exponential convergence for nonlinear diffusion problems with positive lateral boundary conditions it is established that the solution u of ut um 0 with positive initial data positive lateral boundary data and positive exponent m converges exponentially to the solution v of the corresponding stationary equation vm 0 the analysis also provides the form of the leading contribution to the difference u v

the principle of linearized stability for a class of degenerate diffusion equations here u0 is a given bounded nonnegative function the functions b s and f s defined for s 2 0 are smooth for s 0 f 0 0 i o 0 y o co i s 0 for s o and fib is lipschitz continuous for s 30 the precise hypotheses on the data can be found in section 2 since p o co the diffusion is degenerate near points where u 0 in this paper we study the stability of steady state solutions of problem i by linearizing 1 1 near an equilibrium v so let 0 e c d satisfy

variable structure model reference adaptive control using only input and output measurements abstract a variable structure model reference adaptive controller vs mrac using only the input and output measurements of the plant is proposed stability and transient properties are analysed and discussed remarkable transient behaviour is revealed in particular model following can be perfectly achieved in a finite time the effect of plant input disturbance is investigated and the performance of the vs controller is also shown to be qualitatively superior to that obtained with conventional integral adaptation algorithms a link between the vs mrac and the conventional mrac is suggested simulations are presented to highlight the theoretical results

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

global existence and l estimates of solution for doubly nonlinear parabolic equation the existence uniqueness and l estimate of the weak solution to the initial boundary value problem for the doubly nonlinear parabolic equationut div u r u m 2 u au fx t uin 0 with zero boundary condition in a bounded domain rn are established in particular the behavior of the solution as t 0 and t is investigated

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

an asymptotic and numerical description of self similar blow up in quasilinear parabolic equations we study the blow up behaviour of two reaction diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction we show that in each case the blow up is self similar in contrast to the linear diffusion limit of each in which the diffusion is only approximately self similar we then investigate the limit of the self similar behaviour and describe the transition from a stable manifold blow up behaviour for quasilinear diffusion to a centre manifold one for the linear diffusion

the existence of a global fundamental solution for homogeneous hormander operators via a global lifting method we prove the existence of a global fundamental solution x y with pole x for any hormander operator l i 1mxi2 on rn which is homogeneous of degree 2 here homogeneity is meant with respect to a family of non isotropic diagonal maps of the form x 1x1 nxn with 1 1 n due to a global lifting method for homogeneous operators proved by folland comm partial differential equations 2 1977 161 207 there exists a carnot group g and a polynomial surjective map g rn such that l is related to a sub laplacian lg on g we show that it is always possible to perform a global change of variable on g such that the lifting map becomes the projection of g rn rp onto rn we prove that an integration argument over the non compact fibers of provides a fundamental solution for l indeed if g x x y y x y rn x y rp is the fundamental solution of lg we show that g x 0 y y is always integrable with respect to y rp and its y integral is a fundamental solution for l

effective recursive algorithm for judging the positive definiteness of matrices of high dimension this paper presents a new efficient recursive algorithm for judging the positive definiteness of a matrix with this algorithm both a simpler computation process and a decreased memory requirement can be obtained

on solutions to the initial boundary value problem for an equation in the theory of infiltration etude de l existence et du comportement des solutions du probleme de valeur limite initiale

smoothing effects of the initial boundary value problem for logarithmic type quasilinear parabolic equations abstract we give existence theorems of global solutions in l l o c 0 w 0 1 to the initial boundary value problem for quasilinear degenerate parabolic equations of the form u t div u 2 u 0 where the class of v 2 includes the logarithmic case u 2 log 1 u 2 for a typical example we assume that the initial data belong to w 0 1 p 0 p 0 2 or l r r 1 and we derive precise estimates for u t near t 0

normal families of meromorphic functions concerning shared values abstract in this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family f of meromorphic functions in a domain d is normal if for each pair of functions f and g in f f a f n and g a g n share a value b in d where n is a positive integer and a b are two finite constants such that n 4 and a 0 this result is not true when n 3

