on critical phenomena in interacting growth systems part ii bounded growth this paper completes the classification of some infinite and finite growth systems which was started in part i components whose states are integer numbers interact in a local deterministic way in addition to which every component s state grows by a positive integerk with a probability ek 1 e at every moment of the discrete time proposition 1 says that in the infinite system which starts from the state all zeros percentages of elements whose states exceed a given valuek 0 never exceed ce k wherec const proposition 2 refers to finite systems it states that the same inequalities hold during a time which depends exponentially on the system size

estimates on the fundamental solution to heat flows with uniformly elliptic coefficients we obtain estimates from above and below on the fundamental solution to the cauchy initial value problem for parabolic equations having uniformly elliptic coefficients we impose minimal regularity requirements on the coefficients the estimates are expressed in terms of an energy functional associated with the coefficients

on fundamental solutions of generalized schrodinger operators abstract we consider the generalized schrodinger operator where is a nonnegative radon measure in r n n 3 assuming that satisfies certain scale invariant kato conditions and doubling conditions we establish the following bounds for the fundamental solution of in r n ce e 2 d x y x y n 2 x y ce e 1 d x y x y n 2 where d x y is the distance function for the modified agmon metric m x dx2 associated with we also study the boundedness of the corresponding riesz transforms 1 2 on lp r n dx

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

absence of ground state for the nelson model on static space times abstract we consider the nelson model on some static space times and investigate the problem of absence of a ground state nelson models with variable coefficients arise when one replaces in the usual nelson model the flat minkowski metric by a static metric allowing also the boson mass to depend on position we investigate the absence of a ground state of the hamiltonian in the presence of the infrared problem i e assuming that the boson mass m x tends to 0 at spatial infinity using path space techniques we show that if m x c x at infinity for some c 0 and 1 then the nelson hamiltonian has no ground state

lp norms of non critical schrodinger semigroups we consider schrodinger semigroups e th h v on rn with v c x 2 as x 0 c 12 n 2 2 with h 0 we determine the exact power law divergence of e th p p and of some e th q p as maps from lp to lq the results are expressed most naturally in terms of the power for which there exists a positive resonance such that h 0 x x

asymptotic expansion of solutions to the black scholes equation arising from american option pricing near the expiry our aim in this paper is to approximate the price of an american call option written on a dividend paying stock close to expiry using an asymptotic analytic approach we use the heat equation equivalent of the black scholes partial differential equation defined on an unbounded spatial domain and decompose it into inner and outer problems we extend the idea presented in h han and x wu a fast numerical method for the black scholes equation of american options siam journal on numerical analysis 41 6 2003 2081 2095 in which a weakly singular memory type transparent boundary condition tbc is obtained for the special case that the initial condition is equal to zero we first derive this tbc in the general case and then focus on the outer problem in conjunction with an equivalent non singular version of the tbc dubbed etbc which is more tractable for analytical purposes we then obtain the general solution of the outer problem in series form based on the repeated integrals of the complementary error function which also satisfies the introduced etbc as the next step using the machinery of poincare asymptotic expansion and taking time to expiry as the expansion parameter we find the general term of this series in closed form when the risk free interest rate r is less than the dividend yield we also obtain the first five terms in the opposite case r in a systematic manner we also prove the convergence properties of the obtained series rigorously under some general conditions our numerical experiments based on the obtained asymptotic series demonstrate the applicability and effectiveness of the results in valuation of a wide range of american option problems

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

a gaussian estimate for the heat kernel on differential forms and application to the riesz transform let m g be a complete riemannian manifold which satisfies a sobolev inequality of dimension n and on which the volume growth is comparable to the one of mathbb r n for big balls if there is no non zero l 2 harmonic 1 form and the ricci tensor is in l frac n 2 varepsilon cap l infty for an varepsilon 0 then we prove a gaussian estimate on the heat kernel of the hodge laplacian acting on 1 forms this allows us to prove that under the same hypotheses the riesz transform d vardelta 1 2 is bounded on l p for all 1 3

