swapping labeled tokens on graphs consider a puzzle consisting of n tokens on an n vertex graph where each token has a distinct starting vertex and a distinct target vertex it wants to reach and the only allowed transformation is to swap the tokens on adjacent vertices we prove that every such puzzle is solvable in o n 2 token swaps and thus focus on the problem of minimizing the number of token swaps to reach the target token placement we give a polynomial time 2 approximation algorithm for trees and using this obtain a polynomial time 2 approximation algorithm for graphs whose tree spanners can be computed in polynomial time finally we show that the problem can be solved exactly in polynomial time on complete bipartite graphs

second order differential operators with feller wentzell type boundary conditions in this paper we study a basic generation problem concerning the second order differential operator ad2dx2 bddx c in the space c 0 1 of complex continuous functions equipped with feller wentzell type boundary conditions which originates from the work of feller w feller the parabolic differential equations and the associated semi groups of transformations ann of math 2 55 1952 468 519 we prove successfully that the operator under suitable assumptions generates a strongly continuous cosine function on c 0 1 or on a subspace of c 0 1 by means of an operator matrix analysis combined with perturbation approximation and similarity techniques

approximating minimum max stretch spanning trees on unweighted graphs given a graph g and a spanning tree t of g we say that t is a tree t spanner of g if the distance between every pair of vertices in t is at most t times their distance in g the problem of finding a tree t spanner minimizing t is referred to as the minimum max stretch spanning tree mmst problem this paper concerns the mmst problem on unweighted graphs the problem is known to be np hard and the paper presents an o log n approximation algorithm for it furthermore it is established that unless mathrm p mathrm np the problem cannot be approximated additively by any o n term

optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions abstract this article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term both dissipation and the source are fully nonlinear and the growth rate of the source map may include critical exponents for sobolev s embedding h 1 l 2 besides continuity and monotonicity no growth or regularity assumptions are imposed on the damping we analyze the system in the presence of neumann type boundary conditions including the mixed cases dirichlet neumann robin the damping affects a thin layer a collar near a portion of the boundary to cope with the lack of control on the remaining section we develop a special method that accounts for propagation of the energy estimates from the dissipative region onto the entire domain the neumann system does not satisfy the lopatinski condition in higher dimensions hence the study of energy propagation in the absence of damping near the neumann segment requires special geometric considerations

random matrix theory and the zeros of ζ s we study the density of the roots of the derivative of the characteristic polynomial z u z of an n n random unitary matrix with distribution given by haar measure on the unitary group based on previous random matrix theory models of the riemann zeta function s this is expected to be an accurate description for the horizontal distribution of the zeros of s to the right of the critical line we show that as n the fraction of the roots of z u z that lie in the region 1 x n 1 z 1 tends to a limit function we derive asymptotic expressions for this function in the limits x and x 0 and compare them with numerical experiments

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 degenerate diffusion problem with dynamical boundary conditions diffusion problem with dynamical boundary conditions the purpose of this paper is to study the existence the uniqueness and the limit in l 1 omega as t to infty of solutions of general initial boundary value problems of the form partial t u delta w 0 and u in beta w in a bounded domain omega with dynamical boundary conditions of the form partial t rho w partial eta w 0

some nonlinear wave equations with acoustic boundary conditions abstract we prove the existence and uniqueness of global solutions to the mixed problem for the carrier equation u tt m u 2 dx u u t u t 0 with acoustic boundary conditions

a solution to an open problem for wave equations with generalized wentzell boundary conditions in this paper we solve an open problem put forward by a favini g r goldstein j a goldstein and s romanelli concerning the mixed problem for wave equations with generalized wentzell boundary conditions as a consequence we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized wentzell boundary conditions

existence of a solution of the wave equation with nonlinear damping and source terms abstract we study the nonlinear wave equation involving the nonlinear damping term u t u t m 1 and a source term of type u u p 1 for 1 p m we prove a global existence theorem with large initial data for 1 m p a blow up result is established for sufficiently large initial data

abstract wave equations with generalized wentzell boundary conditions we introduce a general framework which allows to verify if abstract wave equations with generalized wentzell boundary conditions are well posed i e are governed by a cosine family as an example we study wave equations for second order differential operators on c 0 1 with non local wentzell type boundary conditions moreover in appendix a we give a perturbation result for sine and cosine families

