the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

isometry invariant solutions to a critical problem on non compact riemannian manifolds abstract we analyse an elliptic equation with critical growth set on a d dimensional d 3 hadamard manifold m g by adopting a variational perspective we prove the existence of non zero non negative solutions invariant under the action of a specific family of isometries our result remains valid when the original nonlinearity is singularly perturbed preserving the same variational approach but considering other groups of isometries we finally show that when m r d d 3 and the nonlinearity is odd there exist at least 1 d d 3 2 pairs of sign changing solutions

the ricci flow of asymptotically hyperbolic mass and applications we consider the evolution of the asymptotically hyperbolic mass under the curvature normalized ricci flow of asymptotically hyperbolic conformally compactifiable manifolds in contrast to asymptotically flat manifolds for which arnowitt deser misner adm mass is constant during ricci flow we show that the mass of an asymptotically hyperbolic manifold of dimension n 3 decays smoothly to zero exponentially in the flow time from this we obtain a no breathers theorem and a ricci flow based modified proof of the scalar curvature rigidity of zero mass asymptotically hyperbolic manifolds we argue that the nonconstant time evolution of the asymptotically hyperbolic mass is natural in light of a conjecture of horowitz and myers and is a test of that conjecture finally we use a simple parabolic scaling argument to produce a heuristic derivation of the constancy of adm mass under asymptotically flat ricci flow starting from our decay formula for the asymptotically hyperbolic mass under the curvature normalized flow

attractors for reaction diffusion equations in unbounded domains abstract in this paper we study the asymptotic behaviour of solutions for parabolic non linear evolution equations in r n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 r n

continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities abstract in this paper we discuss continuation properties and asymptotic behavior of regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions a necessary and sufficient condition for global in time existence of regular solutions is given we also formulate sufficient conditions to construct a piecewise regular solutions continuation beyond maximal time of existence for regular solutions applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed in particular global solvability and the existence of a global attractor for u t t d 1 2 u t d u f u in h 0 1 l 2 is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition similar result is obtained for a 2 m th order semilinear parabolic initial boundary value problem in a hilbert space h 2 b j m

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

nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature abstract let m g be a complete simply connected riemannian manifold with nonpositive curvature k its scalar curvature and k a smooth function on m we obtain a nonexistence result of complete metrics on m conformal to g and with k as their scalar curvature

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

dissipative mechanism of a semilinear higher order parabolic equation in r n it is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems in this paper we investigate the dissipative mechanism for some semilinear fourth order parabolic equations in the spaces of bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor while for second order reaction diffusion equations the dissipativeness mechanism has already been satisfactorily understood see arrieta et al 2004 doi 10 1142 s0218202504003234 for higher order problems in unbounded domains it has not yet been fully developed as shown throughout the paper one of the main differences from the case of reaction diffusion equations stems from the lack of a maximum principle thus we have to rely here on suitable energy estimates for the solutions as in the case of second order reaction diffusion equations we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness thus the dissipative mechanisms in second order and fourth order equations are similar although the lack of a maximum principle makes the proofs more difficult and the results not as complete finally we make essential use of the sharp results of cholewa and rodriguez bernal 2012 doi 10 1016 j na 2011 08 022 on linear fourth order equations with a very large class of linear potentials

on the elliptic equation δ 0 on complete riemannian manifolds and their geometric applications we study the elliptic equation u ku kup 0 on complete noncompact riemannian manifolds with k nonnegative three fundamental theorems for this equation are proved in this paper complete analyses of this equation on the euclidean space rn and the hyperbolic space hn are carried out when k is a constant its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper

on the existence of nontrivial extremal metrics on complete noncompact surfaces in this paper we consider the 2 dimensional local calabi flow on a complete noncompact surface sigma g 0 then based on the harnack type estimate we show the long time existence and asymptotic convergence of a subsequence of solutions of such a flow on sigma g 0 with r 0 leq 0 and r 0 bounded from above by a negative constant on a ball for its applications this will lead to the existence of extremal metrics on a complete noncompact surface of finite topological type in particular there exists an extremal metric of nonconstant gaussian curvature on sigma mathbf h 2 or mathbf r 2

attractors of partial differential evolution equations in an unbounded domain there is a large number of papers in which attractors of parabolic reaction diffusion equations in bounded domains are investigated in this paper these equations are considered in the whole unbounded space and a theory of attractors of such equations is built while investigating these equations specific difficulties arise connected with the noncompactness of operators with the continuity of their spectra etc therefore some new conditions on nonlinear terms arise in this paper weighted spaces are widely applied an important feature of this problem is worth mentioning namely properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ

on the cahn hilliard equation in h 1 r n in this paper we exhibit the dissipative mechanism of the cahn hilliard equation in h 1 r n we show a weak form of dissipativity by showing that each individual solution is attracted in some sense by the set of equilibria we also indicate that strong dissipativity that is asymptotic compactness in h 1 r n cannot be in general expected then we consider two types of perturbations a nonlinear perturbation and a small linear perturbation in both cases we show that for the resulting equations the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor

global robust adaptive path following of underactuated ships we propose a method for designing a global robust adaptive controller that forces an underactuated ship to follow a reference path under both constant and time varying disturbances induced by waves wind and ocean currents both linear and nonlinear damping terms are included to cover both low and high speed applications all nonlinear damping coefficients are assumed unknown but lie in a known compact set the new results are derived using a choice of an appropriate body fixed frame origin a smooth approximation of nonsmooth damping terms several nonlinear coordinate changes the backstepping technique and utilization of the ship dynamic structure experiments on a model ship illustrate the results

