global existence and blow up of solutions to a nonlocal quasilinear degenerate parabolic system abstract this paper investigates the properties of nonnegative solutions of a quasilinear degenerate parabolic system u t div u p 2 u a v x t d x v t div v q 2 v b u x t d x with zero dirichlet boundary conditions in a smooth bounded domain r n n 1 where p q 2 1 and a b 0 are positive constants under appropriate hypotheses we first establish the local existence and uniqueness of solutions then we show that whether or not the solution blows up in finite time depends on the initial data and the relations between and p 1 q 1 in the special case of q 1 and p 1 we conclude that the solution exists globally if p 1 d x q 1 d x 1 a b while if p 1 d x q 1 d x 1 a b then the solution blows up in finite time here x and x denote the unique solution of the following elliptic problem div p 2 1 in x 0 and div q 2 1 in x 0 respectively

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

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

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

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

hardy spaces associated to operators satisfying davies gaffney estimates and bounded holomorphic functional calculus abstract let x be a space of homogeneous type assume that an operator l has a bounded holomorphic functional calculus on l 2 x and the kernel of the heat semigroup e t l t 0 satisfies the davies gaffney estimates without the assumption that l is self adjoint we develop a theory of hardy spaces h l p x 0 p 1 which includes a molecular decomposition an atomic decomposition a square function characterization duality of hardy and lipschitz spaces and a marcinkiewicz type interpolation theorem as applications we show that l has a bounded holomorphic functional calculus on h l p x for all p 0 and certain riesz transforms associated to l are bounded from h l p x to l p x for all 0 p 2

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

second order riesz transforms associated to the schrodinger operator for p 1 abstract let l v be the schrodinger operator on r n where v belongs to the class of reverse holder weights r h q for some q max 2 n 2 we show that the second order riesz transforms 2 l 1 and v l 1 are bounded from the hardy spaces h l p r n associated to l into l p r n for 0 p 1 we show also that the operators 2 l 1 map the classical hardy spaces h p r n into h p r n for a restricted range of p

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

global existence and blow up solutions for doubly degenerate parabolic system with nonlocal source abstract this paper deals with the following nonlocal doubly degenerate parabolic system u t div u m p 2 u m a u 1 x t v 1 x t d x v t div v n q 2 v n b u 2 x t v 2 x t d x with null dirichlet boundary conditions in a smooth bounded domain r n where m n 1 p q 2 i i 0 i 1 2 and a b 0 are positive constants we first get the non existence result for a related elliptic systems of non increasing positive solutions secondly by using this non existence result blow up estimates for above non newton polytropic filtration systems with the homogeneous dirichlet boundary value conditions are obtained then under appropriate hypotheses we establish local theory of the solutions and prove that the solution either exists globally or blows up in finite time in the special case 1 n q 1 2 2 m p 1 1 we also give a criterion for the solution to exist globally or blow up in finite time which depends on a b and x x as defined in the main results

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

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

second order riesz transforms associated with magnetic schrodinger operators abstract let a 1 i a 2 v be a magnetic schrodinger operator on r n where a l loc 2 r n n and 0 v l loc 1 r n we show that l p r n boundedness of l j l k a 1 and v a 1 for some interval of p automatically implies boundedness of the same operators and their commutators on l w p r n for certain muckenhoupt weights w and on the musielak orlicz hardy type spaces

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

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

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

global solutions for some nonlinear parabolic equations with nonmonotonic peturbations publisher summary this chapter discusses the existence uniqueness and decay property of the solutions of two problems it also presents the sobolev s lemma and gagliardo nirenberg inequality

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

periodic solutions of non newtonian polytropic filtration equations with nonlinear sources in this paper the authors first consider the dirichlet boundary value problem to the non newtonian polytropic filtration equation of the form u t div u m p 2 u m h x t u a in wxrwith strong nonlinear sources the existence of nontrivial periodic solutions is established based on topological degree theory the authors also studied the dirichlet boundary value problem to the equation in the form u t div u m 1u p 2 u m 1u b x t u f x t in wxrwith weak nonlinear sources the existence is treated with leray schauder fixed point theory

