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

quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions on considere le probleme suivant a n dimensions u t au t 0 x u x t 0 t 0 x u x t u x t t 0 x u x 0 0 x ou est un domaine de r n n 2 a est un operateur elliptique lineaire independant du temps est une derivee oblique sur et 1 0 est continument differentiable et croissante et 1

incomplete quenching of heat equations with coupled singular logarithms boundary fluxes abstract this paper deals with quenching phenomena for a heat equations with coupled singular logarithms boundary fluxes under appropriate hypotheses the non simultaneous quenching of the solution for the system is proved and the estimates of quenching rates are given then we give a natural continuation of the solution u v after the quenching time when the equations occurs non simultaneous quenching moreover we identify the heat equations verified by the continuation beyond quenching time i e the equations occurs incomplete quenching

fast rate of formation of dead core for the heat equation with strong absorption and applications to fast blow up we consider the dead core problem for the semilinear heat equation with strong absorption ut uxx up with 0 p 1 and positive boundary values we investigate the dead core rate i e the rate at which the solution reaches its first zero surprisingly we find that the dead core rate is faster than the one given by the corresponding ode this stands in sharp contrast with known results for the related extinction quenching and blow up problems moreover we find that the dead core rate is actually quite unstable the ode rate can be recovered if the absorption term is replaced by a t x up for a suitable bounded uniformly positive function a t x

critical exponents for a nonlinear diffusion system abstract this paper deals with critical exponents for a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary flux a precise analysis of the interactions among the multi nonlinearities in the system is given to determine the critical exponents which are simply described via the signs of a pair of parameters 1 2 solving the characteristic algebraic system introduced the solutions are non global for large initial data if 1 1 or 1 2 0 and global if 1 1 1 2 0 while in the critical case of 1 1 1 2 0 0 the solutions can be both global and non global depending on the geometry and the size of the domain

existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain abstract in this paper we study the long time behavior of solutions for the parabolic equation with non linear laplacian principal part in r n we prove the existence of a global l 2 r n l r n attractor when n p and the existence of a global l 2 r n l n p n p r n attractor when n p

multidimensional reaction diffusion equations with nonlinear boundary conditions we consider diffusion models in which distributed nonlinear absorption mechanisms compete with nonlinear boundary sources assuming that the nonlinearities are weak formal asymptotic approximation

a problem of l l campbell on the equivalence of the kramer and shannon sampling theorems abstract a result concerning the equivalence of the sampling theorems of kramer and whittaker shannon kotelnikov is presented in case the associated kernels are bessel functions or jacobi functions the general problem is open a generalized mehler dirichlet formula for the jacobi functions is used to give the proof in this case for the bessel functions a generalization of poisson s integral is needed it turns out that in both cases kramer s theorem gives nothing more than shannon s theorem in the sense that each function that can be sampled by kramer s theorem can also be reconstructed by the classical shannon sampling result

classification of blow up with nonlinear diffusion and localized reaction abstract we study the behaviour of nonnegative solutions of the reaction diffusion equation u t u m x x a x u p in r 0 t u x 0 u 0 x in r the model contains a porous medium diffusion term with exponent m 1 and a localized reaction a x u p where p 0 and a x 0 is a compactly supported symmetric function we investigate the existence and behaviour of the solutions of this problem in dependence of the exponents m and p we prove that the critical exponent for global existence is p 0 m 1 2 while the fujita exponent is p c m 1 if 0 p p 0 every solution is global in time if p 0 p p c all solutions blow up and if p p c both global in time solutions and blowing up solutions exist in the case of blow up we find the blow up rates the blow up sets and the blow up profiles we also show that reaction happens as in the case of reaction extended to the whole line if p m while it concentrates to a point in the form of a nonlinear flux if p m if p m the asymptotic behaviour is given by a self similar solution of the original problem

on the inversion of integral transforms associated with sturm liouville problems consider the sturm liouville boundary value problem 1 1 y q x y t2y a x b 2 2 y a cos y a sin 0 3 3 y b cos y b sin 0 where q x is continuous on a b let x t be a solution of either the initial value problem 1 and 2 or 1 and 3 in this paper we develop two techniques to invert the integral f t abf x x t dx where f x l2 a b one technique is based on the construction of some biorthogonal sequence of functions and the other is based on poisson s summation formula

non simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system abstract this paper deals with finite time quenching for the nonlinear parabolic system with coupled singular absorptions u t u v p v t v u q in 0 t subject to positive dirichlet boundary conditions where p q 0 is a bounded domain in r n with smooth boundary we obtain the sufficient conditions for global existence and finite time quenching of solutions and then determine the blow up of time derivatives and the quenching set for the quenching solutions as the main results of the paper a very clear picture is obtained for radial solutions with b r the quenching is simultaneous if p q 1 and non simultaneous if p 1 q or q 1 p if p q 1 with r 2 n then both simultaneous and non simultaneous quenching may happen depending on the initial data in determining the non simultaneous quenching criteria of the paper some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous dirichlet boundary value conditions in addition to techniques frequently used in the literature

self similar subsolutions and blow up for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the form u t u f u u we give finite time blow up results for large initial data blow up time estimates are also provided these results rely on a new method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results on reaction diffusion equations on generalized burgers equations and on the equation of chipot and weissler this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

paley wiener type theorems for a class of integral transforms abstract a characterization of weighted l 2 i spaces in terms of their images under various integral transformations is derived where i is an interval finite or infinite this characterization is then used to derive paley wiener type theorems for these spaces unlike the classical paley wiener theorem our theorems use real variable techniques and do not require analytic continuation to the complex plane the class of integral transformations considered is related to singular sturm liouville boundary value problems on a half line and on the whole line

on the existence of the compact global attractor for semilinear reaction diffusion systems on rn we show that a class of reaction diffusion systems on rngenerates an asymptotically compact semiflow on the banach space of bounded uniformly continuous functions if such a semiflow is dissipative then a unique non empty compact minimal attractor is known to exist we apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on rnthat contain appropriate weight functions we also state conditions which guarantee that the attractor has finite hausdorff dimension

gradient like parabolic semiflows on buc ℝn we prove that a class of weighted semilinear reaction diffusion equations on rn generates gradient like semiflows on the banach space of bounded uniformly continuous functions on rn if n 1 we show convergence to a single equilibrium the key for getting the result is to show the exponential decay of the stationary solutions which is obtained by means of a decay estimate of the kernel of the underlying semi group

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

effects of reactive gradient term in a multi nonlinear parabolic problem this paper deals with parabolic equation ut u u r aepu subject to nonlinear boundary flux u equ where r 1 p q a 0 there are two positive sources the gradient reaction and the boundary flux and a negative one the absorption in the model it is well known that blow up or not of solutions depends on which one dominating the model the positive or negative sources and furthermore on the absorption coefficient for the balance case of them the aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions we at first determine the critical blow up exponent and then obtain the blow up rate the blow up set as well as the spatial blow up profile for blow up solutions in the one dimensional case it turns out that the gradient term makes a substantial contribution to the formation of blow up if and only if r 2 where the critical r 2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level in addition it is observed that the gradient term with r 2 significantly affects the blow up rate also in fact the gained blow up rates themselves contain the exponent r of the gradient term moreover the blow up rate may be discontinuous with respect to parameters included in the problem due to convection as for the influence of gradient perturbations on spatial blow up profiles we only need some coefficients related to r for the profile estimates while the exponent of the profile itself is r independent this seems natural for boundary blow up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity

study on monostable and bistable reaction diffusion equations by iteration of travelling wave maps abstract in this paper based on the iterative properties of travelling wave maps we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction diffusion equations precisely for dirichlet problems of monostable reaction diffusion equations on the half line by making links between travelling wave maps and integral operators associated with the dirichlet diffusion kernel the latter is not invariant under translation we obtain some iteration properties of the dirichlet diffusion and some a priori estimates on nontrivial solutions of dirichlet problems under travelling wave transformation we then provide the asymptotic behavior of nontrivial solutions in the space time region for dirichlet problems these enable us to develop a unified method to obtain results on heterogeneous steady states travelling waves spreading speeds and asymptotic spreading behavior for dirichlet problem of monostable reaction diffusion equations on r as well as of monostable bistable reaction diffusion equations on r

self similar subsolutions and blowup for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the formut u f u u we prove the nonexistence of global solutions for large initial data we also estimate the maximal existence time to do so we use a method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results onut u up on generalized burgers equations and on other semilinear equations this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