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

existence and nonexistence of the global solution for the semilinear parabolic system on riemannian manifold abstract in this paper we study the global existence and nonexistence of solutions to the following semilinear parabolic system u t u v p in m n 0 v t v u q in m n 0 u x 0 u 0 x in m n v x 0 v 0 x in m n where m n n 3 is a non compact complete riemannian manifold is the laplace beltrami operator we assume that both u 0 x and v 0 x are nonnegative smooth and bounded functions constants p q 1

the form of blow up for nonlinear parabolic equations on considere des equations paraboliques semi lineaires de la forme u t 2 u sf u ou f est positive et 0 ds f s est finie on etudie des problemes a 1 dimension dans la region semi infinie x o et on trouve le temps t

positive solutions and large time behaviors of schrodinger semigroups simon s problem let h v where v is a multiplication operator by a real valued function v x on rn which is uniformly holder continuous and 1 x 2 2 v x l rn for some 4 the relationship between existence of positive solutions with growth conditions of hg 0 and asymptotic behaviors as t of e th is established using it b simon s problem for h on r2 is solved

spectral multipliers via resolvent type estimates on non homogeneous metric measure spaces we describe a simple but surprisingly effective technique of obtaining spectral multiplier results for abstract operators which satisfy the finite propagation speed property for the corresponding wave equation propagator we show that in this setting spectral multipliers follow from resolvent or semigroup type estimates the most notable point of the paper is that our approach is very flexible and can be applied even if the corresponding ambient space does not satisfy the doubling condition or if the semigroup generated by an operator is not uniformly bounded as a corollary we obtain l p spectrum independence for several second order differential operators and recover some known results our examples include the laplace belltrami operator on manifolds with ends and schrodinger operators with strongly subcritical potentials

regularity of weak solutions of elliptic and parabolic equations with some critical or supercritical potentials abstract we prove holder continuity of weak solutions of the uniformly elliptic and parabolic equations 0 1 i a i j x j u x a x 2 u x 0 a 0 0 0 2 i a i j x t j u x t a x 2 u x t t u x t 0 a 0 0 with critical or supercritical 0 order term coefficients which are beyond de giorgi nash moser s theory we also prove in some special cases weak solutions are even differentiable previously p baras and j a goldstein 3 treated the case when a 0 a i j i and 0 for which they show that there does not exist any regular positive solution or singular positive solutions depending on the size of a when a 0 0 and a i j i p d milman and y a semenov 7 8 obtain bounds for the heat kernel

an optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with ricci ge 0 by establishing an optimal comparison result on the heat kernel of the conformal laplacian on open manifolds with nonnegative ricci curvature a we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant b we identify a class of functions with the following property if the manifold has a scalar curvature in this class then there exists a complete conformal metric whose scalar curvature is any given function in this class this class is optimal in some sense c we have identified all manifolds with nonnegative ricci curvature which are uniformly conformal to manifolds with zero scalar curvature even in the euclidean case we obtain a necessary and sufficient condition under which the main existence results in ni1 and kn on prescribing nonnegative scalar curvature will hold this condition had been sought in several papers in the last two decades

a critical exponent of fujita type for a nonlinear reaction diffusion system on a riemannian manifold abstract in this paper we study the global existence and nonexistence of positive solutions to the following nonlinear reaction diffusion system u t u w x v p s x in m n 0 v t v f x u d g x in m n 0 u x 0 u 0 x in m n v x 0 v 0 x in m n where m n n 3 is a non compact complete riemannian manifold is the laplace beltrami operator and s x g x are non negative l l o c 1 functions we assume that both u 0 x and v 0 x are non negative smooth and bounded functions and constants p d 1 when p d there is an exponent p which is critical in the following sense when p 1 p the above problem has no global positive solution for any non negative constants s x g x not identically zero when p p the problem has a global positive solution for some s x g x 0 and u 0 x v 0 x 0

solvability conditions for some semi linear parabolic equations consider the cauchy problem of the semi linear parabolic equation u tu vup w 0 p 1 in rn 0 t t 0 u x 0 u0 x 0 and u0 v w lloc1 rn we establish a necessary condition on the nonlinear potential v v x so that the above has a positive solution for some w w x 0 u0 u0 x 0 an application to the case when v is the important inverse square potential is also given

