strong instability of standing waves for a fourth order nonlinear schrodinger equation with the mixed dispersions abstract in this paper we consider the strong instability of standing waves for a fourth order nonlinear schrodinger equation with the mixed dispersions i t 2 p 0 t x 0 t r n where 0 and 0 this equation arises in describing the propagation of intense laser beams in a bulk medium with kerr nonlinearity we firstly obtain the variational characterization of ground state solutions by using the profile decomposition theory in h 2 then we deduce that if 2 s u 1 0 the ground state standing wave e i t u is strongly unstable by blow up where u x n 2 u x and s is the action this result is a complement to the result of bonheure et al 2019 where the strong instability of standing waves has been studied in the case 0

on the stabilization of a flexible beam with a tip mass we study the stability of a flexible beam that is clamped at one end and free at the other a mass is also attached to the free end of the beam to stabilize this system we apply a boundary control force at the free end of the beam we prove that the closed loop system is well posed and is exponentially stable we then analyze the spectrum of the system for a special case and prove that the spectrum determines the exponential decay rate for the considered case

on the wave equation with hyperbolic dynamical boundary conditions interior and boundary damping and supercritical sources abstract the aim of this paper is to study the problem u t t u p x u t f x u in 0 u 0 on 0 0 u t t u u q x u t g x u on 0 1 u 0 x u 0 x u t 0 x u 1 x in where is a bounded open subset of r n with c 1 boundary n 2 1 is relatively open on denotes the laplace beltrami operator on is the outward normal to and the terms p and q represent nonlinear damping terms while f and g are nonlinear perturbations in the paper we establish local and global existence uniqueness and hadamard well posedness results when source terms can be supercritical or super supercritical

a survey of public key cryptosystems we give an overview of the most important public key cryptosystems and discuss the difficult task of evaluating the merit of such systems

ground state sign changing solutions for fractional kirchhoff equations in bounded domains we study the existence of ground state sign changing solutions for the fractional kirchhoff problem under mild assumptions on the nonlinearity by using some new analytical skills and the non nehari manifold method we prove that the fractional kirchhoff problem possesses a ground state sign changing solution ub moreover we show that the energy of ub is strictly larger than twice that of the ground state solutions of nehari type finally we establish the convergence property of ub as the parameter b 0 our results generalize some results obtained by shuai j differ equations 259 1256 2015 and tang and cheng j differ equations 261 2384 2016

global existence and finite time blow up for a class of semilinear wave equations on r n abstract in this paper we consider the initial value problem for a class of semilinear wave equations with damping and source terms under certain conditions we establish the global existence of solutions and the asymptotic behavior of the solutions by introducing an appropriate lyapunov functions moreover we examine the blow up in finite time when the initial data is sufficiently large

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

a noncollision periodic solution for n body problems using variational minimization methods we prove the existence of one noncollision periodic solution for n body type problems whose potentials are pinched between two homogeneous potentials in rk k 2

decay of solutions of the wave equation with arbitrary localized nonlinear damping abstract we study the problem of decay rate for the solutions of the initial boundary value problem to the wave equation governed by localized nonlinear dissipation and without any assumption on the dynamics i e the control geometric condition is not satisfied we treat separately the autonomous and the non autonomous cases providing regular initial data without any assumption on an observation subdomain we prove that the energy decays at last as fast as the logarithm of time our result is a generalization of lebeau in a boutet de monvel v marchenko eds algebraic and geometric methods in mathematical physics kluwer academic publishers dordrecht the netherlands 1996 pp 73 result in the autonomous case and nakao adv math sci appl 7 1 1997 317 work in the non autonomous case in order to prove that result we use a new method based on the fourier bross iaglintzer fbi transform

existence and multiplicity results for perturbed kirchhoff type schrodinger systems in r3 abstract in this paper we study a class of kirchhoff type schrodinger systems with a positive parameter e by applying variational methods and the ljusternik schnirelmann category theory under some suitable conditions we obtain the existence and multiplicity results for the systems when the parameter e is sufficiently small

asymptotic stability for anisotropic kirchhoff systems we study the question of asymptotic stability as time tends to infinity of solutions of dissipative anisotropic kirchhoff systems involving the p x laplacian operator governed by time dependent nonlinear damping forces and strongly nonlinear power like variable potential energies this problem had been considered earlier for potential energies which arise from restoring forces whereas here we allow also the effect of amplifying forces global asymptotic stability can then no longer be expected and should be replaced by local stability the results are further extended to the more delicate problem involving higher order damping terms