large sets at infinity and maximum principle on unbounded domains for a class of sub elliptic operators abstract maximum principles on unbounded domains play a crucial role in several problems related to linear second order pdes of elliptic and parabolic type in this paper we consider a class of sub elliptic operators l in r n and we establish some criteria for an unbounded open set to be a maximum principle set for l we extend some classical results related to the laplacian by deny hayman and kennedy and to the sub laplacians on stratified lie groups by bonfiglioli and the second named author

gradient estimates for degenerate diffusion equations ii we establish upper and lower bounds for various norms of solutions and their gradients for the equation u t div u m 1 u in n in terms of the norms of the initial data based on the l estimate of u we conclude that u x t is lipschitz continuous in space time for all t 0 whenever u x 0 is in l 1 n

exceptional functions and normal families of meromorphic functions with multiple zeros abstract let k be a positive integer with k 2 let h 0 be a holomorphic function which has no simple zeros in d and let f be a family of meromorphic functions defined in d all of whose poles are multiple and all of whose zeros have multiplicity at least k 1 if for each function f f f k z h z then f is normal in d

asymptotic behaviour of the doubly nonlinear diffusion equation ut δpum on bounded domains abstract we study the homogeneous dirichlet problem for the doubly nonlinear diffusion equation u t p u m where p 1 m 0 posed in a bounded domain in r n with homogeneous boundary conditions and with non negative and integrable initial data in this paper we consider the degenerate case m p 1 1 and the quasilinear case m p 1 1 in the first case we establish the large time behaviour by proving the uniform convergence to a unique asymptotic profile and we also give rates for this convergence the difference in the second case is that the asymptotic profile is unique up to a constant factor that we determine

output feedback model reference sliding mode control of uncertain multivariable systems this note considers the robust output tracking problem using a model reference sliding mode controller for linear multivariable systems of relative degree one it is shown that the closed loop system is globally exponentially stable and the performance is insensitive to bounded input disturbances and parameter uncertainties the strategy is based on output feedback unit vector control to generate sliding mode the only required a priori information about the plant high frequency gain matrix k sub p is the knowledge of a matrix s sub p such that k sub p s sub p is hurwitz which relaxes the positive definiteness requirement usually needed by other methods

characterization of the hilbert ball by its automorphism group let be a bounded convex domain in a separable hilbert space the authors prove a version of the theorem of bun wong which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point then it is biholomorphic to the ball key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new normal families arguments in the construction of the limit biholomorphism

global existence to the initial boundary value problem for a system of nonlinear diffusion and wave equations abstract we prove the global existence of weak solution pair to the initial boundary value problem for a system of m laplacian type diffusion equation and nonlinear wave equation the interaction of two equations is given through nonlinear source terms f u v and g u v

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

nonnegative solutions of the initial dirichlet problem for generalized porous medium equations in cylinders in this paper we study nonnegative solutions u of a class of nonlinear evolutions au at a u whose best known example is the porous medium equation u u m 1 see 9 for a recent survey on these equations we study our solutions u subject to dirichlet boundary conditions in d x 0 t where d is a bounded smooth domain in rn i e uiadx 0 t 0 the corresponding theory for d rn was studied by the authors in 3 we find that there is one special solution x t the friendly giant which tends to infinity as t 1 0 and that all other solutions u x t are in one to one correspondence with suitable pairs of measures u on d and a ou ad we also show that any solution on d x 0 t extends to a solution in d x 0 oo and that the friendly giant f x t governs the asymptotic behavior as t t oo the nonlinearity is assumed to be continuous increasing with p 0 0 and also verifies the growth conditions

regularity of solutions and interfaces of the porous medium equation via local estimates on demontre la regularite c pour des interfaces mobiles de solutions locales de l equation de milieu poreux

on the initial boundary value problem for some quasilinear parabolic equations of divergence form abstract in this paper we give an existence theorem of global classical solution to the initial boundary value problem for the quasilinear parabolic equations of divergence form u t div u 2 u f u u x t where u 2 may not be bounded as u as an application the logarithmic type nonlinearity u 2 log 1 u 2 which is growing as u and degenerate at u 0 is considered under f 0

