proof of the symmetry of the off diagonal hadamard seeley dewitt s coefficients in c lorentzian manifolds by a local wick rotation completing the results achieved in a previous paper we prove the symmetry of hadamard seeley dewitt off diagonal coefficients in smooth d dimensional lorentzian manifolds this result is relevant because it plays a central role in physics in particular in the theory of the stress energy tensor renormalization procedure in quantum field theory in curved spacetime to this end it is shown that in any lorentzian manifold a sort of local wick rotation of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinate are appropriate no time like killing field is necessary such a local wick rotation analytically continues the lorentzian metric in a neighborhood of any point more generally in a neighborhood of a space like cauchy hypersurface into a riemannian metric the continuation locally preserves geodesically convex neighborhoods in order to make rigorous the procedure the concept of a complex pseudo riemannian not hermitian or kahlerian manifold is introduced and some features are analyzed using these tools the symmetry of hadamard seeley dewitt off diagonal coefficients is proven in lorentzian analytical manifolds by analytical continuation of the symmetric riemannian heat kernel coefficients this continuation is performed in geodesically convex neighborhoods in common with both the metrics then the symmetry is generalized to c non analytic lorentzian manifolds by approximating lorentzian c metrics by analytic metrics in common geodesically convex neighborhoods

the microlocal spectrum condition and wick polynomials of free fields on curved spacetimes quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis we discuss a condition on the wave front set for the correspondingn point distributions called microlocal spectrum condition sc on minkowski space this condition is satisfied as a consequence of the usual spectrum condition based on radzikowski s determination of the wave front set of the two point function of a free scalar field satisfying the hadamard condition in the kay and wald sense we construct in the second part of this paper all wick polynomials including the energy momentum tensor for this field as operator valued distributions on the manifold and prove that they satisfy our microlocal spectrum condition

improved conditions for single point blow up in reaction diffusion systems abstract we study positive blowing up solutions of the system u t u v p v t v u q as well as of some more general systems for any p q 1 we prove single point blow up for any radially decreasing positive and classical solution in a ball this improves on previously known results in 3 directions i no type i blow up assumption is made and it is known that this property may fail ii no equidiffusivity is assumed i e any 0 is allowed iii a large class of nonlinearities f u v g u v can be handled which need not follow a precise power behavior as a side result we also obtain lower pointwise estimates for the final blow up profiles

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

on some classes of nonstationary axially symmetric solutions to the navier stokes equations in the paper the navier stokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids the problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t the three functions are sought in the form of power series in r with coefficients depending on t and z for the unknown coefficients recurrence relations are obtained which contain three arbitrary functions the relations are examined in three particular cases in which they give analytical solutions to the navier stokes equations for any values of the coordinates t z and r

on a particular analytical solution to the 3d navier stokes equations and its peculiarity for high reynolds numbers a special class of axially symmetric nonstationary flows of incompressible viscous fluids is examined for it the 3d navier stokes equations are reduced to a nonlinear partial differential equation of the third order and a linear partial differential equation of the second order these equations are studied and their particular analytical solutions are found the obtained particular solution to the navier stokes equations could be used to describe some types of turbulent flows of viscous fluids in the case of high reynolds numbers

rate of convergence for one dimensional quasilinear parabolic problem and its applications abstract based on a comparison principle we derive an exponential rate of convergence for solutions to the initial boundary value problem for a class of quasilinear parabolic equations in one space dimension we then apply the result to some models in population dynamics and image processing

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

on eulerian and lagrangian representation of steady incompressible fluid flow for the description of fluid flow there are in principle two approaches the eulerian approach and the lagrangian approach the first one describes the flow of its velocity u u x v x ujx u x in every point x xi x2 x3 of the domain g containing the fluid the second one uses the trajectory x xi t x2 t x3 t x t x t x0 of a single particle of fluid which at initial time t 0 is located at some point x0 e g this second approach is of great importance for the numerical analysis and computation of fluid flow l 4 while the first one has often also been used in connection with theoretical questions 5 81 in the present paper we consider the stationary motion of a viscous incompressible fluid in a bounded domain g c r3 with a sufficiently smooth boundary s because for steady flow the streamlines and the trajectories of the fluid particles coincide both approaches mentioned above are correlated by the autonomous system of characteristic ordinary differential equations

a particle system with explosions law of large numbers for the density of particles and the blow up time consider a system of independent random walks in the discrete torus with creation annihilation of particles and possible explosion of the total number of particles in finite time rescaling space and rates for diffusion creation annihilation of particles we obtain a strong law of large numbers for the density of particles in the supremum norm the limiting object is a classical solution to the semilinear heat equation t u xx u f u if f u u p 1 p 3 we also obtain a law of large numbers for the explosion time

