critical exponent for damped wave equations with nonlinear memory abstract we consider the cauchy problem in r n n 1 for a semilinear damped wave equation with nonlinear memory global existence and asymptotic behavior as t of small data solutions have been established in the case when 1 n 3 we also derive a blow up result under some positive data in any dimensional space

scaling variables and asymptotic expansions in damped wave equations abstract we study the long time behavior of small solutions to the nonlinear damped wave equationeu u a u n u u u r 0 whereeis a positive not necessarily small parameter we assume that the diffusion coefficienta converges to positive limitsa as and that the nonlinearity n u u u vanishes sufficiently fast asu 0 introducing scaling variables and using various energy estimates we compute an asymptotic expansion of the solutionu in powers of 1 2as and we show that this expansion is entirely determined up to the second order by a linear parabolic equation which depends only on the limiting valuesa in particular this implies that the small solutions of the damped wave equation behave for large like those of the parabolic equation obtained by settinge 0

large time behavior of global solutions of the semilinear damped beam equation with slowly decaying data abstract we discuss the existence of the global solution and its asymptotic profile for the cauchy problem for the nonlinear damped beam equation we show that the equation admits a unique global solution with small slowly decaying data e g 1 x 2 k 2 0 k 1 moreover we show that the solution can be approximated by the solution of the linear heat equation with suitable data the proof is based upon the estimate for the evolution operator of the linearized equation in the homogeneous sobolev space

on the critical nonlinear damped wave equation with large initial data abstract we study the nonlinear damped wave equation 0 1 u t t u t u u u x r n t 0 u 0 x u 0 x u t 0 x u 1 x x r n in the critical case of 2 n our aim is to prove the large time asymptotic formulas for solutions of the cauchy problem 0 1 without any restriction on the size of the initial data and on the spatial dimension

existence and non existence of global solutions of the cauchy problem for higher order semilinear pseudo hyperbolic equations abstract we consider the cauchy problem for a higher order pseudo hyperbolic equation using the l p l q type estimation for the corresponding linear problem the existence and non existence criteria of global solutions are found the existence and uniqueness of smooth global solutions are also investigated we also establish the behavior of solutions and their derivatives as t

inverse nodal problems it is shown that nodal sequences determine the underlying manifold up to scaling within classes of rectangles with dirichlet boundary conditions separable two dimensional tori two dimensional flat klein bottles and flat tori in two and three dimensions

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

optimal decay rate of solutions for cahn hilliard equation with inertial term in multi dimensions abstract the cauchy problem of the cahn hilliard equation with inertial term is considered based on green s function method together with energy estimates we get the global in time existence and optimal decay rate of solutions furthermore the viscous case is investigated in the last section

the number of nodal domains on quantum graphs as a stability index of graph partitions the courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of laplacian type operators in particular it holds for generic eigenfunctions of a quantum graph the theorem stipulates that after ordering the eigenvalues as a non decreasing sequence the number of nodal domains n of the n th eigenfunction satisfies n n here we provide a new interpretation for the courant nodal deficiency d n n n in the case of quantum graphs it equals the morse index at a critical point of an energy functional on a suitably defined space of graph partitions thus the nodal deficiency assumes a previously unknown and profound meaning it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction to demonstrate this connection the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail

asymptotic behaviors of solutions for viscoelastic wave equation with space time dependent damping term abstract in this paper we consider a viscoelastic wave equation with an absorbing term and space time dependent damping term based on the weighted energy method and by assuming that the kernel decaying exponentially we obtain the l 2 decay rates of the solutions more precisely we show that the decay rates are the same as those obtained in lin et al 2010 15 for the semilinear wave equation with absorption term

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

damped wave equation in the subcritical case abstract we study large time asymptotics of small solutions to the cauchy problem for the one dimensional nonlinear damped wave equation 1 v tt v t v xx v 1 0 x r t 0 v 0 x v 0 x v t 0 x v 1 x in the sub critical case 2 3 2 we assume that the initial data v 0 1 x 1 v 1 l l 1 a a 0 1 where l 1 a l 1 l 1 a a l 1 x 1 x 2 also we suppose that the mean value of initial data r v 0 x v 1 x dx 0 then there exists a positive value such that the cauchy problem 1 has a unique global solution v t x c 0 l l 1 a satisfying the following time decay estimate v t l c t 1 for large t 0 here 2 3 2