the stability for a one dimensional wave equation with nonlinear uncertainty on the boundary abstract in this work we are concerned with the boundary stabilization of a one dimensional wave equation subject to boundary nonlinear uncertainty the nonlinear uncertainty is first estimated in terms of the output and then canceled by its estimates we show that this strategy works well when the derivative of the uncertainty is bounded

global existence and nonexistence for nonlinear wave equations with damping and source terms we consider an initial boundary value problem for a nonlinear wave equation in one space dimension the nonlinearity features the damping term u m 1 u t and a source term of the form u p 1 u with m p 1 we show that whenever m p then local weak solutions are global on the other hand we prove that whenever p m and the initial energy is negative then local weak solutions cannot be global regardless of the size of the initial data

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

uniform stabilization of the damped cauchy ventcel problem with variable coefficients and dynamic boundary conditions in this paper the uniform stabilization of the cauchy ventcel problem with variable coefficients is considered and the uniform energy decay rate for the problem is established by riemannian geometry methods

an extension of the concept of gradient semigroups which is stable under perturbation abstract in this article we introduce the concept of a gradient like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup those possessing a lyapunov function see j k hale asymptotic behavior of dissipative systems math surveys monogr vol 25 amer math soc 1989 and a nonlinear semigroup possessing a gradient like attractor we prove that a perturbation of a gradient like nonlinear semigroup remains a gradient like nonlinear semigroup moreover for non autonomous dynamical systems we introduce the concept of a gradient like evolution process and prove that a non autonomous perturbation of a gradient like nonlinear semigroup is a gradient like evolution process for gradient like nonlinear semigroups and evolution processes we prove continuity characterization and pullback and forwards exponential attraction of their attractors under perturbation extending the results of a n carvalho j a langa j c robinson a suarez characterization of non autonomous attractors of a perturbed gradient system j differential equations 236 2007 570 603 on characterization and of a v babin m i vishik attractors in evolutionary equations stud math appl vol 25 north holland amsterdam 1992 on exponential attraction

global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term in this paper we consider the weak viscoelastic wave equation u t t u u t t 0 t g t s u s d s u p 2 u with dynamic boundary conditions and nonlinear delay term first we prove a local existence theorem by using the faedo galerkin approximations combined with a contraction mapping theorem secondly we show that under suitable conditions on the initial data and the relaxation function the solution exists globally in time in using the concept of stable sets finally by exploiting the perturbed lyapunov functionals we extend and improve the previous result from gerbi and said houari 2011

approximation theorems for the propagators of higher order abstract cauchy problems in this paper we present two quite general approximation theorems for the propagators of higher order in time abstract cauchy problems which extend largely the classical trotter kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions then we apply the approximation theorems to deal with the second order dynamical boundary value problems

the moore gibson thompson equation with memory in the critical case abstract we consider the following abstract version of the moore gibson thompson equation with memory t t t u t t t u t a t u t a u t 0 t g s a u t s d s 0 depending on the parameters 0 where a is strictly positive selfadjoint linear operator and g is a convex nonnegative memory kernel in the subcritical case the related energy has been shown to decay exponentially in 19 here we discuss the critical case and we prove that exponential stability occurs if and only if a is a bounded operator nonetheless the energy decays to zero when a is unbounded as well

the multidimensional wave equation with generalized acoustic boundary conditions i strong stability we study the wave equation on a domain of mathbb r d d geq 2 with dynamical boundary control applied on a part of the boundary and a dirichlet boundary condition on the remaining part we prove the asymptotic stability and nonuniform stability of the associated semigroup

optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks we consider the wave equation damped by a nonlinear boundary velocity feedback q ut first we consider the case where q has a linear growth at infinity we prove that the usual decay rate estimates proved by nakao differential integral equations 8 1995 pp 681 688 haraux and zuazua arch rational mech anal 100 1988 pp 191 206 conrad leblond and marmorat in proceedings of the fifth international federation of automatic control symposium on control of distributed parameter systems perpignan 1989 pp 101 116 zuazua siam j control optim 28 1990 pp 466 477 and komornik in control and estimation of distributed parameter systems nonlinear phenomena internat ser numer math 118 birkhauser basel 1994 pp 253 266 when q has a polynomial behavior at zero and by the second author esaim control optim calc var 4 1999 pp 419 444 in the general case are in fact optimal in one space dimension more generally we prove that the energy decays exactly like the solution of an explicit and simple ordinary differential equation next we study the problem when q is bounded at infinity we prove that strong solutions decay exponentially to zero and we exhibit a sequence of weak solutions for which the associated energy decays to zero at infinity as slowly as the iterated logarithms go to infinity at infinity

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

on the zeros of zeta s near the critical line let rho prime beta prime i gamma prime denote the zeros of zeta prime s s sigma it it is shown that there is a positive proportion of the zeros of zeta prime s in 0 t t satisfying beta prime 1 2 ll log t 1 further results relying on the riemann hypothesis and conjectures on the gaps between the zeros of zeta s are also obtained

the lack of exponential stability in certain transmission problems with localized kelvin voigt dissipation in this paper we consider the transmission problem of a material composed of three components one of them is a kelvin voigt viscoelastic material the second is an elastic material no dissipation and the third is an elastic material inserted with a frictional damping mechanism the main result of this paper is that the rate of decay will depend on the position of each component when the viscoelastic component is not in the middle of the material then there exists exponential stability of the solution instead when the viscoelastic part is in the middle of the material there is not exponential stability in this case we show that the corresponding solution decays polynomially as 1 t 2 moreover we show that the rate of decay is optimal over the domain of the infinitesimal generator finally using a second order scheme that ensures the decay of energy newmark beta method we give some numerical examples which demonstrate this asymptotic behavior

weighted l2 estimates for dissipative wave equations with variable coefficients abstract we establish weighted l 2 estimates for the wave equation with variable damping u t t u a u t 0 in r n where a x a 0 1 x with a 0 0 and 0 1 in particular we show that the energy of solutions decays at a polynomial rate t n 2 1 if a x a 0 x for large x we derive these results by strengthening significantly the multiplier method this approach can be adapted to other hyperbolic equations with damping

on a navier stokes allen cahn model with inertial effects abstract a mathematical model describing the flow of two phase fluids in a bounded container is considered under the assumption that the phase transition is influenced by inertial effects the model couples a variant of the navier stokes system for the velocity u with an allen cahn type equation for the order parameter relaxed in time in order to introduce inertia at the microscopic level the resulting model is characterized by second order material derivatives which constitute the main difficulty in the mathematical analysis in order to obtain a tractable problem a viscous relaxation term is then included in the phase equation the mathematical results consist in existence of weak solutions in 3d and under additional assumptions existence and uniqueness of strong solutions in 2d a partial characterization of the long time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed

porous elastic system with nonlinear damping and sources terms abstract we study the long time behavior of porous elastic system focusing on the interplay between nonlinear damping and source terms the sources may represent restoring forces but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest by employing nonlinear semigroups and the theory of monotone operators we obtain several results on the existence of local and global weak solutions and uniqueness of weak solutions moreover we prove that such unique solutions depend continuously on the initial data under some restrictions on the parameters we also prove that every weak solution to our system blows up in finite time provided the initial energy is negative and the sources are more dominant than the damping in the system additional results are obtained via careful analysis involving the nehari manifold specifically we prove the existence of a unique global weak solution with initial data coming from the good part of the potential well for such a global solution we prove that the total energy of the system decays exponentially or algebraically depending on the behavior of the dissipation in the system near the origin we also prove the existence of a global attractor

stabilisation and polynomial decay estimate for distributed semilinear systems in this article we propose a family of feedback controls that guarantee weak and strong stabilisability for distributed semilinear systems in the case of strong stabilisation an optimal decay of the state is estimated applications to wave equations are provided

matrix methods for wave equations in analogy to a characterisation of operator matrices generating c0 semigroups due to r nagel 13 we give conditions on its entries in order that a 2 2 operator matrix generates a cosine operator function we apply this to systems of wave equations to second order initial boundary value problems and to overdamped wave equations