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

convergence in a quasilinear parabolic equation with neumann boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t k t u 1 t t 0 where k are smooth functions which are periodic in both t and u brunovsk et al proved in their paper brunovsk et al 1992 8 that if a time global solution u is bounded then it converges to a periodic solution we prove that if u is unbounded from above then it converges to a periodic traveling wave v x t c t in case k k t or k k u where v is a time periodic function and c 0 in addition the periodic traveling wave is unique up to space shifts or time shifts it is stable and asymptotically stable the average traveling speed c and the instantaneous speed v t c are also studied

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

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

convergence for degenerate parabolic equations abstract we prove that any bounded global solution to a degenerate parabolic problem in one spatial dimension converges to a unique stationary state

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

local causal structures hadamard states and the principle of local covariance in quantum field theory in the framework of the algebraic formulation we discuss and analyse some new features of the local structure of a real scalar quantum field theory in a strongly causal spacetime in particular we use the properties of the exponential map to set up a local version of a bulk to boundary correspondence the bulk is a suitable subset of a geodesic neighbourhood of an arbitrary but fixed point p of the underlying background while the boundary is a part of the future light cone having p as its own tip in this regime we provide a novel notion for the extended algebra of wick polynomials on the aforesaid cone and on the one hand we prove that it contains the information of the bulk counterpart via an injective homomorphism while on the other hand we associate to it a distinguished state whose pull back in the bulk is of hadamard form the main advantage of this point of view arises if one uses the universal properties of the exponential map and of the light cone in order to show that for any two given backgrounds m and m and for any two subsets of geodesic neighbourhoods of two arbitrary points it is possible to engineer the above procedure such that the boundary extended algebras are related via a restriction homomorphism this allows for the pull back of boundary states in both spacetimes and thus to set up a machinery which permits the comparison of expectation values of local field observables in m and m

a criterion for uniqueness of lagrangian trajectories for weak solutions of the 3d navier stokes equations foias guillope temam showed in 1985 that for a given weak solution u in l infty 0 t l 2 cap l 2 0 t h 1 of the three dimensional navier stokes equations on a domain one can define a trajectory mapping phi omega times 0 t rightarrow omega that gives a consistent choice of trajectory through each initial condition a in omega xi a t phi a t and that respects the volume preserving property one would expect for smooth flows the uniqueness of this mapping is guaranteed by the theory of renormalised solutions of non smooth odes due to diperna lions

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

a galerkin characteristic unified finite element method for moving thermal fronts in porous media abstract we investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non isothermal flow conditions the governing equations consist of coupling the darcy equation for the pressure to two convection diffusion reaction equations for the temperature and depth of conversion the aim is to develop a non oscillatory unified galerkin characteristic method for efficient simulation of moving fronts in porous media the method is based on combining the modified method of characteristics with a galerkin finite element discretization of the governing equations the main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure velocity temperature and concentration analysis of convergence and stability is also presented in this study and error estimates in the l 2 norm are established for the numerical solutions in addition due to the lagrangian treatment of convection terms the standard courant friedrichs lewy condition is relaxed and the time truncation errors are reduced in the diffusion reaction part we verify the method for the benchmark problem of moving fronts around an array of cylinders the numerical results obtained demonstrate the ability of the proposed method to capture the main flow features

dimension prints and the avoidance of sets for flow solutions of non autonomous ordinary differential equations we provide a criterion for a generalised flow solution of a non autonomous ordinary differential equation to avoid a subset of the phase space this improves on that established by aizenman for the autonomous case where avoidance is guaranteed if the underlying vector field is sufficiently regular and the subset has sufficiently small box counting dimension we define the r codimension print of a subset s rn 0 t which is a subset of 0 2 that encodes the dimension of s in a way that distinguishes spatial and temporal detail we prove that the subset s is avoided by a generalised flow solution with underlying vector field in lp 0 t lq rn with 1 p q if the holder conjugates q p are in the r codimension print of s