nonlinear balance for reaction diffusion equations under nonlinear boundary conditions dissipativity and blow up the authors consider a reaction diffusion equation in a bounded smooth domain r n with nonlinear flux terms on the boundary of they derive suitable conditions on the nonlinear terms of the problem which imply its dissipativity the authors explore different situations in which there exist solutions of the problem that blow up in finite time the main tools in the proof of the results are either energy estimates or sub and super solution techniques

on the blow up of solutions of a convective reaction diffusion equation we study the blow up of positive solutions of the cauchy problem for the semilinear parabolic equation where u is a scalar function of the spatial variable x n and time t 0 a v a 0 1 p and 1 q we show that a if p 1 and 1 q p there always exist solutions which blow up in finite time b if 1 q p min 1 2 n 1 2 q n 1 or if q 1 and 1 p l 2 n then all positive solutions blow up in finite time c if q 1 and p min 1 2 n 1 2 q n 1 then global solutions exist d if q 1 and p 1 2 n then global solutions exist

asymptotic behavior and attractors for reaction diffusion equations in unbounded domains abstract in this paper we give general and flexible conditions for a reaction diffusion equation to be dissipative in an unbounded domain the functional setting is based on standard lebesgue and sobolev lebesgue spaces we show how the reaction and diffusion mechanisms have to work together to obtain the asymptotic compactness of solutions and therefore the existence of the compact attractor in particular cases our results allow us to improve some previous known results

porous medium equation with absorption and a nonlinear boundary condition where is a bounded domain with smooth boundary is the outer normal derivative m 1 p and q are positive parameters and u0 is in l problems of this form arise in mathematical models in a number of areas of science for instance in models for gas or uid ow in porous media 3 and for the spread of certain biological populations 13 in the semilinear case that is for m 1 there is an extensive literature about global existence and blow up results for this type of problems see among others 5 9 16 and the literature therein for the degenerate case that is for m 1 with a nonlinear boundary condition local existence and uniqueness of weak solutions which are limit of solutions of nondegenerate problems has been established in 1 also in 2 existence and uniqueness of global weak solutions for a similar

quenching behavior of solutions to heat equations with coupled boundary singularities abstract this paper deals with finite time quenching for one dimensional heat equations coupled via negative singular boundary sources in the more general product form of two component powers u p 1 v q 1 and u p 2 v q 2 differently from the generally considered there are two additional singular factors namely u p 1 and v q 2 for the boundary sources of u and v respectively we at first propose a criterion to distinguish simultaneous and non simultaneous quenching and then determine all possible blow up rates in the model three for simultaneous quenching and two for non simultaneous quenching under different nominations this shows real contributions of the two additional singular factors to the quenching behavior of solutions

attractors for partly dissipative reaction diffusion systems in rn in this paper we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in 2 n 2 n l l 2000 academic press

existence multiplicity shape and attractivity of heterogeneous steady states for bistable reaction diffusion equations in the plane abstract we consider a class of bistable reaction diffusion equations in the plane first we introduce a partition of the plane into infinitely many sectors and consider dirichlet problems in these sectors by establish some a priori estimates for nontrivial solutions to these sub systems we obtain the existence and attractivity of a heterogeneous steady state of the dirichlet problem in each of the sectors and prove the existence of a maximum positive steady state and describe the asymptotic behaviours of positive steady states at the infinities we also estimate limit sets at the vicinities of the boundaries of the sectors near origin and at infinities further assuming the sub linearity for the reaction term we obtain the uniqueness and attractivity of a heterogeneous steady state by applying the dynamical and sliding methods these results help us describe the multiplicity shape and attractivity of the heterogeneous steady states for the equation

quenching for a reaction diffusion system with logarithmic singularity abstract in this paper we study the quenching phenomenon for a reaction diffusion system with singular logarithmic source terms and positive dirichlet boundary conditions some sufficient conditions for quenching of the solutions in finite time are obtained and the blow up of time derivatives at the quenching point is verified furthermore under appropriate hypotheses the non simultaneous quenching of the system is proved and the estimates of quenching rate is given

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

the continuous α β jacobi transform and its inverse when α β 1 is a positive integer definition de la transformation de jacobi continue determination de la transformation inverse application a une variante du theoreme d echantillonnage