global blow up of a nonlinear heat equation solutions to semilinear parabolic equations of the form u t u f u x in which blow up at some finite time t are investigated for slowly growing functions f for nonlinearities such as f s 2 s ln 2 s 1 b with 0 u becomes infinite throughout as t t it is alsofound that for marginally more quickly growing functions e g f s 2 s ln 2 s 2 u is unbounded on some subset of which has positive measure and is unbounded throughout if is a small enough region

comportement des noyaux de la chaleur des operateurs de schrodinger et applications a certaines equations paraboliques semi lineaires abstract we consider general schrodinger operators on domains of riemannian manifolds with possibly exponential volume growth we prove sharp large time gaussian upper bounds these bounds are then used to prove new l p l p estimates for the corresponding semigroups applications to semi linear parabolic equations are given

on a new family of radial basis functions mathematical analysis and applications to option pricing abstract in this paper we introduce a new family of infinitely smooth and nearly locally supported radial basis functions rbfs derived from the general solution of a heat equation arising from the american option pricing problem these basis functions are expressed in terms of the repeated integrals of the complementary error function and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model we introduce an integral operator with a function dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the rbf theory we then show that using the introduced functions as expansion bases in the context of an rbf based meshless collocation scheme we could exactly impose the transparent boundary condition accompanying the heat equation we prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular rbf families used in current literature some other properties of these bases such as their fourier transforms as well as some useful representations in terms of positive borel measures will also be discussed

isoperimetric properties of higher eigenvalues of elliptic operators we prove in the setting of a measure energy space m e f that if the smallest eigenvalue 1 of the generator of the dirichlet form e in any precompact open set m admits the estimate 1 where is a measure absolutely continuous with respect to and 0 then a similar estimate holds for the kth smallest eigenvalue k const k as an application we obtain an upper estimate of the stability index of a minimal surface in r 3 via the total curvature

simple one dimensional interaction systems with superexponential relaxation times finite one dimensional random processes with local interaction are presented which keep some information of a topological nature about their initial conditions during time the logarithm of whose expectation grows asymptotically at least asm3 wherem is the size of the setr m of states of one component actuallyr m is a circle of lengthm at every moment of the discrete time every component turns into some kind of average of its neighbors after which it makes a random step along this circle all these steps are mutually independent and identically distributed in the present version the absolute values of the steps never exceed a constant the processes are uniform in space time and the set of states this estimation contributes to our awareness of what kind of stable behavior one can expect from one dimensional random processes with local interaction

the heat equation with a singular potential of concern is the singular problem du dt au c x 2 iz x tz x 0 u0 x and its generalizations here c 0 x g rn t 0 and and t 0 are nonnegative and not both identically zero there is a dimension dependent constant c n such that the problem has no solution for c ct n for c s ct a necessary and sufficient conditions are found for and u0 so that a nonnegative solution exists

uniform blow up profile for a degenerate parabolic system with nonlocal source in this paper we establish the local existence of the solution and the finite time blow up result for the following system ut ur1 u a updx forx ohm t 0 ut ur2 u b uqdx forx ohm t 0 where p q 1 and 0 r1 r2 1 moreover it is proved that the solution has global blow up and limt t t t p 1 r2 yu x t 1 r1 11 r1 a a y p 1 r2 y q 1 r1 py xa p 1 r2 b1 r2 y limt t t t q 1 r1 yu x t 1 r2 11 r2 b a y q 1 r1 y p 1 r2 qy xb q 1 r1 a1 r1 y uniformly on compact subsets of where pq 1 r1 1 r2 and t is the blow up time

the quantizing effect of potentials on the critical number of reaction diffusion equations we give an almost complete characterization of the relation between the critical fujita number of the equation u vu up ut 0 and the potentials v behaving like a 1 d x b here a 0 and b is any real number the only type of v not covered is when v x a 1 d x b with b 2 and a is not small it is interesting to note that when v a 1 d x 2 we are at a border line case where the critical exponent depending on a can vary from 1 to for all other v except the ones the theorem does not cover the critical fujita number takes only three discrete values 1 and 1 2n in the euclidean case we also obtain global estimates of schrodinger heat kernels and a liouville theorem for a semilinear elliptic equation

