exponential attractors and finite dimensional reduction for non autonomous dynamical systems we suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly holder continuous with respect to the variation of the dynamical system in some natural large class moreover we extend this construction to non autonomous dynamical systems dynamical processes treating in that case the exponential attractor as a uniformly exponentially attracting finite dimensional and time dependent set in the phase space in particular this result shows that for a wide class of non autonomous equations of mathematical physics the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is we illustrate the main results of this paper on the model example of a non autonomous reaction diffusion system in a bounded domain

on a class of degenerate parabolic equations with dynamic boundary conditions we consider a quasi linear parabolic possibly degenerate equation with nonlinear dynamic boundary conditions the corresponding class of initial and boundary value problems has already been studied previously proving well posedness of weak solutions and the existence of the global attractor assuming that the nonlinearities are subcritical to a given exponent the goal of this article is to show that the previous analysis can be redone for supercritical nonlinearities by proving an additional l estimate on the solutions in particular we derive new conditions which reflect an exact balance between the internal and the boundary mechanisms involved even when both the nonlinear sources contribute in opposite directions then we show how to construct a trajectory attractor for the weak solutions of the associated parabolic system and prove that any solution belonging to the attractor is bounded which implies uniqueness finally we also prove for the semilinear reaction diffusion equation with nonlinear dynamic boundary conditions that the fractal dimension of the global attractor is of the order n 1 as diffusion 0 in any space dimension n 2 improving some recent results in gal 2012 23

elliptic 1 laplacian equations with dynamical boundary conditions abstract this paper is concerned with an evolution problem having an elliptic equation involving the 1 laplacian operator and a dynamical boundary condition we apply nonlinear semigroup theory to obtain existence and uniqueness results as well as a comparison principle our main theorem shows that the solution we found is actually a strong solution we also compare solutions with different data

uniform attractors for the closed process and applications to the reaction diffusion equation with dynamical boundary condition abstract in this paper we first introduce the concept of a closed process in a banach space and we obtain the structure of a uniform attractor of the closed process by constructing a skew product flow on the extended phase space then the properties of the kernel section of closed process are investigated moreover we prove the existence and structure of the uniform attractor for the reaction diffusion equation with a dynamical boundary condition in l p without any restriction on the growth order of the nonlinear term

on the iwasawa invariants of elliptic curves suppose that e is an elliptic curve defined over q and that p is a prime where e has good ordinary reduction under the assumption that e is modular one can define nonnegative integers e alg e anal e and anal e the algebraic iwasawa invariants alg e and e are defined in terms of the structure of the p primary subgroup sele q p of the selmer group for e over the cyclotomic zp extension q of q the definition of the analytic invariants e and anal e is in terms of the p adic l function for e constructed by mazur and swinnerton dyer msd74 we will recall these definitions below our purpose in this article is to prove in certain cases that e anal e 0 and that alg e anal e these equalities together with a deep theorem of kato imply the main conjecture for e over q in this introduction we will discuss the nature of our results and give an outline of the proofs for the case of modular elliptic curves we want to point out however that the theorems proven in the text apply to modular forms as well as to elliptic curves with multiplicative reduction at p we will not attempt to state the most general versions here if k is any algebraic extension of q then the selmer group for e over k is a certain subgroup of h gk e q tors where gk gal q k the selmer group fits into an exact sequence 0 e k q z sele k xe k

global attractors for p laplacian equation abstract the existence of a l 2 w 0 1 p l q global attractor is proved for the p laplacian equation u t div u p 2 u f u g on a bounded domain r n n 3 with dirichlet boundary condition where p 2 the nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c 1 u q k f u u c 2 u q k and f u l where q 2 is arbitrary there is no other restriction on p and q the asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method called asymptotic a priori estimate

evolution equations with dynamic boundary conditions in this paper we study boundary problems with dynamic boundary conditions that is with boundary operators containing time derivatives the equations under consideration are transformed into abstract cauchy problems x cx f and x 0 x 0 abstract theoretical results concerning the operators c are obtained by the study of a naturally arising pseudodifferential operator for existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations we then apply the theory of semigroups in banach spaces some examples of semilinear and quasilinear problems to which our results apply are given

