcing at the boundary abstract under the influence of a sufficiently weak nonlinear source term it is by now well known that a degenerate diffusion equation is globally solvable a similar result is known when the nonlinear source is present as a forcing term at the boundary such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle however these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present in this work we obtain the global existence of solutions for such problems by deriving a differential inequality involving thelr 1norms of solutions the result is similar to that described above for equations with a nonlinear source term and furthermore establishes the global existence of small solutions even for strong forcings the technique also applies to a more general class of problems involving nonlinear reaction diffusion and convection

interactions among multi nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux abstract this paper deals with a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary conditions two parameters which solve an introduced matrix equation covering all the eight nonlinear exponents of the system are used to get a simple description for the interactions among the multi nonlinearities in the system blow up or not of the solutions is determined by the signs of the two parameters in order to obtain the critical properties of the solutions for the critical sign of the parameters some further analysis on the absorption coefficients and the geometry of is given in addition to the discussion for the eight nonlinear exponents in the system

stability analysis of uncertain genetic sum regulatory networks this paper addresses the problem of establishing robust stability of uncertain genetic networks with sum regulatory functions specifically we first consider uncertain genetic networks where the regulation occurs at the transcriptional level and we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity we hence show that this condition can be formulated as a convex optimization through polynomial lyapunov functions and polynomial descriptions of the bounding set by exploiting the square matricial representation smr of polynomials which allows to establish whether a polynomial is a sum of squares sos via a linear matrix inequality lmi then we propose a method for computing a family of bounding sets by means of convex optimizations it is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed as being the solution of a system of parameter dependent nonlinear equations and is hence unknown lastly the proposed approach is extended to models where the regulation occurs at different levels and both mrna and protein dynamics are nonlinear

critical exponents for the fast diffusion equation with a nonlinear boundary condition abstract in this paper we consider the fast diffusion equation t u u m x t 0 with a nonlinear boundary condition u m u p x t 0 where 0 m 1 p 0 r n is a smooth domain and n 1 we prove that p 0 m 1 2 is the critical global existence exponent for the cases r n b 1 n 2 and b 1 x r n x 1 n 1

stochastic delay differential equations for genetic regulatory networks time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation and protein diffusion between the cytosol and nucleus in this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks based on recent developments with the delay stochastic simulation algorithm the delay chemical master equation and the delay reaction rate equation are developed for describing biological reactions with time delay which leads to stochastic delay differential equations derived from the langevin approach two simple genetic regulatory networks are used to study the impact of intrinsic noise on the system dynamics where there are delays

stabilization of solutions of nonlinear and degenerate evolution equations on etudie le comportement asymptotique des solutions bornees des equations de la forme u t u f x u dans r u x o u 0 x dans u x t u 1 x t dans r ou est un domaine borne de r n avec de classe c 2 0 1