well posedness and stability of a hinged plate equation with a localized nonlinear structural damping abstract we consider an n dimensional plate equation in a bounded domain with a locally distributed nonlinear dissipation involving the laplacian the dissipation is effective in a neighborhood of a suitable portion of the boundary when the space dimension equals two the associated linear equation corresponds to the plate equation with a localized viscoelastic or structural damping first we prove existence uniqueness and smoothness results then using an appropriate perturbed energy coupled with multiplier technique we directly prove exponential and polynomial decay estimates for the underlying energy to the author s best knowledge the perturbed energy approach is new in the framework of stabilization of second order evolution equations with locally distributed damping

general decay rate estimates and numerical analysis for a transmission problem with locally distributed nonlinear damping in this paper we obtain very general decay rate estimates associated to a wavewave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in lasiecka and tataru 1993 and we shall present explicit decay rate estimates as considered in alabau boussouira 2005 and cavalcanti etal 2007 in addition we implement a precise and efficient code to study the behavior of the transmission problem when k1k2 and when one has a nonlinear frictional dissipation g ut more precisely we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper

alternative solution of the inhomogeneous linear differential equation of order two abstract in this paper we present an alternative method for solving the general inhomogeneous linear ordinary differential equation ode of order two the solution appears in the standard form as the sum of the solution of the equivalent homogeneous problem and the particular solution of the inhomogeneous problem at hand the main advantage of the method exposed herein is that the particular solution is computable from two different integrals this allows the problem solver to choose the simplest integral with which to work with in order to get the final solution for illustrative purposes we employ the method presented to aid in the solution of some example problems including the inhomogeneous klein gordon equation

non autonomous perturbation of autonomous semilinear differential equations continuity of local stable and unstable manifolds abstract in this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a banach space the situation considered is such that the perturbed dynamical system is non autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium then we prove that the local unstable and stable manifolds associated to them are given as graphs roughness of dichotomy plays a fundamental role here moreover we prove the continuity of this local unstable and stable manifolds with respect to the perturbation with that result we conclude the lower semicontinuity of pullback attractors

moore gibson thompson equation with memory part ii general decay of energy abstract we study a temporally third order moore gibson thompson equation with a memory term previously it was known that in non critical regime the global solutions exist and the energy functionals decay to zero more precisely it is known that the energy has exponential decay if the memory kernel decays exponentially the current work is a generalization of the previous one part i in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does exponentially or not

gaps between zeros of ζ s and the distribution of zeros of ζ s abstract we assume the riemann hypothesis in this paper we settle a conjecture of farmer and ki in a stronger form roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta function s if and only if there is a positive proportion of zeros of s lying very closely to the half line our work has applications to the siegel zero problem we provide a criterion for the non existence of the siegel zero solely in terms of the distribution of the zeros of s finally on the riemann hypothesis and the pair correlation conjecture we obtain near optimal bounds for the number of zeros of s lying very closely to the half line such bounds are relevant to a deeper understanding of levinson s method allowing us to place one third of the zeros of the riemann zeta function on the half line

existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term the wave equation with a source term is considered utt u u uin 0 we prove the existence and uniform decay rates of the energy by assuming a nonlinear feedback ut acting on the boundary provided that has necessarily not a polynomial growth near the origin to obtain the existence of global solutions we make use of the potential well method combined with the faedo galerkin procedure and constructing a special basis furthermore we prove that the energy of the system decays uniformly to zero and we obtain an explicit decay rate estimate adapting the ideas of lasiecka and tataru differential integral equations 6 3 1993 507 and patrick martinez esain control optimisation calc var 4 1999 419 the resulting problem generalizes martinez results and complements the works of lasiecka and tataru 1993 and vitillaro glasgow math j 44 2002 375

on a mixed problem with a nonlinear acoustic boundary condition for a non locally reacting boundaries abstract we discuss global solvability to the mixed problem for a nonlinear wave equation in domains with non locally reacting boundaries and nonlinear acoustic boundary condition we prove the existence and uniqueness of solution additionally using the method of nakao we demonstrate the decay of the associated energy stability