a note on on the existence of solutions of quasilinear wave equations with viscosity abstract in the note we study the existence and uniqueness of the global generalized solution for the following initial boundary value problem 1 u t t t x div u t x 2 u t x u t x u t t x u t t x p 1 u t t x u t x q 1 u t x x t 0 t 2 u t x 0 t 0 t 3 u 0 x u 0 x u t 0 x u 1 x x where is a bounded domain in r n with smooth boundary p 1 q 1 0 0 denotes the laplacian operator in r n denotes the gradient operator and s is a given nonlinear function

gradient estimates for solutions of parabolic equations and systems 1 statement of the problem let 52 c r n 2 be a bounded domain with smooth boundary r 2 in this note we study scalar quasilinear parabolic differential equations u div y g ivul vu f u 0 1 u a u f u o 2 and systems u da u f u 0 3 on q x 0 co with initial conditions u 0 u0 resp u 0 u0 and homogeneous dirichlet boundary conditions u i x o co o for 1 and 2 resp 4 u1r x 0 00 0 for 3 5

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

on global attractor for nonlinear parabolic equations of m laplacian type abstract existence and some regularity results of global attractor in l q q 1 for m laplacian type quasilinear parabolic equation with a perturbation like a x u u u u f x with 0 a x 0 are proved for the proofs moser s technique is used extensively

removable singularities of solutions of degenerate nonlinear elliptic equations on the boundary of a domain the removability of singularities of solutions for the dirichlet problem for degenerate nonlinear elliptic equations on the boundary of a domain is studied a method based on a priori energetic estimates of solutions to elliptic boundary value problems is used the growth in the vicinity of a boundary point finite or at infinity for generalized solutions is studied

exact results for the nonlinear diffusion equations delta u delta t delta delta x u 4 3 delta u delta x and delta u delta t delta delta x u 2 3 delta u delta x the author exploits local and non local symmetries of the two equations of the title and of some related equations some new exact solutions are derived and some specific boundary value problems of physical significance are considered

variable structure model reference adaptive control vs mrac using only input and output measurements the general case a globally asymptotically stable vs mrac for the general case of plants with arbitrary relative degree is presented it is shown that the vs approach leads to general adaptive controllers with outstanding performance the controllers use only i o measurements and are free of differentiators the results indicate that several of the usual drawbacks of conventional mracs e g unsatisfactory transient behavior lack of robustness difficulties in designing controllers for time varying or nonlinear plants can be circumvented

existence of global solution for reaction diffusion systems with density dependent diffusion on demontre l existence de solutions globales pour un systeme de reaction diffusion a diffusion dependant de la densite du type u id t u i mi f i u x t 0 i 1 2 m u u 1 u 2 u m avec des conditions aux limites de derichlet

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

long time existence of smooth solution for the porous medium equation in a bounded domain in this paper we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 ut umin 0 u x 0 u0 0in u x t 0for x where m 1 is the permeability the proof is based on the short time existence of cs2 smooth solution the global cs1 estimate the holder estimate of divergence type degenerate equation with measurable coefficients and cs1 estimate of mixed type equation with lipschitz coefficients

global positivity estimates and harnack inequalities for the fast diffusion equation abstract we investigate local and global properties of positive solutions to the fast diffusion equation u t u m in the range d 2 d m 1 corresponding to general nonnegative initial data for the cauchy problem posed in the whole euclidean space r d we prove sharp local positivity estimates weak harnack inequalities and elliptic harnack inequalities we use them to derive sharp global positivity estimates and a global harnack principle for the mixed initial and boundary value problem posed in a bounded domain of r d with homogeneous dirichlet condition we prove weak and elliptic harnack inequalities our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow

the first boundary value problem for u λup on etudie le probleme ui6 u u p 1 pour 0 0 sont des constantes reelles fixees on considere l existence de solutions pour 0 0