critical exponents of fujita type for inhomogeneous parabolic equations and systems abstract we consider the large time behavior of sign changing solutions of inhomogeneous parabolic equations and systems for example for u t u u p w x in r n 0 t we show the following if n 3 and r n w x dx 0 and 1 p n n 2 then all solutions blow up in finite time while if p n n 2 there are both global and nonglobal solutions we show by example that global solutions exist for all p 1 and w satisfying r n w x dx n 1 2 and r n w x dx 0 no solution can exist for all time extensions of the above result to various geometries and some other problems are indicated

large time behavior of the heat kernel in this paper we study the large time behavior of the minimal heat kernel kpm x y t of a general time independent parabolic operator lu ut p x x u which is defined on a noncompact manifold m more precisely we prove that limt e 0tkpm x y t always exists here 0 is the generalized principal eigenvalue of the operator p in m

functions with bounded variation on a class of riemannian manifolds with ricci curvature unbounded from below after establishing some new global facts like a measure theoretic structure theorem and a new invariant characterization of sobolev functions about complex valued functions with bounded variation on arbitrary noncompact riemannian manifolds we extend results of miranda the second author paronetto preunkert and of carbonaro mauceri on the heat semigroup characterization of the variation of mathsf l 1 functions to a class of riemannian manifolds with possibly unbounded from below ricci curvature

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

the critical exponent of a reaction diffusion equation on some lie groups we shall study the global existence and blow up of the following semilinear parabolic cauchy problem 1 1 hu h0u up x i aijxju tu up 0 in rn 0 u x 0 u0 x 0 in rn where aij aij x t are bounded measurable functions and the matrix a aij is uniformly elliptic xj j 1 k are smooth vector fields in rn and x j is the formal adjoint of xj basic assumptions i we assume that xj satisfy the hormander s condition for hypoellipticity i e the rank of the lie algebra generated by x1 xp equals n given two points x y rn d x y will be the x1 xp control distance x denotes the euclidean distance d x d x 0 b x r will represent the metric ball y rn d x y 6 r properties of the distance metric balls and the fundamental solutions of h0 have been studied extensively in nsw fes ks1 2 and sa ii there are positive constants q b c and b such that 1 2 b x r br lim x d x lim d x x

large time behavior of reaction diffusion equations with bessel generators abstract we investigate explosion in finite time of one dimensional semilinear equations of the form u t t x 1 2 2 u t x 2 x x x u t x x a x 2 u t x u t 1 x with initial value 0 where c 2 r is positive and a 0 0 are constants in the free case a 0 we provide conditions on under which any positive nontrivial solution is non global in the case a 0 and x x 1 2 r which includes in the special case 1 2 the equation u t t x 1 2 2 u t x 2 x a x 2 u t x u t 1 x we use the feynman kac formula for bessel processes to give conditions on the equation parameters ensuring finite time blowup and existence of nontrivial positive global solutions

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

a new approach to pointwise heat kernel upper bounds on doubling metric measure spaces abstract on doubling metric measure spaces endowed with a strongly local regular dirichlet form we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale invariant inequalities that correspond respectively to the nash inequality and to a gagliardo nirenberg type inequality when the volume growth is polynomial this yields a new proof and a generalisation of the well known equivalence between classical heat kernel upper bounds and relative faber krahn inequalities or localised sobolev or nash inequalities we are able to treat more general pointwise estimates where the heat kernel rate of decay is not necessarily governed by the volume growth a crucial role is played by the finite propagation speed property for the associated wave equation and our main result holds for an abstract semigroup of operators satisfying the davies gaffney estimates

on the critical exponent for reaction diffusion equations in this paper we study the initial boundary value problem for u t u h t u p with homogeneous dirichlet boundary conditions where h t t q for large t let s sup s positive solutions w of u t u such that mathop lim sup limits t to infty t s parallel w cdot t parallel infty infty then for p 1 q 1 s we show if p p there are global positive solutions that decay to zero uniformly for t if 1 p p then all nontrivial solutions blow up in finite time we determine p for some conical domains in r 2 and r 3 a similar result is derived for a bounded domain if h t e t for large t