linear and semilinear higher order parabolic equations in rn abstract in this paper we consider some fourth order linear and semilinear equations in r n and make a detailed study of the solvability of the cauchy problem for the linear equation we consider some weakly integrable potential terms and for any 1 p prove that for a suitable family of bessel potential spaces h p r n the linear equation defines a strongly continuous analytic semigroup using this result we prove that the nonlinear problems we consider can be solved for initial data in l p r n and in h p 2 r n we also find the corresponding critical exponents that is the largest growth allowed for the nonlinear terms for these classes of initial data

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

path tracking of underactuated ships with general form of dynamics abstractthis article considers the path tracking problem for underactuated ships with general form of dynamics including uncertainties by introducing certain two polar coordinate transformations the ship s tracking kinematics and dynamics can be transformed into certain two inputs two outputs nonlinear strict feedback form to avoid possible singularity problem in the backstepping control design we introduce an asymptotic modification of orientation concept presented tracking scheme can guarantee the uniformly ultimately boundedness of closed loop system in terms of polar coordinates for the convenience of comparison with previous related works where all tracking schemes were discussed in the cartesian frame we carry out some case studies to investigate the conditions under which the proposed tracking method can guarantee the same stability and convergence properties of tracking errors in the cartesian frame numerical simulation studies are also carried out to demonstrate the effectiveness of pres

nonlinear model predictive control for constrained output path following we consider the tracking of geometric paths in output spaces of nonlinear systems subject to input and state constraints without pre specified timing requirements commonly referred to as constrained output path following problems specifically we propose a predictive control approach to constrained path following problems with and without velocity assignments we present sufficient convergence conditions based on terminal regions and end penalties furthermore we analyze the geometric nature of constrained output path following problems and thereby provide insight into the computation of suitable terminal control laws and terminal regions we draw upon an example from robotics to illustrate our findings

prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature let m n g be a complete noncompact riemannian manifold with the curvature bounded between two negative constants given a function k on m n in terms of the behaviors of k at infinite we give a fairly complete answer to when the k can be the scalar curvature function of a complete metric g 1 which is conformal to g

critical and supercritical higher order parabolic problems in rn due to the lack of the maximum principle the analysis of higher order parabolic problems in rn is still not as complete as the one of the second order reaction diffusion equations while the critical exponents and then a dissipative mechanism in the subcritical case have already been satisfactorily described see cholewa and rodriguez bernal 2012 for problems in the critical or supercritical regime the questions concerning well or illposedness as well as possible dissipative properties of the solutions have not yet been satisfactorily answered this article is devoted to the analysis of the higher order parabolic problems in r n in the latter case focusing on the critical and supercritical regimes we give sufficient good sign conditions proving that the problem is then globally well posed in l 2 r n and even possesses a compact global attractor on the other hand for supercritically growing bad signed nonlinearities we show that the problem is ill posed

extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations abstract we consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors we then show some applications of the abstract scheme to various evolutionary problems from odes and retarded functional differential equations to parabolic and hyperbolic pdes in particular we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in r n with nonlinearities depending on the gradient of the solution we consider as well systems of reaction diffusion equations in r n and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in r n we further discuss some nonlocal and quasilinear problems as well as the fourth order cahn hilliard equation

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

semistable extremal ground states for nonlinear evolution equations in unbounded domains in this paper we show that dissipative reaction diffusion equations in unbounded domains posses extremal semistable ground states equilibria which bound asymptotically the global dynamics uniqueness of such positive ground state and their approximation by extremal equilibria in bounded domains is also studied the results are then applied to the important case of logistic equations

negative scalar curvature metrics on noncompact manifolds in this paper we prove that every noncompact smooth manifold admits a complete metric of constant negative scalar curvature

tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics a continuous time varying tracking controller is designed that globally exponentially forces the position orientation tracking error of an underactuated surface vessel to a neighborhood about zero that can be made arbitrarily small i e global uniformly ultimately boundedness guub the result is facilitated by fusing a filtered tracking error transformation with a dynamic oscillator design we also illustrate that the proposed tracking controller yields a guub result for the regulation problem

technical communique point to point navigation of underactuated ships this paper considers point to point navigation of underactuated ships where only surge force and yaw moment are available in general a ship s sway motion satisfies a passive boundedness property which is expressed in terms of a lyapunov function under this kind of consideration a certain concise nonlinear scheme is proposed to guarantee the closed loop system to be uniformly ultimately bounded uub a numerical simulation study is also performed to illustrate the effectiveness of the proposed scheme

brief backstepping for nonsmooth systems the paper presents a constructive control design for integrator backstepping in nonsmooth systems the approach is based on non smooth analysis and lyapunov stability for nonsmooth systems and is similar in spirit with the robust control designs that have appeared in literature but is applicable to a larger class of systems the backstepping controller is first applied to the case of a unicycle driven by a new discontinuous kinematic controller yielding global asymptotic convergence with bounded inputs then it is used to implement a sliding mode controller in a hybrid system simulations results not only verify the convergence properties but also reveal the ability of the new backstepping controller to suppress chattering