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

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

global existence nonexistence and asymptotic behavior of solutions for the cauchy problem of semilinear heat equations abstract in this paper we study the cauchy problem of semilinear heat equations by introducing a family of potential wells we first prove the invariance of some sets and isolating solutions then we obtain a threshold result for the global existence and nonexistence of solutions finally we discuss the asymptotic behavior of the solution

dynamic theory of quasilinear parabolic equations i abstracts evolution equations on etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity we consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter as the parameter crosses some critical values there appear certain resonances in the equation producing solutions that bifurcate from infinity we study the bifurcation branches characterize when they are sub or supercritical and analyse the stability type of the solutions furthermore we apply these results and techniques to obtain landesman lazer type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti maximum principle

non autonomous reaction diffusion model with dynamic boundary conditions abstract we investigate the dynamics of a non autonomous reaction diffusion model with dynamic boundary conditions we first show that under the same assumptions the known l 2 l 2 pullback d attractor indeed can attract in l 2 l 2 norm for any 0 then we prove the continuity of the solution in h 1 h 1 2 with respect to the initial data and finally show that such attractor can also attract in h 1 h 1 2 norm under a slightly stronger integrability condition on the time dependent external forcing term the proofs are based on a new nash moser alikakos type a priori estimate about the difference of solutions near the initial time

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

extension of the eulerian lagrangian description to nonlinear perturbations in an arbitrary inviscid flow abstract a new theoretical formulation is proposed to describe the propagation of large amplitude perturbations in an arbitrary flow of inviscid fluid this formulation relies on the mixed eulerian lagrangian description for which exact nonlinear equations are derived using properties of hamiltonian formalism generalized expressions are found for the acoustic pseudo energy density and flux they verify a conservative energy balance all these expressions only depend on the reference flow variables and the displacement vector of the fluid particles due to the perturbation when the small perturbation assumption is added the linear expressions from the literature are retrieved

well posedness and long time behavior for p laplacian equation with nonlinear boundary condition abstract in this paper we study the homogeneous nonlinear boundary value problem for the p laplacian equation u t p u a x t u 2 u b x t u 2 u h x t we prove the existence of weak solutions which is global or local in time in dependence on the relation between the exponent of nonlinear part in boundary value and p boundedness of weak solution is proved we established conditions of uniqueness we prove also the properties of extinction in a finite time finite speed propagation and waiting time lastly by using the energy method we obtain sufficient conditions that the solutions of this problem with non positive initial energy blow up in finite time

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

bifurcation of positive solutions to scalar reaction diffusion equations with nonlinear boundary condition abstract the bifurcation of non trivial steady state solutions of a scalar reaction diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems the existence and stability of positive steady state solutions are proved using a unified approach the general results are applied to a laplace equation with nonlinear boundary condition and bistable nonlinearity and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions

efficient approximation of the solution of certain nonlinear reaction diffusion equations with large absorption we study the positive stationary solutions of a standard finite difference discretization of the semilinear heat equation with nonlinear neumann boundary conditions we prove that if the absorption is large enough compared with the flux in the boundary there exists a unique solution of such a discretization which approximates the unique positive stationary solution of the continuous equation furthermore we exhibit an algorithm computing an e approximation of such a solution by means of a homotopy continuation method the cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required

on the controllability of the heat equation with nonlinear boundary fourier conditions in this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form y n f y 0 and distributed controls with support in a small set we show that when the function f is globally lipschitz continuous the system is approximately controllable we also show that the system is locally null controllable and null controllable for large time when f is regular enough and f 0 0 for the proofs of these assertions we use controllability results for similar linear problems and appropriate fixed point arguments in the case of the local and large time null controllability results the arguments are rather technical since they need among other things holder estimates for the control and the state