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

decay of energy for second order boundary hemivariational inequalities with coercive damping abstract in this article we consider the asymptotic behavior of solutions to second order evolution inclusions with the boundary multivalued term u t a t u t b u t j t u t 0 and u t a t u t b u t j t u t 0 where a is a possibly nonlinear coercive and pseudomonotone operator b is linear continuous symmetric and coercive is the trace operator and j is a locally lipschitz integral functional with denoting the clarke generalized gradient taken with respect to the second variable for both cases we provide conditions under which the appropriately defined energy decays exponentially to zero as time tends to infinity we discuss assumptions and provide examples of multivalued laws that satisfy them

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

collective additive tree spanners of bounded tree breadth graphs with generalizations and consequences in this paper we study collective additive tree spanners for families of graphs enjoying special robertson seymour s tree decompositions and demonstrate interesting consequences of obtained results we say that a graph g admits a system of m collective additive tree r spanners resp multiplicative tree t spanners if there is a system t g of at most m spanning trees of g such that for any two vertices x y of g a spanning tree t t g exists such that d t x y d g x y r resp d t x y t d g x y when m 1 one gets the notion of additive tree r spanner resp multiplicative tree t spanner it is known that if a graph g has a multiplicative tree t spanner then g admits a robertson seymour s tree decomposition with bags of radius at most t 2 in g we use this to demonstrate that there is a polynomial time algorithm that given an n vertex graph g admitting a multiplicative tree t spanner constructs a system of at most log 2n collective additive tree o tlogn spanners of g that is with a slight increase in the number of trees and in the stretch one can turn a multiplicative tree spanner into a small set of collective additive tree spanners we extend this result by showing that if a graph g admits a multiplicative t spanner with tree width k 1 then g admits a robertson seymour s tree decomposition each bag of which can be covered with at most k disks of g of radius at most t 2 each this is used to demonstrate that for every fixed k there is a polynomial time algorithm that given an n vertex graph g admitting a multiplicative t spanner with tree width k 1 constructs a system of at most k 1 log 2n collective additive tree o tlogn spanners of g

stability of parabolic problems with nonlinear wentzell boundary conditions abstract of concern is the nonlinear uniformly parabolic problem u t div a u u 0 x f x u t a u x u q lb u 0 for x r n and t 0 the last equation holds on the boundary here a a i j x i j is a real hermitian uniformly positive definite n n matrix c with 0 r r q 0 lb is the laplace beltrami operator on the boundary and a u is the conormal derivative of u with respect to a and everything is sufficiently regular the solution of this wellposed problem depends continuously on the ingredients of the problem namely a q f this is shown using semigroup methods in g m coclite a favini g r goldstein j a goldstein s romanelli continuous dependence on the boundary parameters for the wentzell laplacian semigroup forum 77 1 2008 101 108 here we prove explicit stability estimates of the solution u with respect to the coefficients a q and the initial condition f moreover we cover the singular case of a problem with q 0 which is approximated by problems with positive q

regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source abstract wave equation driven by a nonlinear dissipative source and subjected to a nonlinear damping that is localized on a small region near the boundary is considered while finite energy h 1 l 2 solutions to this problem are bounded uniformly for all times t 0 this property generally fails for higher energy norms h 2 h 1 this is due to a generation of higher energy by the source and the ultimate loss of dissipativity however the presence of the damping may turn things around by counteracting the sources also at the higher energy levels the benefits of this counteraction depend on the nonlinear characteristics of dissipation any deviation from linearity be it origin or infinity causes degradation of the damping hence of decay rates of finite energy this in turn is shown to have an adverse effect on the stability of higher energies the main aim of this paper is to provide a quantitative analysis of the interaction between nonlinearity of the damping and nonlinearity of the source we show that under some correlation between growth rates of the damping and the source the norms of topological order above the finite energy level remain globally bounded for all times

rate of decay of solutions of the wave equation with arbitrary localized nonlinear damping abstract we study the rate of decay of solutions of the wave equation with localized nonlinear damping without any growth restriction and without any assumption on the dynamics providing regular initial data the asymptotic decay rates of the energy functional are obtained by solving nonlinear ode moreover we give explicit uniform decay rates of the energy more precisely we find that the energy decays uniformly at last as fast as 1 ln t 2 2 0 when the damping has a polynomial growth or sublinear and for an exponential damping at the origin the energy decays at last as fast as 1 ln ln t e 2 2 0