boundedness and blow up for a semilinear reaction diffusion system abstract we consider the semilinear parabolic system s u t u p t u q where x rnn 1 t 0 and p q are positive real numbers at t 0 nonnegative continuous and bounded initial values u0 v0 x are prescribed the corresponding cauchy problem then has a nonnegative classical and bounded solution u t x v t x in some strip st 0 t r n 0 t sup t 0 u v remain bounded in s t we show in this paper that if 0 t so that solutions can be continued for all positive times when pq 1 and 1 pq 1 n 2 with max p q one has t for every nontrivial solution u v t is then called the blow up time of the solution under consideration finally if 1 pq 1 n 2 both situations coexist since some nontrivial solutions remain bounded in any strip s while others exhibit finite blow up times

on critical phenomena in interacting growth systems part i general components which are placed in a finite or infinite space have integer numbers as possible states they interact in a discrete time in a local deterministic way in addition to which all the components states are incremented at every time step by independent identically distributed random variables we assume that the deterministic interaction function is translation invariant and monotonic and that its values are between the minimum and the maximum of its arguments theorems 1 and 2 based on propositions which we give in a separate part ii give sufficient conditions for a system to have an invariant distribution or a bounded mean other statements proved herein provide background for them by giving conditions when a system has no invariant distribution or the mean of its components states tends to infinity all our main results use one and the same geometrical criterion

on the fujita exponent for a semilinear heat equation with a potential term abstract we consider the existence and nonexistence of positive global solutions for the cauchy problem t u u v x u u p in r n 0 u x 0 x 0 in r n where p 1 and v behaves like x 2 1 o 1 with 0 as x in this paper we determine the so called fujita exponent p for this cauchy problem furthermore for the critical case p p we prove that the cauchy problem has no global positive solutions

on diagonal estimates on schrodinger semigroup kernels and reduced heat kernels we prove various estimates for the kernels of semigroups generated by schrodinger operators with magnetic field and potential of polynomial growth we also investigate the reduced heat kernels

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

nine ways to implement the binomial method for option valuation in matlab in the context of a real life application that is of interest to many students we illustrate how the choices made in translating an algorithm into a high level computer code can affect the execution time more precisely we give nine matlab programs that implement the binomial method for valuing a european put option the first program is a straightforward translation of the pseudocode in figure 10 4 of the mathematics of financial derivatives by p wilmott s howison and j dewynne cambridge university press 1995 four variants of this program are then presented that improve the efficiency by avoiding redundant computation vectorizing and accessing subarrays via matlab s colon notation we then consider reformulating the problem via a binomial coefficient expansion here a straightforward implementation is seen to be improved by vectorizing avoiding overflow and underflow and exploiting sparsity overall the fastest of the binomial method programs has an execution time that is within a factor 2 of direct evaluation of the black scholes formula one of the vectorized versions is then used as the basis for a program that values an american put option the programs show how execution times in matlab can be dramatically reduced by using high level operations on arrays rather than computing with individual components a principle that applies in many scientific computing environments the relevant files are downloadable from the world wide web

on a parabolic equation with a singular lower order term abstract we obtain the existence of the weak green s functions of parabolic equations with lower order coefficients in the so called parabolic kato class which is being proposed as a natural generalization of the kato class in the study of elliptic equations as a consequence we are able to prove the existence of solutions of some initial boundary value problems moreover based on a lower and an upper bound of the green s function we prove a harnack inequality for the non negative weak solutions

deforming the metric on complete riemannian manifolds soit m g ij x une variete de riemann a n dimensions complete non compacte de tenseur de complexe riemannien r ijkl satisfaisant r ijkl 2 k 0 sur m ou 0 0 dependant seulement de n et de k 0 telle que l equation d evolution g ij x t t 2r ij x t sur m g ij x o g ij x x m ait une solution lisse g ij x t 0 pour un temps court 0 t t u k 0 et satisfait l estimation suivante pour un entier m 0 il existe des constantes c m 0 dependant seulement de n m et k 0 telles que sup x m v m r ijkl x t 2 c m t m 0 t t u k 0

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0