subharmonic functions on carnot groups many results of classical potential theory are extended to sub laplacians on carnot groups some characterizations of subharmonicity representation formulas of poisson jensen s kind and nevanlinna type theorems are proved we also characterize the riesz measure related to bounded above subharmonic functions in n

analysis and design of i o based variable structure adaptive control this paper presents an overview of a technique for the design of variable structure model reference adaptive control vs mrac systems using only input output data the main ideas underlying the design and the analysis as well as the main properties of the controller are described having as point of departure a well known parameter adaptive mrac scheme then asymptotic properties of the vs mrac system are established taking into account the effect of the averaging filters necessary to implement some equivalent control signals in particular global exponential stability of the associated error system with respect to a small residual set is demonstrated the effect of linear zones in the relay functions is assessed showing that they can alleviate the chattering phenomena this is related with the fact that the vs mrac has a high gain stability property a procedure for noise sensitivity reduction is also proposed simulations illustrate the very satisfactory performance of the vs mrac in adverse operating conditions

convergence to equilibrium in degenerate parabolic equations with delay abstract in 11 busenberg huang 1996 showed that small positive equilibria can undergo supercritical hopf bifurcation in a delay logistic reaction diffusion equation with dirichlet boundary conditions consequently stable spatially inhomogeneous time periodic solutions exist previously in 12 badii diaz tesei 1987 considered a similar logistic type delay diffusion equation but differing in two important respects firstly by the inclusion of nonlinear degenerate diffusion of so called porous medium type and secondly by the inclusion of an additional dominating instantaneous negative feedback where terms local in time majorize the delay terms in some sense sufficient conditions were given ensuring convergence of non negative solutions to a unique positive equilibrium a natural question to ask and one which motivated the present work is can one still ensure convergence to equilibrium in delay logistic diffusion equations in the presence of nonlinear degenerate diffusion but in the absence of dominating instantaneous negative feedback the present paper considers this question and provides sufficient conditions to answer in the affirmative in fact the results are much stronger establishing global convergence for a much wider class of problems which generalize the porous medium diffusion and delay logistic terms to larger classes of nonlinearities furthermore the results obtained are independent of the size of the delay

smooth solution for the porous medium equation in a bounded domain abstract in this paper we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain 0 1 u t u m m 1 with zero boundary condition on the fixed boundary the flux u m is nonzero while the gradient of pressure u m 1 is zero as a consequence the parabolic equation above becomes degenerate the proof is based on a definition of weighted space c s 2 corresponding to the given degeneracy and the schauder estimates in its linearized equation 0 2 w t x n a i j x d i j w in c s 2 for 0 m 1 m 1

normal families of meromorphic functions with multiple values let f be a family of meromorphic functions defined in a domain d let 0 be a holomorphic function in d and k be a positive integer suppose that for every function f f f 0 f k 0 and all zeros of f k z have multiplicities at least k 2 k if for k 1 has only zeros with multiplicities at most 2 and for k 2 has only simple zeros then f is normal in d this improves and generalizes the related results of gu fang and chang yang schwick et al

finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption the energy solutions to the equation have finite speed of propagation if l or l if 1 vq t is uniformly bounded for t 0 localisation property and if q sharp upper bounds of the interface or free boundary are obtained we use a weighted energy method the weights being powers of the distance to a variable half space we also study decay rates as t and extinction in finite time for bounded and unbounded domains with null dirichlet boundary conditions our equation includes the porous media equation with absorption analogous results hold if m is replaced by an appropriate quasilinear elliptic operator

boundary regularity for the porous medium equation we study the boundary regularity of solutions to the porous medium equation u t delta u m in the degenerate range m 1 in particular we show that in cylinders the dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values provided that the spatial domain satisfies the elliptic wiener criterion this condition is known to be optimal and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general not necessarily cylindrical domains in bf r n 1 one of our fundamental tools is a new strict comparison principle between sub and superparabolic functions which makes it essential for us to study both nonstrict and strict perron solutions to be able to develop a fruitful boundary regularity theory several other comparison principles and pasting lemmas are also obtained in the process we obtain a rather complete picture of the relation between sub superparabolic functions and weak sub supersolutions