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

on a singularly perturbed wave equation with dynamic boundary conditions in this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping we show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions conditions are given for uniform convergence in the energy space

pullback exponential attractors for the viscous cahn hilliard navier stokes system with dynamic boundary conditions abstract this paper is concerned with the existence of pullback exponential attractors for the viscous cahn hilliard navier stokes system with dynamic boundary conditions due to the lack of the holder continuity in time of the semigroup and the smoothing property of the difference of two weak solutions we construct a pullback exponential attractor in the original phase space by the l trajectories methods

global solutions for quasilinear parabolic systems we present an approach for proving the global existence of classical solutions of certain quasilinear parabolic systems with homogeneous dirichlet boundary conditions in bounded domains with a smooth boundary c 2003 elsevier inc all rights reserved

diffusion effect and stability analysis of a predator prey system described by a delayed reaction diffusion equations in this paper we consider a delayed reaction diffusion equations which describes a two species predator prey system with diffusion terms and stage structure by using the linearization method and the method of upper and lower solutions we study the local and global stability of the constant equilibria respectively the results show that the free diffusion of the delayed reaction diffusion equations has no effect on the populations when the diffusion is too slow otherwise the free diffusion has a certain influence on the populations however the influence can be eliminated by improving the parameters to satisfy some suitable conditions

semigroups and nonlinear evolution equations abstract in the main part of this paper we survey some recent results on a semigroup approach to parabolic equations under nonlinear boundary conditions in particular we give a detailed motivation and a heuristic derivation of a generalized variation of constant formula which is the basis for the whole method in the appendix we prove some new results about semigroups in interpolation and extrapolation spaces

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

long time behavior of reaction diffusion equations with dynamical boundary condition abstract in this paper we study the long time behavior of the reaction diffusion equation with dynamical boundary condition where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order some asymptotic regularity of the solution has been proved as an application of the asymptotic regularity results we can not only obtain the existence of a global attractor a in h 1 l p l q immediately but also can show further that a attracts every l 2 l 2 bounded subset with h 1 l p l q norm for any 0

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

primeness semiprimeness and localisation in iwasawa algebras necessary and sufficient conditions are given for the completed group algebras of a compact p adic analytic group with coefficient ring the p adic integers or the field of p elements to be prime semiprime and a domain necessary and sufficient conditions are found for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups some information is obtained about the krull and global dimensions of the localisations the results extend and complete work of a neumann and j coates et al

blow up rate for radially symmetric solutions of some parabolic equations with nonlinear boundary conditions abstract we consider the following parabolic equations with nonlinear boundary conditions u t u a u p 1 u in b 1 0 t u u q 1 u on b 1 0 t where p q 1 a 0 and b 1 is the unit ball we study the blow up rate of radial symmetric solutions u r t without any assumptions on the initial data furthermore we show the uniqueness of positive solutions of one dimensional backward self similar solutions as a consequence we can determine the asymptotic behavior of blow up solutions near the blow up time

coleman gurtin type equations with dynamic boundary conditions abstract we present a new formulation and generalization of the classical theory of heat conduction with or without fading memory as a special case we investigate the well posedness of systems which consist of coleman gurtin type equations subject to dynamic boundary conditions also with memory nonlinear terms are defined on the interior of the domain and on the boundary and subject to either classical dissipation assumptions or to a nonlinear balance condition in the sense of gal 2012 additionally we do not assume that the interior and the boundary share the same memory kernel

critical exponents in a parabolic system with inner absorption and coupled nonlinear boundary flux this paper deals with critical exponents for a parabolic system with inner absorption and coupled nonlinear boundary flux a detail analysis is given to the interactions among the multi nonlinearities in the system the critical exponent of the system is simply described via a characteristic matrix equation introduced