exponential decay of energy of evolution equations with locally distributed damping consider an evolution equation with energy dissipation frac partial partial t y x t ay x t by x t on a bounded x domain omega where by signifies a dissipative perturbation to an otherwise energy conserving system this dissipation may be due to medium impurities viscous effects or artificially imposed dampers and stabilizers it is distributed over only part of the domain omega the question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined because of the locally distributed nature of energy dissipation the problem lacks coercivity and is not directly solvable by energy identities thus to get conditions sufficient for uniform exponential decay a different approach needs to be taken provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorbon the separated eigenmodes or clustered eigenmodes ofa the main theorems are general enough to treat the wave and bea

uniform energy decay of a variable coefficient wave equation with nonlinear acoustic boundary conditions abstract a variable coefficient wave equation with nonlinear damped acoustic boundary conditions is considered the riemannian geometry method is applied to deal with the variable coefficients under some checkable conditions on the coefficients the uniform decay of the energy is achieved without any geometrical conditions on the shape of the dissipative portion of the boundary

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

the distribution of zeros of ζ s and gaps between zeros of ζ s abstract assume the riemann hypothesis we establish a local structure theorem for zeros of the riemann zeta function s and its derivative s as an application we prove a stronger form of half of a conjecture of radziwill 18 concerning the global statistics of these zeros roughly speaking we show that on the riemann hypothesis if there occurs a small gap between consecutive zeta zeros then there is exactly one zero of s lying not only very close to the critical line but also between that pair of zeta zeros this refines a result of zhang 22 some related results are also shown for example we prove a weak form of a conjecture of soundararajan and suggest a repulsion phenomena for zeros of s

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

energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms abstract in this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g u t and a source term f u the proposed approach allows to consider in an unified way much more general classes of hyperbolic problems than addressed before in the literature these generalizations refer to both geometric and topological aspects of the problem the method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary the distinct features of the model include i neumann boundary conditions are assumed and ii no growth conditions are imposed on the damping g s it is well known that neumann boundary does not satisfy lopatinski condition and therefore the analysis of propagation of energy in the absence of the damping on the neumann part of the boundary requires special geometric considerations in addition the sole conditions assumed on g s are monotonicity continuity and g 0 0 in particular no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity the asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ode several examples illustrate the theory by exhibiting various decay rates exponential algebraic rational logarithmic etc for the energy functional an important corollary of our energy decay theorem is a stability result which shows that under certain conditions when dissipation is sublinear at infinity the solution of the system remains uniformly bounded for all time in the norms above the finite energy level even in the presence of a nonlinear source term

optimal control of fractional diffusion equation in this paper we apply the classical control theory to a fractional diffusion equation in a bounded domain the fractional time derivative is considered in a riemann liouville sense we first study the existence and the uniqueness of the solution of the fractional diffusion equation in a hilbert space then we show that the considered optimal control problem has a unique solution interpreting the euler lagrange first order optimality condition with an adjoint problem defined by means of right fractional caputo derivative we obtain an optimality system for the optimal control

frequency domain approach for the polynomial stability of a system of partially damped wave equations in this paper we study the stability of a system of wave equations which are weakly coupled and partially damped using a frequency domain approach based on the growth of the resolvent on the imaginary axis we establish the polynomial energy decay rate for smooth initial data we show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations

nonautonomous semilinear second order evolution equations with generalized wentzell boundary conditions abstract of concern are the initial boundary value problems for nonautonomous semilinear second order evolution equations with generalized wentzell boundary conditions we succeed in establishing a global wellposedness theorem in a classical sense for these problems via a specifically designed operator theoretic approach moreover we obtain sharp estimations for the evolution of the solution along the characteristic curves which enable us to derive the uniform exponential decay of the associated energies the results obtained in this paper are still new even for the autonomous case

note on intrinsic decay rates for abstract wave equations with memory in this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g h g 0 s 0 where h s is a given continuous positive increasing and convex function such that h 0 0 we shall develop an intrinsic method based on the main idea introduced by lasiecka and tataru uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation differential and integral equations 6 507 533 1993 for determining decay rates of the energy given in terms of the function h s this will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ode we shall show that the obtained result while generalizing previous results obtained in the literature is also capable of proving optimal decay rates for polynomially decaying memory kernels h s sp and for the full range of admissible parameters p 1 2 while such result has been known for certain restrictive ranges of the parameters p 1 3 2

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

landau siegel zeros and zeros of the derivative of the riemann zeta function abstract we show that if the derivative of the riemann zeta function has sufficiently many zeros close to the critical line then the zeta function has many closely spaced zeros this gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields

numerical analysis for the wave equation with locally nonlinear distributed damping in this paper we present spectral methods in order to solve wave equation subject to a locally distributed nonlinear damping thanks to the efficiency and the accuracy of spectral method we can check that discrete energy decreases to zero as time goes to infinity uniformly with respect to the mesh size when the damping is supported in a suitable subset of the domain of consideration we prove the convergence of the full fourier galerkin discretization thus we apply our schemes to illustrate the uniform discrete energy decay rates of the solution for a wide range of damping functions

stabilization of the euler bernoulli plate with variable coefficients by nonlinear internal feedback we consider the energy decay for solutions of the euler bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain the riemannian geometric method is used to deal with variable coefficient problems when the feedback region has a structure similar to that for the wave equation with constant coefficients we establish the stabilization of the system in the case of fixed boundary conditions several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity we further show that unlike for the case of constant coefficients choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a riemannian metric based on the coefficients of the system

uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure this paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed u tt delta u a x g u t 0 hbox in omega times 0 infty subject to dirichlet boundary conditions where omega subset mathbb r n n geq 2 is an unbounded open set with finite measure and unbounded smooth boundary partial omega gamma the function a x responsible for the localized effect of the dissipative mechanism is assumed to be nonnegative and essentially bounded and in addition a x geq a 0 0 hbox a e in omega where omega omega cup x in omega x r r 0 and omega is a neighborhood in omega of the closure of partial omega cap b r where b r x in omega x r

stability estimates for parabolic problems with wentzell boundary conditions abstract of concern is the uniformly parabolic problem u t div a u u 0 x f x u t a u u q lb u 0 for x r n and t 0 here a a i j x i j is a real hermitian uniformly positive definite n n matrix c with 0 q 0 and a u is the conormal derivative of u with respect to a and everything is sufficiently regular the solution of this well posed problem depends continuously on the ingredients of the problem namely a q f this is shown using semigroup methods in g m coclite a favini g r goldstein j a goldstein s romanelli continuous dependence on the boundary parameters for the wentzell laplacian semigroup forum in press more precisely if we have a sequence of such problems with solutions u n and if a n a n etc in a suitable sense then u n u the solution of the limiting problem the abstract analysis associated with operator semigroup theory gives this conclusion but no rate of convergence determining how fast the convergence of the solutions is requires detailed estimates such estimates are provided in this paper

boundary stabilization for wave equations with damping only on the nonlinear wentzell boundary abstract we are concerned with the problem of boundary stabilization for wave equations with damping only on the nonlinear wentzell boundary and the damping is also nonlinear we obtain boundary stabilization results explicit energy decay rates as well as the wellposedness for the systems our method consists in establishing a new type of non uniform integral inequality of energy and exploiting it to derive the decay rates the results are set up in all space dimensions even for the one dimensional linear system the result is new and we also indicate that the obtained decay rate is optimal in this case that is the obtained decay rate for linear strings is optimal

global well posedness and stability of semilinear mindlin timoshenko system abstract we study long term behavior of reissner mindlin timoshenko rmt plate systems focusing on the interplay between nonlinear viscous damping and source terms the sources may represent restoring forces but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest this work complements 28 which established local well posedness of this problem global well posedness when damping dominates the sources in an appropriate sense and a blow up in the complementary scenario assuming negative total initial energy the current paper develops the potential well theory for the rmt system it proves global existence for potential well solutions without restricting the source exponents derives explicit energy decay rates dependent on the order of the damping exponents and verifies a blow up result for positive total initial energy