upper semicontinuity of the pullback attractors of non autonomous damped wave equations with terms concentrating on the boundary abstract in this paper we analyze the asymptotic behavior of the pullback attractors of a non autonomous damped wave equation when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary as a parameter e goes to zero we prove a result of regularity of the pullback attractors and that this family of attractors is upper semicontinuous at e 0

divisors in residue classes constructively let r s n be integers satisfying 0 r 1 4 and let gcd r s 1 lenstra showed that the number of integer divisors of n equivalent to r mod s is upper bounded by o 1 4 2 we reexamine this problem showing how to explicitly construct all such divisors and incidentally improve this bound to o 1 4 3 2

a potential well theory for the wave equation with a nonlinear boundary condition soit d un sous ensemble connexe ouvert borne de r n a frontiere de lipschitz d soit d l union de deux sous varietes disjointes a n 1 dimensions et et leur confluence de lipschitz on considere le probleme aux valeurs limites et initiales 2 u t 2 u dans d o t u x o u x u x o t v x dans d u x t o sur o t u n f u x t sur o t

blow up at infinity of solutions for integro differential equation in this paper we consider the blow up at infinity of solutions for the integro differential equation with strong damping term as well as a time dependent nonlinear dissipative function q and a driving force term f under the homogeneous dirichlet boundary conditions some interesting applications are given in particular subcases of q and f this improves earlier results in the literatures

global bifurcation and positive solution for a class of fully nonlinear problems in this paper we study global bifurcation phenomena for the following kirchhoff type problem m u x 2 d x u f x u in u 0 on where m is a continuous function under some natural hypotheses we show that 1 a m 0 0 is a bifurcation point and there is a global continuum c emanating from 1 a m 0 0 where 1 a denotes the first eigenvalue of the above problem with f x s a x s as an application of the above result we study the existence of positive solution for this problem with asymptotically linear nonlinearity

kirchhoff systems with dynamic boundary conditions abstract we are interested in the study of the global non existence of solutions of hyperbolic nonlinear problems governed by the p kirchhoff operator under dynamic boundary conditions when p p n with p n 2 the systems involve nonlinear external forces and may be affected by a perturbation of the type u p 2 u several models already treated in the literature are covered in special subcases and concrete examples are provided for the source term f and the external nonlinear boundary damping q

existence of nontrivial solutions for kirchhoff type variational inclusion system in r n abstract in this paper we consider the following kirchhoff type variational inclusion system a b r n u 2 dx u v x u 1 f u v in r n c d r n v 2 dx v v x v 2 f u v in r n with u v h 1 r n where n 2 and f r 2 r is a locally lipschitz function and i f u v i 1 2 are the partial generalized gradients in the sense of clarke under proper growth conditions on the nonlinearity f the existence of nontrivial solutions for the above system is established our approach is variational based on the theories of nonsmooth analysis

asymptotic stability for intermittently controlled second order evolution equations motivated by several works on ordinary differential equations we are interested in the asymptotic stability of intermittently controlled partial differential equations we give a condition of asymptotic stability for second order evolution equations uniformly damped by an on off feedback this result extends to the case of partial differential equations a previous result of r a smith concerning ordinary differential equations

on the laplace equation with non linear dynamical boundary conditions the main part of the paper deals with local existence and global existence versus blow up for solutions of the laplace equation in bounded domains with a non linear dynamical boundary condition more precisely we study the problem consisting in 1 the laplace equation in 0 infty times omega 2 a homogeneous dirichlet condition 0 infty times gamma 0 3 the dynamical boundary condition frac partial u partial nu u t m 2 u t u p 2 u on 0 infty times gamma 1 4 the initial condition u 0 x u 0 x on partial omega here omega is a regular and bounded domain in mathbb r n with n ge 1 and gamma 0 and gamma 1 endow a measurable partition of partial omega moreover m 1 2 le p r where r 2 n 1 n 2 when n ge 3 r infty when n 1 2 and u 0 in h 1 2 partial omega u 0 0 on gamma 0 the final part of the paper deals with a refinement of a global non existence result by levine park and serrin which is applied to the previous problem

on conditions for asymptotic stability of dissipative infinite dimensional systems with intermittent damping abstract we study the asymptotic stability of a dissipative evolution in a hilbert space subject to intermittent damping we observe that even if the intermittence satisfies a persistent excitation condition if the hilbert space is infinite dimensional then the system needs not being asymptotically stable not even in the weak sense exponential stability is recovered under a generalized observability inequality allowing for time domains that are not intervals weak asymptotic stability is obtained under a similarly generalized unique continuation principle finally strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition evaluating their total contribution to the decay of the trajectories of the damped system our results are discussed using the example of the wave equation schrodinger s equation and for strong stability also the special case of finite dimensional systems