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

extinction decay and blow up for keller segel systems of fast diffusion type abstract we consider the quasi linear keller segel system of singular type where the principal part u m represents a fast diffusion like 0 m 1 we first construct a global weak solution with small initial data in the scaling invariant norm l n q m 2 for all dimensions n 2 and all exponents q 2 as for the large initial data we show that there exists a blow up solution in the case of n 2 in the second part the decay property in l r with 1 r for 1 2 n m 1 with the mass conservation is shown on the other hand in the case of 0 m 1 2 n the extinction phenomenon of solution is proved it is clarified that the case of m 1 2 n exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1 2 n for the borderline case of m 1 2 n our solution decays in l r exponentially as t

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

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

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

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

riesz transform gaussian bounds and the method of wave equation for an abstract self adjoint operator l and a local operator a we study the boundedness of the riesz transform al on lp for some 0 a very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation we also discuss the relation between the gaussian bounds and the finite speed propagation property using the wave equation methods we obtain a new natural form of the gaussian bounds for the heat kernels for a large class of the generating operators we describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature

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

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

existence of solutions to the slow diffusion equation with a nonlinear source abstract in this paper we consider the existence of solutions to cauchy problem for the slow diffusion equation with a source term t u u m u p x r n t 0 u x 0 x 0 where m 1 and p m our purpose is to obtain lower estimates of the existence time of solutions in the uniformly local lebesgue spaces furthermore we obtain the estimate of blow up time blow up rate of solutions and blow up of critical norm

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

on sign changing and multiple solutions of the p laplacian abstract in this paper we construct the pseudo gradient vector field in w 0 1 p by which we obtain the positive and negative cones of w 0 1 p are both invariant sets of the descent flow of the corresponding functional then we use differential equations theory in banach spaces and dynamics theory to study p laplacian boundary value problems with jumping nonlinearities at zero or infinity and get new multiple solutions and sign changing solutions theorems of p laplacian

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

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

global existence and blow up solutions and blow up estimates for a non local quasilinear degenerate parabolic system this paper deals with p laplacian systems ut div u p 2 u a u 1 x t v 1 x t dx x t 0 vt div v q 2 v b u 2 x t v 2 x t dx x t 0 with null dirichlet boundary conditions in a smooth bounded domain rn where p q 1 i i 0 i 1 2 and a b 0 are positive constants we first get the non existence result for a related elliptic systems of non increasing positive solutions secondly by using this non existence result blow up estimates for above p laplacian systems with the homogeneous dirichlet boundary value conditions are obtained under br x rn x 0 then under appropriate hypotheses we establish local theory of the solutions and obtain that the solutions either exists globally or blow up in finite time

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

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

a comparison between star products on regular orbits of compact lie groups in this paper an algebraic and a differential star product defined on a regular coadjoint orbit of a compact semisimple group are compared it has been proved that there is an injective algebra homomorphism between the algebra of polynomials with the algebraic star product and the algebra of differential functions with the differential star product structure

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

global existence and decay properties for a degenerate keller segel model with a power factor in drift term abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u q 1 v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where m 1 q 2 0 or 1 and n 1 the aim of this paper is to prove the existence of a time global weak solution u v of ks with the l 0 l r n bound such a global bound is obtained in the case of i m q 2 n for large initial data and ii 1 m q 2 n for small initial data in the case of ii the decay properties of the solution u v are also discussed

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

global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where 0 or 1 we show here that the system of ks with m 1 has a time global weak solution u v with a uniform bound in time when u 0 v 0 is a nonnegative function and in l 1 l r n l 1 h 1 w 1 r n u 0 m h 1 r n the decay properties of the solution with small initial data are also discussed

on tangential star products for the coadjoint poisson structure we derive necessary conditions on a lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the lie algebra for the coadjoint poisson structure which is both differential and tangential to all the coadjoint orbits in particular we show that when the lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its lie algebra

periodic solutions of evolutionp laplacian equations with nonlinear sources abstract in this paper we are concerned with the time periodic solutions to the evolution p laplacian equations of the form formula and formula with the dirichlet boundary value condition where p 2 0 is a bounded convex domain in r n with smooth boundary m x t is continuous on r periodic in t with period and positive in r the existence of nontrivial nonnegative periodic solutions is established provided that n 1 and p 1 p 1 p n

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