normal families of functions for subelliptic operators and the theorems of montel and koebe abstract a classical theorem of montel states that a family of holomorphic functions on a domain c uniformly bounded on the compact subsets of is a normal family the aim of this paper is to obtain a generalization of this result in the subelliptic setting of families of solutions u to l u 0 where l belongs to a wide class of real divergence form pdos comprising sub laplacians on carnot groups subelliptic laplacians on arbitrary lie groups as well as the laplace beltrami operator on riemannian manifolds to this end we extend another remarkable result due to koebe we characterize the solutions to l u 0 as fixed points of suitable mean value operators with non trivial kernels a suitable substitute for the cauchy integral formula is also provided finally the local boundedness assumption is relaxed by replacing it with l loc 1 boundedness

on uniform and decay estimates for unbounded solutions of partial differential equations abstract we prove that if a function u satisfies certain integral estimates of energy type then it verifies estimates of the type u t l r c u 0 l r 0 0 t 1 t 0 where r and r 0 are exponents that appear in these integral estimates and 1 and 0 are positive constants that can be expressed in terms of these constants we will see that in some cases r r 0 supercontractive estimates and hence a regularizing effect on u appears while in other cases this improvement of regularity does not appear since r r 0 in any case we prove that the l r norm of u decays in time for t large since it verifies the previous estimate we show how to apply this result to obtain new estimates of this kind for the solutions of many nonlinear parabolic equations this new method allows also to explain the reason of the similar behavior of the solutions of very different parabolic problems finally we study sufficient conditions for extinction in finite time

existence uniqueness and qualitative properties of the solution of a degenerate nonlinear parabolic system abstract in this paper we are concerned with the study of the existence of solutions of a nonlinear parabolic system with density dependent diffusion subject to homogeneous third type boundary conditions a comparison result is shown that allows the study of the asymptotic behavior of the solutions via the same methods used for systems with classical diffusion finally an application of the results to an epidemic system is presented sufficient conditions are given for the positivity of the solution of this system

stabilization of solutions for a class of degenerate equations in divergence form in one space dimension abstract by extending the concept of the liapunov functional and an idea due to dafermos we establish an invariance principle for a class of not necessarily strongly monotone systems this principle is then applied to continuous contraction strongly monotone systems on a hubert space and to a class of l 1 contraction monotone semigroups generated by degenerate equations in divergence form in one space dimension

design and stability analysis of an indirect variable structure model reference adaptive control this paper presents the stability and robustness analysis of an indirect adaptive control based variable structure algorithm for uncertain plants with relative degree one the indirect approach has been proposed as a way to get an easier design of the switching laws compared to the direct relationship between the plant transfer function parameters and the physical parameters of the system here a singular perturbation approach is used to establish the robustness of the controller in the presence of unmodelled dynamics and disturbances it is shown that for sufficiently small parasitics the system remains stable with respect to some small residual set

quasilinear parabolic and elliptic equations with nonlinear boundary conditions abstract this paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both dirichlet and nonlinear neumann boundary conditions the equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient the consideration of the class of equations is motivated by some heat transfer problems where the heat capacity and thermal conductivity are both temperature dependent the aim of the paper is to show the existence and uniqueness of a global time dependent solution of the parabolic problem existence of maximal and minimal steady state solutions of the elliptic problem including conditions for the uniqueness of a solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions applications are given to some heat transfer problems and an extended logistic reaction diffusion equation

large time asymptotics for nonlinear diffusions the initial boundary value problem in this paper we investigate the large time behavior of solutions to the first initial boundary value problem for the nonlinear diffusion ut um xx m 0 in particular we prove exponential decay of u x t towards its own steady state in l1 norm for long times and we give an explicit upper bound for the rate of decay the result is based on a new application of entropy estimates and on detailed lower bounds for the entropy production in this situation