stability of solutions for some classes of nonlinear damped wave equations we consider two classes of semilinear wave equations with nonnegative damping which may be of type on off or integrally positive in both cases we give a sufficient condition for the asymptotic stability of the solutions in the case of integrally positive damping we show that such a condition is also necessary

on a p kirchhoff equation via fountain theorem and dual fountain theorem abstract in this paper we show the existence of infinite solutions to the kirchhoff type quasilinear elliptic equation m u p x u p d x p 1 p u x u p 2 u f x u in a smooth bounded domain r n with nonlinear boundary condition u p 2 u u p 2 on the method we used here is based on fountain theorem and dual fountain theorem

collisionless periodic solutions to some three body problems we provide sufficient conditions for the existence of periodic solutions to some three body problems periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions

global existence for the wave equation with nonlinear boundary damping and source terms abstract the paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms the typical problem studied is u tt u 0 in 0 u 0 on 0 0 u u t m 2 u t u p 2 u on 0 1 u 0 x u 0 x u t 0 x u 1 x on where r n n 1 is a regular and bounded domain 0 1 m 1 2 p r where r 2 n 1 n 2 when n 3 r when n 1 2 and the initial data are in the energy space we prove local existence of the solutions in the energy space when m r r 1 p or n 1 2 and global existence when p m or the initial data are inside the potential well associated to the stationary problem

asymptotic stability for a strongly coupled klein gordon system in an inhomogeneous medium with locally distributed damping abstract we consider a strongly coupled klein gordon system posed in an inhomogeneous medium with smooth boundary subject to a local damping distributed around a neighborhood of the boundary according to the geometric control condition we show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase space for this purpose refined microlocal analysis arguments are considered by exploiting ideas due to burq and gerard 16 by using sharp l p l q carleman estimates we prove a unique continuation property for coupled systems we also consider a strongly coupled system of wave equations with localized nonlinear damping acting only on one equation considering a compact riemannian manifold we show that the energy of the coupled system goes uniformly to zero

global nonexistence for abstract evolution equations with positive initial energy abstract in this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the form pu tt q t u t a t u f t u t j 0 wherepandqare linear self adjoint operators anda t u andf t u are respectively a linear operator inu typically of differential type and a nonlinear driving force our principal concern is the noncontinuation or blow up of solutions when the initial energy is positive but appropriately bounded

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

global existence and blow up phenomena for a nonlinear wave equation abstract in this paper we study the solution to the initial boundary value problem for a class of nonlinear wave equations with damping and source term by using the energy compensation method we prove that under some conditions on the initial value the growth orders of the nonlinear strain term the damping term and the source term the solution to the problem exists globally and blows up respectively and the estimate for the blow up time is given

kirchhoff type system with linear weak damping and logarithmic nonlinearities abstract for the nonlinear kirchhoff type wave system with logarithmic nonlinearities and weak dissipation the global well posedness of initial boundary value problem is analyzed focusing on the interplay between kirchhoff terms and logarithmic sources we investigate the kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest by employing potential well method and concavity method we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy which is used to classify initial data for global existence and finite time blow up finally via careful analysis involving the unstable invariant set under supercritical initial energy we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

energy decay of solutions to the cauchy problem for a nondissipative wave equation in this paper we consider energy decay for nondissipative wave equation in unbounded domain the usual energy is not decreasing we use an approach introduced by guesmia which leads to decay estimates known in the dissipative case when the integral inequalities method by komornik exact controllability and stabilization the multiplier method masson wiley paris 1994 cannot be applied due to the lack of dissipativity first we study the stability of a wave equation with a weak nonlinear dissipative term based on the equation u xu 2 x u t g u t h xu 0 in irn we consider the general case with a function h satisfying a smallness condition and we obtain decay of solutions under weak growth assumptions on the feedback function and without any control of the sign of the derivative of the energy related to the above equation in the second case we consider the case h u u we prove some precise decay estimates of equivalent energy

higher nonlocal problems with bounded potential abstract the aim of this paper is to study a class of nonlocal fractional laplacian equations depending on two real parameters more precisely by using an appropriate analytical context on fractional sobolev spaces due to servadei and valdinoci we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals we emphasize that the dependence of the underlying equation from one of the real parameters is not necessarily of affine type

existence of a global attractor for semilinear dissipative wave equations on rn abstract we consider the semilinear hyperbolic problem utt ut x u f u x x r n t 0 with the initial conditions u x 0 u0 x and ut x 0 u1 x in the case where n 3 and x 1 g x lies in ln 2 r n the energy space x 0 d 1 2 r n l2g r n is introduced to overcome the difficulties related with the noncompactness of operators which arise in unbounded domains we derive various estimates to show local existence of solutions and existence of a global attractor in x 0 the compactness of the embedding d 1 2 r n l2g r n is widely applied