local wick polynomials and time ordered products of quantum fields in curved spacetime in order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime it is necessary to construct wick polynomials and their time ordered products for the noninteracting theory a construction of these quantities has recently been given by brunetti fredenhagen and kohler and by brunetti and fredenhagen but they did not impose any locality or covariance condition in their constructions as a consequence their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters in this paper we construct an extended wick polynomial algebra large enough to contain the wick polynomials and their time ordered products by generalizing a construction of dutsch and fredenhagen to curved spacetime we then define the notion of a local covariant quantum field and seek a definition of local wick polynomials and their time ordered products as local covariant quantum fields we introduce a new notion of the scaling behavior of a local covariant quantum field and impose scaling requirements on our local wick polynomials and their time ordered products as well as certain additional requirements such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric for a given polynomial order in powers of the field we prove that these conditions uniquely determine the local wick polynomials and their time ordered products up to a finite number of parameters these parameters correspond to the usual renormalization ambiguities occurring in minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature we also prove existence of local wick polynomials however the issue of existence of local time ordered products is deferred to a future investigation

blow up of solutions with sign changes for a semilinear diffusion equation abstract this paper is concerned with the initial boundary value problem formula with the dirichlet neumann or periodic boundary condition here 0 is a parameter and f is an odd function of u satisfying f 0 0 and some convexity condition let z u be the number of times of sign changes for u c 0 1 it is shown that there exists an increasing sequence of positive numbers k k 0 1 2 such that any solution with z u 0 k blows up in finite time if k and there exists a global solution with z u 0 k if k

symmetry of the blow up set of a porous medium type equation we study the blow up set of a porous medium type equation with source under some technical conditions we prove that if the blow up set is a bounded smooth region then it must be a ball with a certain radius this problem can be reduced to a sublinear elliptic equation coupled with an overdetermined boundary condition roughly speaking the overdetermined boundary condition forces the domain to be a ball because the nonlinear term is sublinear and then non lipschitz many difficulties arise if one wants to use the moving plane method to reach the goal in particular the hopf boundary lemma is not applicable to this problem instead we investigate various related problems in a half space and a problem in the first quadrant of the entire space and then use the symmetry results obtained for these problems to overcome the obstacles encountered 1995 john wiley sons inc

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

convergence in time periodic quasilinear parabolic equations in one space dimension abstract we consider time periodic quasilinear parabolic equations in the domain t x r 2 0 x r t t 0 where the right boundary r t of the spatial interval is a positive function which might be periodic or asymptotically periodic or a function tending to infinity or infinity we show that in the first case that is r t is a periodic function any bounded solution of the equations converges as t to a periodic one in the other three cases any positive bounded solution converges as t to a nonnegative periodic one using such a result we study the long time dynamics of the initial boundary value problem on the half line as well as the stefan free boundary problem of a general heterogeneous reaction diffusion equation also we use the convergence result to study the long time dynamics of the initial boundary value problem for a time periodic mean curvature flow equation

existence and stability of time periodic solutions in a model for spherical flames with time periodic heat losses the problem under investigation is the study of the effect of periodicity when time periodic heat losses are added in a model for spherical flames introduced by g joulin in the present context the flame either quenches in finite or infinite time or its radius converges to an upper stable time periodic solution or to a lower unstable time periodic solution

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

asymptotic behavior of zeros of solutions for parabolic equations abstract let p 1 0 and l r l 1 r change its sign finite times this paper is concerned with a cauchy problem formula define the set of zeros of a solution u by z t x r u x t 0 for t 0 in the case of 0 we show that the set z t is contained in ct ct for large t 0 with some c 0 and that this order of t is best possible when 0 we also give estimates of z t for global solutions and prove that z t k k for all t 0 t with some k 0 for each blowup solution where t is the blowup time

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

existence of local covariant time ordered products of quantum fields in curved spacetime we establish the existence of local covariant time ordered products of local wick polynomials for a free scalar field in curved spacetime our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem so our construction essentially completes the analysis of the existence uniqueness and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime as a byproduct of our analysis we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times lorentz invariant distributions plus a remainder term of arbitrarily low scaling degree

blow up of sign changing solutions for a one dimensional nonlinear diffusion equation abstract this paper is concerned with a spatially inhomogeneous semilinear diffusion equation on a bounded interval under the neumann or dirichlet boundary conditions assuming that the nonlinearity satisfies a rather general growth condition we consider the blow up and global existence of sign changing solutions it is shown that for some nonnegative integer k depending on the linearized operator at a trivial solution the solution blows up in finite time if an initial value changes its sign at most k times whereas there exist stationary solutions with more than k zeros the proof is based on the intersection number principle combined with a topological method