on space time means and strong global solutions of nonlinear hyperbolic equations on demontre l existence de solutions globales appartenant a l loc r 2 l 2 2 r n pour des equations non lineaires de la forme t u u b x d u f u 0 t x r r n u 0 x 0 x t u 0 x 1 x x r n

an optimal estimate for the difference of solutions of two abstract evolution equations abstractikehata and nishihara have established that the difference between any solution u of alinearly damped abstract wave equation and a certain solution v of a related abstract heatequation decays at least like t 1 logt 12 e as time tends to in nity they conjectured that thedecayisinfactliket 1 weproveherethevalidityofthisconjecturebyrelyingonthespectraltheorem for unbounded self adjoint operators we also establish the optimality of thisestimate for the wave equation in an exterior domain r 2003 elsevier science usa all rights reserved msc primary34d05 secondary35l90keywords abstract heatequation abstract waveequation diffusionphenomenon 1 introductionthe main objective of this work is to establish an estimate of the differencebetween the solution u of the abstract dissipative wave equationu 00 u 0 au 0 u 0 u 0 u 0 0 u 1 1 1 and the solution v of the abstract heat equationv 0 av 0 v 0 u 0 u 1 1 2 article in press corresponding author e mail address haraux ann jussieu fr a haraux 0022 0396 03 seefrontmatterr 2003 elsevierscience usa allrights reserved doi 10 1016 s0022 0396 03 00057 3

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

global asymptotics of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u 1 u 0 t x r r n the behavior of u as t is expected to be same as that for the corresponding heat equation t 1 0 t x r r n in the subcritical case 1 c n 1 2 n there exists a similarity solution w b t x with the form t 1 1 f x t depending on b lim x x 2 1 f x 0 our first aim is to show the decay rates u t l 2 u t l 1 u t l 2 o t 1 1 n 4 t 1 1 n 2 1 t 1 1 1 2 n 4 provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order the decay rates are sharp in the sense that they are same as those of the similarity solution the second aim is to show that the gauss kernel is the asymptotic profile in the supercritical case which has been shown in case of one dimensional space by hayashi kaikina and naumkin n hayashi e i kaikina p i naumkin asymptotics for nonlinear damped wave equations with large initial data preprint 2004 we show this assertion in two and three dimensional space to prove our results both the weighted l 2 energy method and the explicit formula of solutions will be employed the weight is an improved one originally developed in y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

the lp lq estimates of solutions to one dimensional damped wave equations and their application to the compressible flow through porous media we first obtain the lp lq estimates of solutions to the cauchy problem for one dimensional damped wave equation vtt vxx vt 0 v vt t 0 v0 v1 x x t r r corresponding to that for the parabolic equation t xx 0 t 0 v0 v1 x the estimates are shown by v t e t 2v0 t v0 t 2lp ct 121q 1p 1 v0 v1 lq t 1 etc for 1 q p to show the explicit formula of the damped wave equation will be used to apply the estimates to nonlinear problems is the second aim we will treat the system of a compressible flow through porous media the solution is expected to behave as the diffusion wave which is the solution to the porous media equation due to the darcy law when the initial data has the same constant state at a sharp lp convergence rate for p 2 has been recently obtained by nishihara proc roy soc edinburgh sect a 133a 2003 1 20 by choosing a suitably located diffusion wave we will show the l1 convergence applying

asymptotic behaviors of solutions for time dependent damped wave equations abstract in this paper we consider the cauchy problem for the wave equation with time dependent damping b t u t and absorbed semilinear term u 1 u here b t b 0 1 t with 1 1 and b 0 0 using the weighted energy method we obtain the l 1 and l 2 decay rates of the solution which coincide to those for self similar solutions to the corresponding parabolic equation when 1 f n 1 2 n

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

controls insensitizing the norm of the solution of a semilinear heat equation abstract we consider here a semilinear heat equation with partially known initial and boundary conditions the insensitizing problem consists in finding a control function such that some functional of the state is locally insensitive to the perturbations of these initial and boundary data in this paper the insensitizing control of the norm of the observation of the solution in an open subset of the domain is studied under appropriate assumptions on the nonlinearity and the observation subset it is shown that the insensitivity conditions are equivalent to a particular nonlinear exact controllability problem for parabolic equations due to the smoothing effects of this type of equation exact controllability is very hard to achieve and this is why it seems natural to introduce the idea of approximately insensitizing control and then to solve a nonlinear approximate controllability problem of a special type that is done using a linearization and fixed point method solving the linear problem leads to the proof of a nontrivial uniqueness property which is abo used to characterize a particular subset of the admissible controls this characterization is made thanks to a convex duality theorem and allows us to solve a fixed point problem and get the result for the nonlinear case various comments and conclusions are eventually given with other approximately insensitizing problems that can be solved by our methods

resolving isospectral drums by counting nodal domains several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different one important class consists of laplace beltrami operators for pairs of flat tori in r n with n 4 we propose that the spectral ambiguity can be resolved by comparing the nodal sequences the numbers of nodal domains of eigenfunctions arranged by increasing eigenvalues in the case of isospectral flat tori in four dimensions where a four parameter family of isospectral pairs is known we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count thus one can count the shape of a drum if it is designed as a flat torus in four dimensions less

decay property of solutions for damped wave equations with space time dependent damping term abstract we consider the cauchy problem for the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 using the weighted energy method we can obtain the l 2 decay rate of the solution which is almost optimal in the case c n 1 2 n combining this decay rate with the result that we got in the paper j lin k nishihara j zhai l 2 estimates of solutions for damped wave equations with space time dependent damping term j differential equations 248 2010 403 422 we believe that c n is a critical exponent note that when 0 c n coincides to the fujita exponent f n 1 2 n the new points include the estimate in the supercritical exponent and for not necessarily compactly supported data

damped wave equation with a critical nonlinearity in higher space dimensions abstract we study the cauchy problem for nonlinear damped wave equations with a critical defocusing power nonlinearity u 2 n u where n denotes the space dimension for n 1 2 3 global in time existence of small solutions was shown in 4 in this paper we generalize the results to any spatial dimension via the method of decomposition of the equation into the high and low frequency components under the assumption that the initial data are small and decay rapidly at infinity furthermore we present a sharp time decay estimate of solutions with a logarithmic correction

global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth order in this paper we focus on the cauchy problem for a nonlinear wave equation of fourth order in n dimensional space n 1 the decay structure of which is of regularity loss property based on the decay estimate of solutions to the linear problem we introduce a set of time weighted sobolev spaces by using the contraction mapping theorem we obtain the global in time existence and the optimal decay estimates of solutions to the cauchy problem under smallness assumption on the initial data

the pointwise estimates of solutions for semilinear dissipative wave equation in multi dimensions abstract in this paper we study the global in time existence and the pointwise estimates of solutions to the cauchy problem for the dissipative wave equation in multi dimensions using the fixed point theorem we obtain the global existence of the solution in addition the pointwise estimates of the solution are obtained by the method of the green function furthermore we obtain the l p 1 p convergence rate of the solution

global existence and asymptotic behavior of solutions for a semi linear wave equation abstract in this paper we investigate the initial value problem for a semi linear wave equation in n dimensional space based on the decay estimate of solutions to the corresponding linear equation we define a set of time weighted sobolev spaces under small condition on the initial value we prove the global existence and asymptotic behavior of the solution in the corresponding sobolev spaces by the contraction mapping principle

damped wave equation with a critical nonlinearity we study large time asymptotics of small solutions to the cauchy problem for nonlinear damped wave equations with a critical nonlinearity 2 t u t u u u 1 2 n 0 x r n t 0 u 0 x eu 0 x t u 0 x eu 1 x x r n where e 0 and space dimensions n 1 2 3 assume that the initial data uo h 0 n h 0 u 1 e h 1 0 n h 1 where n 2 weighted sobolev spaces are h l m e l2 m l x l 2 1 x 2 also we suppose that 2 n 0 u 0 x dx 0 where then we prove that there exists a positive e 0 such that the cauchy problem above has a unique global solution u c 0 oo h 0 satisfying the time decay property u t e g t x e t lp ce 1 2 n g 1 n 2 t n 2 1 1 p for all t 0 1 p where e 0 e 0

large time behavior and lp lq estimate of solutions of 2 dimensional nonlinear damped wave equations abstract we show the asymptotic behavior of the solution to the cauchy problem of the two dimensional damped wave equation it is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the lp lq type estimate this is a two dimensional generalization of the three dimensional result due to nishihara math z 244 2003 631 to show this we use the fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations by using the lp lq estimate we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity u u our result covers the whole super critical case 1 where the 1 is well known as the fujita exponent when n 2

existence and nonexistence of time global solutions to damped wave equation on half line on the half line r 0 infinity the initial boundary value problems with null dirichlet boundary for both the semilinear heat equation and damped wave equation are considered the critical exponent p c n k of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space d n d k r k rn k is given by p c n k 1 2 n k j appl math phys 39 1988 135 149 arch rational mech anal 109 1990 63 71 since the damped wave equation is expected to be close to the heat equation j differential equations 191 2003 445 469 math z 244 2003 631 649 the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation however there is no blow up result on the halved space for the damped wave equation in this paper the exponent pc l 1 2 is shown to be critical for the existence and nonexistence of time global solution to both the semilinear heat equation and damped wave equation on the half line r together with the derivation of the blow up time for the proof the explicit formulas of solutions are used in a similar fashion to those in li and zhou discrete continuous dynamic systems 1 1995 503 520 c 2005 elsevier ltd all rights reserved

small data global existence for the semilinear wave equation with space time dependent damping abstract in this paper we consider the critical exponent problem for the semilinear wave equation with space time dependent damping when the damping is effective it is expected that the critical exponent agrees with that of only the space dependent coefficient case we shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent moreover we do not assume that the data are compactly supported however it is still open whether there exists a blow up solution if the power of nonlinearity is smaller than the expected exponent

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

diffusion phenomenon for linear dissipative wave equations in an exterior domain abstract under the general condition of the initial data we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain in order to derive the diffusion phenomenon for dissipative wave equations the time integral method which was developed by ikehata and matsuyama sci math japon 55 2002 33 plays an effective role

nonlinear dissipative wave equations with space time dependent potential abstract we study the long time behavior of solutions for damped wave equations with absorption these equations are generally accepted as models of wave propagation in heterogeneous media with space time dependent friction a t x u t and nonlinear absorption u p 1 u ikawa 2000 17 we consider 1 p n 2 n 2 and separable a t x x t with x 1 x and t 1 t satisfying conditions a1 or a2 which are given the main results are precise decay estimates for the energy l 2 and l p 1 norms of solutions we also observe the following behavior if 0 1 1 1 and 0 1 there are three different regions for the decay of solutions depending on p if 0 and 1 1 there are only two different regions for the decay of the solutions depending on p

global existence of solutions for semilinear damped wave equations in rn with noncompactly supported initial data abstract we consider the n dimensional cauchy problem in r n for a semilinear damped wave equation with a power type nonlinearity u p for a noncompactly supported initial data which has a small energy we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 2 n p n n 2 this generalizes a previous result due to todorova yordanov j differential equations 174 2001 464 489 which dealt with a solution in the framework of compactly supported initial data

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

on the initial value problem of the semilinear beam equation with weak damping i smoothing effect abstract this paper is devoted to the study of the initial value problem for some semilinear damped beam equation we shall prove the unique global existence of a small solution for the equation and also obtain a certain smoothing effect of the solution our method is based on the expansion formula for the propagator to decompose into the dissipative part and the dispersive part to control the dispersive part without regularity loss we apply the stationary phase method developed by van der corput as a result we observe the smoothing effect of the solution which does not occur on the solution for damped wave equations

l2 estimates of solutions for damped wave equations with space time dependent damping term abstract in this paper we consider the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 based on the local existence theorem we obtain the global existence and the l 2 decay rate of the solution by using the weighted energy method the decay rate coincides with the result of nishihara k nishihara decay properties for the damped wave equation with space dependent potential and absorbed semilinear term preprint in the case of 0 and coincides with the result of nishihara and zhai k nishihara j zhai asymptotic behaviors of time dependent damped wave equations preprint in the case of 0

comment on resolving isospectral drums by counting nodal domains in gnutzmann et al 2005 j phys a math gen 38 8921 33 the authors studied the 4 parameter family of isospectral flat 4 tori t a b c d discovered by conway and sloane with a particular method of counting nodal domains they were able to distinguish these tori numerically by computing the corresponding nodal sequences relative to a few explicit tuples a b c d in this note we confirm the expectation expressed in gnutzmann et al 2005 j phys a math gen 38 8921 33 by proving analytically that their nodal count distinguishes any 4 tuple of distinct positive real numbers