asymptotic stability for nonlinear damped kirchhoff systems involving the fractional p laplacian operator abstract this paper is devoted to the question of global and local asymptotic stability for nonlinear damped kirchhoff systems with homogeneous dirichlet boundary conditions under fairly natural assumptions on the external force f f t x u the distributed damping q q t x u u t the perturbation term u p 2 u and the dissipative term t m u s p u t p 2 u t with 0 and in l loc 1 r 0 when the initial data are in a special region here u u 1 u n u t x represents the vectorial displacement with n 1 particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in section 5 finally the results are extended to problems where the fractional p laplacian is replaced by a more general elliptic nonlocal integro differential operator the paper extends in several directions recent theorems and covers also the so called degenerate case that is the case in which m is zero at zero

positive solution for a superlinear kirchhoff type problem with a parameter abstract this paper is devoted to a nonlinear kirchhoff type problem depending on a real function and a nonnegative parameter on a smooth bounded domain of r n n 2 we show that if the nonlinearity is subcritical and superlinear at zero and infinity then there exists a positive value such that the kirchhoff type problem has at least one positive solution for any value of the parameter less than the positive value the main tools are variational methods critical point theory and iterative techniques

existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping abstract this paper concerns with the cauchy problem for the damped nonlinear hyperbolic system u t t u u t v p t x r 1 r n v t t v v t u q t x r 1 r n u 0 x u 0 x u t 0 x u 1 x x r n v 0 x v 0 x v t 0 x v 1 x x r n where p q 1 and satisfy p q 1 we shall prove that if max 1 p p q 1 1 q p q 1 n 2 for n 1 or 3 then the solution with small initial data exists globally in time and satisfies the decay estimates u t c 1 t and v t c 1 t for some positive constants and while if max 1 p p q 1 1 q p q 1 n 2 for n 1 then every solution with initial data having positive average value does not exist globally

global nonexistence theorems for a class of evolution equations with dissipation we study abstract evolution equations with nonlinear damping terms and source terms including as a particular case a nonlinear wave equation of the type ba cl u tt delta u b u t m 2 u t c u p 2 u t x in 0 t times omega 6pt u t x 0 t x in 0 t times partial omega 6pt u 0 cdot u 0 in h 0 1 omega quad u t 0 cdot v 0 in l 2 omega es ea where 0 p 2 m 1 we prove a global nonexistence theorem for positive initial value of the energy when 1 1

blow up for semilinear wave equation with boundary damping and source terms abstract in this paper we consider the semilinear wave equation with boundary damping and source terms this work is devoted to prove a finite time blow up result under suitable condition on the initial data and positive initial energy

existence and multiplicity of solutions for a superlinear kirchhoff type equations with critical sobolev exponent in r n abstract in this article we consider a class of superlinear kirchhoff type equations with critical growth a b r n u 2 d x u u 2 1 k x u q 1 x r n u d 1 2 r n where 0 n 4 2 q 2 2 2 n n 2 a b 0 and a b 0 k satisfies some conditions by using the variational method the multiplicity of positive solutions and infinitely many pairs of solutions are obtained

geometric characterization for variational minimization solutions of the 3 body problem with fixed energy abstract in this paper for any given positive masses we prove that the constrained variational minimization solutions of the fixed energy or fixed mean potential energy problem for the 3 body type problems with homogeneous potentials in r 3 or r 2 are precisely the planar equilateral triangle circular solutions found by j lagrange in 1772

a new approach of stabilization of nondissipative distributed systems in this paper we propose a new approach to prove the nonlinear internal or boundary stabilization of certain nondissipative distributed systems the usual energy is not decreasing this approach leads to decay estimates known in the dissipative case when the integral inequalities method due to komornik exact controllability and stabilization the multiplier method masson paris john wiley chichester uk 1994 cannot be applied due to the lack of dissipativity first we study the stability of a semilinear wave equation with a nonlinear damping based on the equation u delta u h nabla u f u g u 0 we consider the general case with a function h satisfying a smallness condition and we obtain uniform decay of strong and weak solutions under weak growth assumptions on the feedback function and without any control of the sign of the derivative of the energy related with the above equation in the second part we consider the case h nabla u nabla phi cdot nabla u with phi in w 1 infty omega we prove some precise decay estimates exponential or polynomial of equivalent energy without any restriction on phi the same results will be proved in the case of boundary feedback finally we comment on some applications of our approach to certain nondissipative distributed systems some results of this paper were announced without proof in a guesmia c r acad sci paris ser i math 332 2001 pp 633 636