algebraic quantum field theory perturbation theory and the loop expansion the perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory we show that the algebra of interacting fields is additive i e fully determined by its subalgebras associated to arbitrary small subregions of minkowski space we also give an algebraic formulation of the loop expansion by introducing a projective system n of observables up to n loops where 0 is the poisson algebra of the classical field theory finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of green s functions

comments on the stress energy tensor operator in curved spacetime the technique based on a algebra of wick products of field operators in curved spacetime in the local covariant version proposed by hollands and wald is strightforwardly generalized in order to define the stress energy tensor operator in curved globally hyperbolic spacetimes in particular the locality and covariance requirement is generalized to wick products of differentiated quantum fields within the proposed formalism there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress energy tensor are assumed to be related to the interface between the quantum and classical formalism the proposed stress energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion the definition is based on the existence of convenient counterterms given by certain local wick products of differentiated fields these terms are independent from the arbitrary length scale and any quantum state and they classically vanish on solutions of the klein gordon equation considering the averaged stress energy tensor with respect to hadamard quantum states the presented definition turns out to be equivalent to an improved point splitting renormalization procedure which makes use of the nonambiguous part of the hadamard parametrix only that is determined by the local geometry and the parameters which appear in the klein gordon operator in particular no extra added by hand term g q and no arbitrary smooth part of the hadamard parametrix generated by some arbitrary smooth term 0 are involved the averaged stress energy tensor obtained by the point splitting procedure also coincides with that found by employing the local function approach whenever that technique can be implemented

conformal generally covariant quantum field theory the scalar field and its wick products in this paper we generalize the construction of generally covariant quantum theories given in bfv03 to encompass the conformal covariant case after introducing the abstract framework we discuss the massless conformally coupled klein gordon field theory showing that its quantization corresponds to a functor between two certain categories at the abstract level the ordinary fields could be thought of as natural transformations in the sense of category theory we show that the wick monomials without derivatives wick powers can be interpreted as fields in this generalized sense provided a non trivial choice of the renormalization constants is given a careful analysis shows that the transformation law of wick powers is characterized by a weight and it turns out that the sum of fields with different weights breaks the conformal covariance at this point there is a difference between the previously given picture due to the presence of a bigger group of covariance it is furthermore shown that the construction does not depend upon the scale appearing in the hadamard parametrix used to regularize the fields finally we briefly discuss some further examples of more involved fields

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

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

energy conserving lagrangian approximation for the equations of navier stokes on considere les equations de navier stokes stationnaires dans un domaine cylindrique borne 0 t ou t 0 et r 3 a une frontiere lisse on construit un quotient aux differences lagrangien conservant l energie qui approche le terme convectif non lineaire

sharp upper bound for lifespan of solutions to some critical semilinear parabolic dispersive and hyperbolic equations via a test function method abstract this paper is concerned with the blowup phenomena for initial boundary value problem 0 1 t 2 u x t u x t a x t u x t u x t p x t c 0 t u x t 0 x t c 0 t u x 0 e f x x c t u x 0 e g x x c where c is a cone like domain in r n n 2 defined as c int r r n r 0 with a connected open set in s n 1 with smooth boundary if n 1 then we only consider two cases c 0 and c r here a x is a non zero coefficient of t u which could be complex valued and space dependent is a fixed constant and e 0 is a small parameter the constants 0 1 switch the parabolicity and hyperbolicity of the problem 0 1 the result proposes an argument for sharp upper lifespan estimates of solutions to 0 1 by a test function method based on mitidieri and pokhozhaev 2001 the crucial idea is to introduce an ordinary differential inequality with a parameter as a variable

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional exterior domain abstract in this paper we consider the following initial boundary value problem with the power type nonlinearity u p with 1 p 2 in a two dimensional exterior domain 0 1 t 2 u x t u x t e i t u x t u x t p x t 0 t u x t 0 x t 0 t u x 0 e f x x t u x 0 e g x x where is given by x r 2 x 1 2 2 c and 0 1 switches the parabolicity dispersivity and hyperbolicity remark that 2 1 2 n is well known as the fujita exponent if p 2 then there exists a small global in time solution of 0 1 for some initial data small enough see ikehata 11 and if p 2 then global in time solutions cannot exist for any positive initial data see ogawa takeda 22 and lai yin 14 the result is that for given initial data f g h 0 1 l 2 satisfying f g log x l 1 with some requirement the solution blows up at finite time and moreover the upper bound for lifespan of solutions to 0 1 is given as the following double exponential type when p 2 lifespan u exp exp c e 1 the crucial idea is to use test functions which approximates the harmonic function log x satisfying dirichlet boundary condition and the technique modified from 9