characteristic elements in noncommutative iwasawa theory let p be a prime number which for simplicity we shall always assume odd in the iwasawa theory of an elliptic curve e over a number field k one has to distinguish between curves which do or do not admit complex multiplication cm for cmelliptic curves their deep arithmetic properties and the link between their selmer group and special values of their hasse weil l functions are not only described by the one variable main conjecture corresponding to the cyclotomic zp extension kcyc of k but also by the two variable main conjecture corresponding to the extension k k ep which arises by adjoining the p power division points ep of e moreover both conjectures are proven by rubin 36 in the case that k is imaginary quadratic and e has cm by the ring of integers ok of k also for non cm elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k but for lack of both an algebraic as well as analytic p adic l function this has not been achieved the aim of this paper is to establish under certain conditions the existence of an algebraic p adic l function viz as an element of the first k group k1 t t t t of a localization t of the usual iwasawa algebra g of the galois group g g k k here for a ring r we denote by r its group of units by the weil pairing kcyc is contained in k and we put h g k kcyc and g kcyc k furthermore we write m h for the kernel of the canonical surjective ring homomorphism

attractors for nonclassical diffusion equations with dynamic boundary conditions abstract in this paper we study the asymptotic behavior of solutions for a class of nonclassical diffusion equation t u e u u u f u g in a smooth bounded domain in r n n 3 with dynamic boundary condition t u e u n e t u u u f u g on under a sobolev type growth condition of the nonlinear functions we prove the existence and uniqueness of weak solution for the problem by using the galerkin approximation method and then establish the existence and upper semicontinuity of global attractors a e at e 0 0 1 by using the asymptotic a priori estimate method

complete and energy blow up in parabolic problems with nonlinear boundary conditions we study the possible continuation of solutions of a nonlinear parabolic problem after the blow up time the nonlinearity in the equation is dissipative and blow up is caused by the nonlinear boundary condition of the form u u q 1u where q 1 is subcritical in h1 if the dissipative term in the equation is linear then we show that blow up of positive solutions is complete if the dissipative term is superlinear then the solution can be continued inside the spatial domain on the other hand we find sufficient conditions on the nonlinearities guaranteeing that no reasonable continuation can be expected on the boundary

pullback attractors for non autonomous reaction diffusion equations with dynamical boundary conditions abstract in this paper we prove the existence and uniqueness of a weak solution for a non autonomous reaction diffusion model with dynamical boundary conditions after that a continuous dependence result is established via an energy method including in particular some compactness properties finally the precedent results are used in order to ensure the existence of minimal pullback attractors in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition the relation among these families is also discussed

a mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description a nonstandard wave equation established by galbrun in 1931 is used to study sound propagation in nonuniform flows galbrun s equation describes exactly the same physical phenomenon as the linearized euler s equations lee but is derived from an eulerian lagrangian description and written only in term of the lagrangian perturbation of the displacement this equation has interesting properties and may be a good alternative to the lee only acoustic displacement is involved even in nonhomentropic cases it provides exact expressions of acoustic intensity and energy and boundary conditions are easily expressed because acoustic displacement whose normal component is continuous appears explicitly in this paper galbrun s equation is solved using a finite element method in the axisymmetric case with standard finite elements the direct displacement based variational formulation gives some corrupted results instead a mixed finite element satisfying the inf sup condition is proposed to avoid this problem a first set of results is compared with semianalytical solutions for a straight duct containing a sheared flow obtained from pridmore brown s equation a second set of results concerns a more complex duct geometry with a potential flow and is compared to results obtained from a multiple scale method which is an adaptation for the incompressible case of rienstra s recent work

approximation of the solution of certain nonlinear odes with linear complexity we study the positive stationary solutions of a standard finite difference discretization of the semilinear heat equation with nonlinear neumann boundary conditions we prove that there exists a unique solution of such a discretization which approximates the unique positive stationary solution of the continuous equation furthermore we exhibit an algorithm computing an e approximation of such a solution by means of a homotopy continuation method the cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required

global attractors in stronger norms for a class of parabolic systems with nonlinear boundary conditions abstract for a class of quasilinear parabolic systems with nonlinear robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity we further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space the approach relies on maximal l p regularity with temporal weights for the linearized problem an inherent smoothing effect due to the weights is employed for obtaining gradient estimates in several applications we can improve the convergence to an attractor by one regularity level