large time behaviour of solutions to a class of non autonomous degenerate parabolic equations we consider a class of non autonomous degenerate parabolic equations and we study the asymptotic behaviour of the solutions even if the equation depends explicitly upon the time we prove that several asymptotic properties valid for the autonomous case are preserved in this more general situation to our knowledge it is the first time that the asymptotic behaviour of solutions to non autonomous equations is studied

quasilinear parabolic and elliptic systems with mixed quasimonotone functions abstract this paper deals with a class of quasilinear parabolic and elliptic systems with mixed quasimonotone reaction functions the boundary condition in the system may be dirichlet nonlinear or a combination of these two types the elliptic operators in the system are allowed to be degenerate the aim is to show the existence and uniqueness of a classical solution to the parabolic system the existence of maximal and minimal solutions or quasisolutions of the elliptic system and the asymptotic behavior of the solution of the parabolic system this consideration leads to a global attractor of the parabolic system as well as an one sided stability of the maximal and minimal solutions applications of these results are given to three models arising from biology and ecology where diffusion coefficients are density dependent and are degenerate these applications exhibit quite distinct dynamical behavior of the population species between degenerate density dependent diffusion and constant diffusion

well posedness of the dirichlet problem for the non linear diffusion equation in non smooth domains we investigate the dirichlet problem for the parablic equation u t u m m 0 in a non smooth domain r n 1 n 2 in a recent paper u g abdulla j math anal appl 260 2 2001 384 403 existence and boundary regularity results were established in this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial boundary data in particular we prove l 1 contraction estimation in general non smooth domains

on solutions of some doubly nonlinear degenerate parabolic equations with absorption abstract the existence uniqueness regularity and behavior of solutions to the initial boundary value problem for the equation u t i 1 n x i u x i p 2 u x i u 0 with zero dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in r n where 0 and p 1

new approach to robust model reference adaptive control for a class of plants motivated by recent advances in designing robust adaptive controllers and in dealing with uncertain dynamical systems a new model reference adaptive control which is robust to a class of unmodelled dynamics and bounded output disturbances in the case of relative degree one is presented the implementation of the controllers includes a switching mechanism which plays a crucially important role in functions of stabilizing as well as tracking it is shown that global stability of the overall system is achieved under no assumption of persistency of excitation and tracking errors will converge to a residual set whose size can be directly related to the size of unmodelled dynamics and of output disturbances explicitly in the ideal case the residual set degenerates to a single null point and convergence can be achieved in finite time without any requirement of persistency of excitation

global existence and gradient estimates for the quasilinear parabolic equations of m laplacian type with a nonlinear convection term abstract in this paper we derive precise estimates for u t including smoothing effects near t 0 and decay as t as well as global existence of the solutions u t to the initial boundary value problem in a bounded domain in r n for the quasilinear parabolic equation of the m laplacian type with a nonlinear convection term b u u for the initial data u 0 we only assume u 0 l q 1 q

normal families and shared values of meromorphic functions abstract let k 2 be a positive integer let f be a family of meromorphic functions in a domain d all of whose zeros have multiplicity at least k 1 and let a z 0 h z 0 be two holomorphic functions on d if for each f f f a z f k h z then f is normal in d

contractive properties for the heat equation in sobolev spaces abstract consider the heat equation r u x u 0 in a cylinder 0 t r n 1 smooth lateral boundary under zero neumann or dirichlet conditions geometric conditions for are given that guarantee that for a given p x u t l p will be non increasing for any solution decay rates are also given for arbitrary and p it is shown how to construct an equivalent l p norm such that x t is non increasing in this norm

normal families and value distribution in connection with composite functions abstract we prove a value distribution result which has several interesting corollaries let k n let c and let f be a transcendental entire function with order less than 1 2 then for every nonconstant entire function g we have that f g k has infinitely many zeros this result also holds when k 1 for every transcendental entire function g we also prove the following result for normal families let k n let f be a transcendental entire function with f 1 k and let a 0 a k 1 a be analytic functions in a domain then the family of analytic functions g such that f g k z j 0 k 1 a j z f g j z a z in is a normal family

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