micro local approach to the hadamard condition in quantum field theory on curved space time for the two point distribution of a quasi free klein gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space time we prove the equivalence of 1 the global hadamard condition 2 the property that the feynman propagator is a distinguished parametrix in the sense of duistermaat and hormander and 3 a new property referred to as the wave front set spectral condition wfssc because it is reminiscent of the spectral condition in axiomatic quantum field theory on minkowski space results in micro local analysis such as the propagation of singularities theorem and the uniqueness up toc of distinguished parametrices are employed in the proof we include a review of kay and wald s rigorous definition of the global hadamard condition and the theory of distinguished parametrices specializing to the case of the klein gordon operator on a globally hyperbolic space time as an alternative to a recent computation of the wave front set of a globally hadamard two point distribution on a globally hyperbolic curved space time given elsewhere by kohler to correct an incomplete computation in 32 we present a version of this computation that does not use a deformation argument such as that used in fulling narcowich and wald and is independent of the cauchy evolution argument of fulling sweeny and wald both of which are relied upon in kohler s proof this leads to a simple micro local proof of the preservation of hadamard form under cauchy evolution first shown by fulling sweeny and wald relying only on the propagation of singularities theorem in another paper 33 the equivalence theorem is used to prove a conjecture by kay that a locally hadamard quasi free klein gordon state on any globally hyperbolic curved space time must be globally hadamard

convergence of solutions to the equation of quasi static approximation of viscoelasticity with capillarity the long time behavior of solutions to the quasi steady approximation of the equation of viscoelasticity with capillarity is investigated we consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one dimensional we prove that any solution converges to an equilibrium point we admit some kind of initial data with infinite energy we use tools based on the assumption that the nonlinear term is real analytic

symmetry and convergence properties for non negative solutions of nonautonomous reaction diffusion problems nonautonomous parabolic equations of the form u t u f u t on a symmetric domain are considered using the moving hyperplane method it is proved that any bounded nonnegative solution symmetrises as t this is then used to show that for nonlinearities periodic in t any non negative bounded solution approaches a periodic solution

on a simple maximum principle technique applied to equations on the circle abstract for a class of general quasilinear equations on s 1 we show that by a very simple maximum principle technique as long as the solution stays finite all of its derivatives also remain finite some specific examples are given under suitable assumptions we also derive exponential decay of the derivatives of the solution

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

analytic solutions of some nonlinear diffusion equations on considere u t u u 2 x n u 0 avec des fonctions initiales non constantes est un domaine borne de r n a frontiere lisse

remarks for the axisymmetric navier stokes equations abstract we are concerned with 3 d incompressible navier stokes equations when the initial data and the domain are cylindrically symmetric we show that there exists a solution in a weighted space and certain weighted norms of vorticity of the solution remain finite if they are finite initially consequently we can estimate the growth rate of the solution both spatially and temporally

microlocal analysis and interacting quantum field theories renormalization on physical backgrounds we present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic curved space times we develop a purely local version of the stuckelberg bogoliubov epstein glaser method of renormalization by using techniques from microlocal analysis relying on recent results of radzikowski kohler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space times we construct time ordered operator valued products of wick polynomials of free fields they serve as building blocks for a local perturbative definition of interacting fields renormalization in this framework amounts to extensions of expectation values of time ordered products to all points of space time the extensions are classified according to a microlocal generalization of steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes as a result we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space times finite renormalizations are deferred to a subsequent paper as byproducts we describe a perturbative construction of local algebras of observables present a new definition of wick polynomials as operator valued distributions on a natural domain and we find a general method for the extension of distributions which were defined on the complement of some surface

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

integral inequalities and global solutions of semilinear evolution equations a criterion for the nonexplosion of solutions to semilinear evolution equations on banach spaces is proved the result is obtained by applying a modification of the bihari type inequality to the case of a weakly singular nonlinear integral inequality

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

on the quenching behavior of the solution of a semilinear parabolic equation where 0 e 0 0 6 u0 0 there exists an e b 0 such that u quenches at finite time t if e e b we say that a is a quenching point for u if there exists a sequence x t with x a and t r t such that u x t 1 as n co throughout this paper we assume that u quenches at finite time t and that u0 is smooth and satisfies

static large deviations for a reaction diffusion model we consider the superposition of a symmetric simple exclusion dynamics speeded up in time with a spin flip dynamics in a one dimensional interval with periodic boundary conditions we prove the large deviations principle for the empirical measure under the stationary state we deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable