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

global well posedness for nonlinear wave equations with supercritical source and damping terms abstract we prove the global well posedness of weak solutions for nonlinear wave equations with supercritical source and damping terms on a three dimensional torus t 3 of the prototype u t t u u t m 1 u t u p 1 u x t t 3 r u 0 u 0 h 1 t 3 l m 1 t 3 u t 0 u 1 l 2 t 3 where 1 p min 2 3 m 5 3 m notably p is allowed to be larger than 6

global smooth solutions and stabilization of nonlinear elastodynamic systems with locally distributed dissipation we consider the existence of global solutions of the nonlinear elastodynamic system with a locally distributed damping in a bounded domain we assume that the energy function satisfies the strong ellipticity condition at the zero equilibrium the local dissipation is in the form a x u where the nonnegative function a x is only positive on a small portion of the domain we show the existence of global smooth solutions when initial data are small in particular we obtain the exponential decay of the energy which implies the exponential stabilization of the system by internal feedback

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

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

systems of nonlinear wave equations with damping and supercritical boundary and interior sources utt u g1 ut f1 u v vtt v g2 vt f2 u v in a bounded domain r with robin and dirichlet boundary conditions on u and v respectively the nonlinearities f1 u v and f2 u v are with supercritical exponents representing strong sources while g1 ut and g2 vt act as damping in addition the boundary condition also contains a nonlinear source and a damping term 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

relative efficiency of unequal cluster sizes in stepped wedge and other trial designs under longitudinal or cross sectional sampling background a cluster trial with unequal cluster sizes often has lower precision than one with equal clusters with a corresponding inflation of the design effect for parallel group trials adjustments to the design effect are available under sampling models with a single intracluster correlation design effects for equal clusters under more complex scenarios have appeared recently including stepped wedge trials under cross sectional or longitudinal sampling we investigate the impact of unequal cluster size in these more general settings results assuming a linear mixed model with an exchangeable correlation structure that incorporates cluster and subject autocorrelation we compute the relative efficiency re of a trial with clusters of unequal size under a size stratified randomization scheme as compared to an equal cluster trial with the same total number of observations if there are no within cluster time effects the re exceeds that for a parallel trial in general the re is a weighted average of the re for a parallel trial and the re for a crossover trial in the same clusters existing approximations for parallel designs are extended to the general setting increasing the cluster size by the factor 1 cv2 where cv is the coefficient of variation of cluster size leads to conservative sample sizes as in a popular method for parallel trials conclusion methods to assess experimental precision for single period parallel trials with unequal cluster sizes can be extended to stepped wedge and other complete layouts under longitudinal or cross sectional sampling in practice the loss of precision due to unequal cluster sizes is unlikely to exceed 12

global existence to the initial boundary value problem for a system of semilinear wave equations abstract we prove the global existence of finite energy and h 2 solutions for a system of nonlinear wave equations in a bounded domain one equation of the system has a dissipative term while another equation has no dissipative mechanism to derive the delicate a priori estimates we employ a loan method

mixed problem for some semi linear wave equation with a nonhomogeneous condition on considere le probleme aux valeurs limites et initiales suivant u tt u xx u t sign u t 0 x t 0 1 0 t u x 0 t g t u 1 t 0 u x 0 u 0 x u t x 0 u 1 x avec 0 1 avec u 0 h 1 0 1 u 1 dans l 2 0 1 et g t dans l 2 0 t un theoreme d existence globale et d unicite est demontre

regularity theory of hyperbolic equations with non homogeneous neumann boundary conditions ii general boundary data abstract this paper studies the regularity of solutions of general mixed second order time dependent hyperbolic problems of neumann type in a previous paper i lasiecka and r triggiani ann mat pura appl iv clvii 1990 285 367 using pseudo differential calculus we have provided sharp regularity results of the solutions and their traces when the non homogeneous data are in l 2 now we complement this study by providing a regularity when the non homogeneous data are more regular than as well as less regular than l 2 in contrast with our previous paper a functional analytic approach based on the l 2 results of our previous paper is used throughout

classification of manifolds with weakly 1 4 pinched curvatures we show that a compact riemannian manifold with weakly pointwise 1 4 pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form more generally we classify all compact locally irreducible riemannian manifolds m with the property that m r 2 has non negative isotropic curvature

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping abstract we consider the wave equation with interior and boundary nonlinear sources and damping and we are interested in local and global wellposedness of finite energy solutions the main difficulty is represented by the fact that the lopatinski condition fails to hold unless the dim 1 and thus the analysis of the boundary nonlinearities becomes a subtle issue

blow up of a hyperbolic equation of viscoelasticity with supercritical nonlinearities abstract we investigate a hyperbolic pde modeling wave propagation in viscoelastic media under the influence of a linear memory term of boltzmann type and a nonlinear damping modeling friction as well as an energy amplifying supercritical nonlinear source u t t k 0 u 0 k s u t s d s u t m 1 u t u p 1 u in 0 t u x t u 0 x t in 0 where is a bounded domain in r 3 with a dirichlet boundary condition the relaxation kernel k is monotone decreasing and k 1 we study blow up of solutions when the source is stronger than dissipations i e p max m k 0 under two different scenarios first the total energy is negative and the second the total energy is positive with sufficiently large quadratic energy this manuscript is a follow up work of the paper 30 in which hadamard well posedness of this equation has been established in the finite energy space the model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite time memory

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

a note on a strongly damped wave equation with fast growing nonlinearities a strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered the main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved the initial boundary value problem for the considered equation is globally well posed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space these results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global lyapunov function

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

statistical efficiency and optimal design for stepped cluster studies under linear mixed effects models in stepped cluster designs the intervention is introduced into some or all clusters at different times and persists until the end of the study instances include traditional parallel cluster designs and the more recent stepped wedge designs we consider the precision offered by such designs under mixed effects models with fixed time and random subject and cluster effects including interactions with time and explore the optimal choice of uptake times the results apply both to cross sectional studies where new subjects are observed at each time point and longitudinal studies with repeat observations on the same subjects the efficiency of the design is expressed in terms of a cluster mean correlation which carries information about the dependency structure of the data and two design coefficients which reflect the pattern of uptake times in cross sectional studies the cluster mean correlation combines information about the cluster size and the intra cluster correlation coefficient a formula is given for the design effect in both cross sectional and longitudinal studies an algorithm for optimising the choice of uptake times is described and specific results obtained for the best balanced stepped designs in large studies we show that the best design is a hybrid mixture of parallel and stepped wedge components with the proportion of stepped wedge clusters equal to the cluster mean correlation the impact of prior uncertainty in the cluster mean correlation is considered by simulation some specific hybrid designs are proposed for consideration when the cluster mean correlation cannot be reliably estimated using a minimax principle to ensure acceptable performance across the whole range of unknown values 2016 the authors statistics in medicine published by john wiley sons ltd

information content of cluster period cells in stepped wedge trials stepped wedge and other multiple period cluster randomized trials which collect data from multiple clusters across multiple time periods are being conducted with increasing frequency statistical research into these designs has not kept apace in particular some stepped wedge designs with missing cluster period cells have been proposed without any formal justification indeed there are no general guidelines regarding which cells of a stepped wedge design contribute the least information toward estimation of the treatment effect and correspondingly which may be preferentially omitted in this article we define a metric of the information content of cluster period cells entire treatment sequences and entire periods of the standard stepped wedge design as the increase in variance of the estimator of the treatment effect when that cell sequence or period is omitted we show that the most information rich cells are those that occur immediately before or after treatment switches but also that there are additional cells that contribute almost as much to the estimation of the treatment effect however the information content patterns depend on the assumed correlation structure for the repeated measurements within a cluster

local hadamard well posedness for nonlinear wave equations with supercritical sources and damping abstract we consider the wave equation with supercritical interior and boundary sources and damping terms the main result of the paper is local hadamard well posedness of finite energy weak solutions the results obtained 1 extend the existence results previously obtained in the literature by allowing more singular sources 2 show that the corresponding solutions satisfy hadamard well posedness conditions during the time of existence this result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping

global well posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping linear strong damping and logarithmic nonlinearity abstract to understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity we aim to study the global well posedness of nonlinear fourth order wave equations with logarithmic source term where the dispersive the nonlinear weak damping and linear strong damping are taken into account based on the potential well method the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow up with subcritical initial energy moreover we extend the results to the critical initial energy which is realized by introducing scaling technique finally surrounding the blow up at arbitrarily high initial energy we compare and discuss the research program stemming from two different strategies in the first one it is assumed that the blow up result is bound to the original logarithmic source by weakening the dispersive dissipative structure while the second one is based on the nonlinear wave equation with complete dispersive dissipative structure but the logarithmic source is replaced by an enhanced version through these two strategies we explain the mechanism of these two structures respectively and establish two kinds of sufficient conditions for initial data leading to high energy blow up

global solutions to a structure acoustic interaction model with nonlinear sources abstract this article focuses on a structural acoustic interaction system consisting of a semilinear wave equation defined on a smooth bounded domain r 3 which is strongly coupled with a berger plate equation acting only on a flat part of the boundary of in particular the source terms acting on the wave and plate equations are allowed to have arbitrary growth order we employ a standard galerkin approximation scheme to establish a rigorous proof of the existence of local weak solutions in addition under some conditions on the parameters in the system we prove such solutions exist globally in time and depend continuously on the initial data

weak solutions and blow up for wave equations of p laplacian type with supercritical sources this paper investigates a quasilinear wave equation with kelvin voigt damping utt pu ut f u in a bounded domain 3 and subject to dirichlet boundary conditions the operator p 2 p 3 denotes the classical p laplacian the nonlinear term f u is a source feedback that is allowed to have a supercritical exponent in the sense that the associated nemytskii operator is not locally lipschitz from w01 p into l2 under suitable assumptions on the parameters we prove existence of local weak solutions which can be extended globally provided the damping term dominates the source in an appropriate sense moreover a blow up result is proved for solutions with negative initial total energy

highly efficient stepped wedge designs for clusters of unequal size the stepped wedge design swd is a form of cluster randomized trial usually comparing two treatments which is divided into time periods and sequences with clusters allocated to sequences typically all sequences start with the standard treatment and end with the new treatment with the change happening at different times in the different sequences the clusters will usually differ in size but this is overlooked in much of the existing literature this paper considers the case when clusters have different sizes and determines how efficient designs can be found the approach uses an approximation to the variance of the treatment effect which is expressed in terms of the proportions of clusters and of individuals allocated to each sequence of the design the roles of these sets of proportions in determining an efficient design are discussed and illustrated using two swds one in the treatment of sexually transmitted diseases and one in renal replacement therapy cluster balanced designs which allocate equal numbers of clusters to each sequence are shown to have excellent statistical and practical properties suggestions are made about the practical application of the results for these designs the paper concentrates on the cross sectional case where subjects are measured once but it is briefly indicated how the methods can be extended to the closed cohort design

global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms abstract we focus on the global well posedness of the system of nonlinear wave equations u t t u d u k e v l u t m 1 u t f 1 u v v t t v d v e u v t r 1 v t f 2 u v in a bounded domain r n n 1 2 3 with dirichlet boundary conditions the nonlinearities f 1 u v and f 2 u v act as a strong source in the system under some restriction on the parameters in the system we obtain several results on the existence of local solutions global solutions and uniqueness in addition we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

hadamard well posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping abstract presented here is a study of a viscoelastic wave equation with supercritical source and damping terms we employ the theory of monotone operators and nonlinear semigroups combined with energy methods to establish the existence of a unique local weak solution in addition it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense

well posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction we establish subject to some natural additional assumptions imposed on the relation between the source and the damping both well posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation the theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources

superclose and superconvergence of finite element discretizations for the stokes equations with damping a class of conforming mixed finite element methods fems are applied to the stokes equations with damping with the help of the prior estimates for the exact and approximated solutions and the appropriate choice of the parameters a n and r appeared in the problem the superclose and superconvergence results for the velocity in h 1 norm and the pressure in l 2 norm are derived to our best knowledge this is the first superconvergence analysis of conforming mixed fems for this kind nonlinear problem

the lifespan for 3d quasilinear wave equations with nonlinear damping terms abstract in this paper we study the initial boundary value problem for a system of nonlinear wave equations involving nonlinear damping terms in a bounded domain the nonexistence of global solutions is discussed under some conditions on the given parameters estimates on the lifespan of solutions are also given our results extend and generalize the recent results in k agre m a rammaha system of nonlinear wave equations with damping and source terms differential integral equations 19 2006 1235 1270 especially the blow up of weak solutions in the case of non negative energy

global existence and global nonexistence of solutions of the cauchy problem for a nonlinearly damped wave equation abstract in this paper we consider the long time behavior of solutions of the initial value problem for semilinear wave equations of the form formula

blow up for nonlinear dissipative wave equations in rn abstract we consider the cauchy problem u t t u u t m 1 u t u p 1 u t x 0 r n u 0 x u 0 x u t 0 x v 0 x for 1 m p p n n 2 for n 3 we prove that for any given numbers 0 0 there exist infinitely many data u 0 v 0 in the energy space such that the initial energy e 0 the gradient norm u 0 2 and the solution of the above cauchy problem blows up in finite time

on nonlinear wave equations with degenerate damping and source terms in this article we focus on the global well posedness of the differential equation u tt u u k j u t u p 1 u in x 0 t where j is a sub differential of a continuous convex function j under some conditions on j and the parameters in the equations we obtain several results on the existence of global solutions uniqueness nonexistence and propagation of regularity under nominal assumptions on the parameters we establish the existence of global generalized solutions with further restrictions on the parameters we prove the existence and uniqueness of a global weak solution in addition we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k m and the initial energy is negative we also address the issue of propagation of regularity specifically under some restriction on the parameters we prove that solutions that correspond to any regular initial data such that u 0 h 2 h 1 0 u 1 h 1 0 are indeed strong solutions

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

two grid mfeas for the incompressible stokes type variational inequality with damping abstract in this paper we study two grid mixed finite element algorithms mfeas for the stationary incompressible stokes equations with damping and nonlinear slip boundary conditions we set up the weak formulation and obtain the well posedness then employing the mixed finite element method some optimal error estimates between the exact solution and the approximation solution are derived two kinds of two grid mfeas based on the stokes correction and newton type correction are established some error estimates for two grid mfeas are obtained several numerical results are presented to confirm our theoretical findings

critical blowup for systems of semilinear wave equations in low space dimensions abstract we consider the cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data del santo et al geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 140 birkhauser boston 1997 have studied the existence and nonexistence of global classical solutions of the cauchy problem except for the critical case in this paper we study the critical case and we show the nonexistence of global classical solutions and also give the upper bounds of the life span

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

constrained matrix li yau hamilton estimates on kahler manifolds we derive a family of constrained matrix li yau hamilton estimates on kahler manifolds as a result we first get a constrained matrix li yau hamilton estimate for heat equation on a kahler manifold with fixed kahler metric secondly we get such estimate for forward conjugate heat equation on kahler manifolds with time dependent kahler metrics evolving under the kahler ricci flow

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

power calculation for cross sectional stepped wedge cluster randomized trials with variable cluster sizes standard sample size calculation formulas for stepped wedge cluster randomized trials sw crts assume that cluster sizes are equal when cluster sizes vary substantially ignoring this variation may lead to an under powered study we investigate the relative efficiency of a sw crt with varying cluster sizes to equal cluster sizes and derive variance estimators for the intervention effect that account for this variation under a mixed effects model a commonly used approach for analyzing data from cluster randomized trials when cluster sizes vary the power of a sw crt depends on the order in which clusters receive the intervention which is determined through randomization we first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power we then obtain an expected power based on a first order approximation to the variance formula where the expectation is taken with respect to all possible randomization sequences finally we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation instead of exact cluster sizes are known in the design stage we evaluate our methods through simulations and illustrate that the average power of a sw crt decreases as the variation in cluster sizes increases and the impact is largest when the number of clusters is small

on wave equations of the p laplacian type with supercritical nonlinearities abstract this article focuses on a quasilinear wave equation of p laplacian type u t t p u u t f u in a bounded domain r 3 with a sufficiently smooth boundary subject to a generalized robin boundary condition featuring boundary damping and a nonlinear source term the operator p 2 p 3 denotes the classical p laplacian the interior and boundary terms f u h u are sources that are allowed to have a supercritical exponent in the sense that their associated nemytskii operators are not locally lipschitz from w 1 p into l 2 or l 2 under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time provided the damping terms dominate the corresponding sources in an appropriate sense moreover a blow up result is proved for solutions with negative initial total energy

global existence and blow up of solutions for a system of nonlinear viscoelastic wave equations with damping and source abstract in this paper we investigate the global existence and finite time blow up of solutions to the system of nonlinear viscoelastic wave equations u t t u 0 t g 1 t u d u t m 1 u t f 1 u v v t t v 0 t g 2 t v d v t r 1 v t f 2 u v in 0 t with initial and dirichlet boundary conditions where is a bounded domain in r n n 1 2 3 under suitable assumptions on the functions g i f i i 1 2 the initial data and the parameters in the equations we establish several results concerning local existence global existence uniqueness and finite time blow up property

the canonical expanding soliton and harnack inequalities for ricci flow we introduce the notion of canonical expanding ricci soliton and use it to derive new harnack inequalities for ricci flow this viewpoint also gives geometric insight into the existing harnack inequalities of hamilton and brendle