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

blow up with logarithmic nonlinearities we study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half line with a nonlinear boundary condition ut uxx u 1 logp u 1 x t r 0 t ux 0 t u 1 logq u 1 0 t t 0 t u x 0 u0 x x r with p q 0 we describe in terms of p q and when the solution is global in time and when it blows up in finite time for blow up solutions we find the blow up rate and the blow up set and we describe the asymptotic behaviour close to the blow up time showing that a phenomenon of asymptotic simplification takes place we finally study the appearance of extinction in finite time

a low mach number model for time harmonic acoustics in arbitrary flows this paper concerns the finite element simulation of the diffraction of a time harmonic acoustic wave in the presence of an arbitrary mean flow considering the equation for the perturbation of displacement due to galbrun we derive a low mach number formulation of the problem which is proved to be of fredholm type and is therefore well suited for discretization by classical lagrange finite elements numerical experiments are done in the case of a potential flow for which an exact approach is available and a good agreement is observed

descent calculations for the elliptic curves of conductor 11 let a be one of the three elliptic curves over q with conductor 11 we show that a has mordell weil rank zero over its field of 5division points in each case we also compute the 5 primary part of the tate shafarevich group our calculations make use of the galois equivariance of the cassels tate pairing

complete second order differential equations in banach spaces with dynamic boundary conditions abstract in this paper we deal with the mixed initial boundary value problem for complete second order in time linear differential equations in banach spaces in which time derivatives occur in the boundary conditions general wellposedness theorems are obtained for the first time which are used to solve the corresponding inhomogeneous problems examples of applications to initial boundary value problems for partial differential equations are also presented

a note on quenching for parabolic equations with dynamic boundary conditions we present a quenching result for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary

semilinear evolution equations and fractional powers of a closed pair of operators the nonlinear evolution problem bu t a t bu u f t bu with b a constant linear operator and a a t bu a time dependent nonlinear operator from one banach space to another is studied existence and uniqueness results are obtained by making use of the theory of b evolutions and the fractional powers of a and b two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions

interactions among multi nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux abstract this paper deals with a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary conditions two parameters which solve an introduced matrix equation covering all the eight nonlinear exponents of the system are used to get a simple description for the interactions among the multi nonlinearities in the system blow up or not of the solutions is determined by the signs of the two parameters in order to obtain the critical properties of the solutions for the critical sign of the parameters some further analysis on the absorption coefficients and the geometry of is given in addition to the discussion for the eight nonlinear exponents in the system

attractors for parabolic equations with dynamic boundary conditions abstract in this paper we prove existence of global solutions and l 2 l 2 h 1 l p l p global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary where there is no other restriction on p 2

newton s method and a mesh independence principle for certain semilinear boundary value problems we exhibit an algorithm which computes an approximation of the positive solutions of a family of boundary value problems with neumann boundary conditions such solutions arise as the stationary solutions of a family of semilinear parabolic equations with neumann boundary conditions the algorithm is based on a finite dimensional newton iteration associated with a suitable discretized version of the problem under consideration to determine the behavior of such a discrete iteration we establish an explicit mesh independence principle we apply a homotopy continuation algorithm to compute a starting point of the discrete newton iteration and the discrete newton iteration until an approximation of the stationary solution is obtained the algorithm performs roughly o 1 1 2 flops and function evaluations

asymptotic regularity for some dissipative equations abstract this paper is devoted to proving some asymptotic regularity for both reaction diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity which excel the sharp regularity allowed by the corresponding stationary equations equilibrium points based on this regularity the existence of the finite dimensional global and exponential attractors can be obtained easily