vanishing of vacuum states and blow up phenomena of the compressible navier stokes equations the navier stokes systems for compressible fluids with density dependent viscosities are considered in the present paper these equations in particular include the ones which are rigorously derived recently as the saint venant system for the motion of shallow water from the navier stokes system for incompressible flows with a moving free surface 14 these compressible systems are degenerate when vacuum state appears we study initial boundary value problems for such systems for both bounded spatial domains or periodic domains the dynamics of weak solutions and vacuum states are investigated rigorously

compressible navier stokes equations with degenerate viscosity coefficient and vacuum in this paper we consider the compressible navier stokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support the viscosity coefficient is assumed to be a power function of the density so that the cauchy problem is well posed new global existence results are established when the density function connects to the vacuum states continuously for this some new a priori estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum we will also give a non global existence theorem of regular solutions when the initial data are of compact support in eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum

controllability and observability of linear retarded systems in banach spaces abstract controllability and observability are studied for linear retarded systems in a banach space x both the x exact and the x approximate controllability are considered and a number of necessary and sufficient conditions in terms of the coefficient operators and or the fundamental solution are presented various corresponding concepts for observability are introduced and the duality between controllability and observability is clarified an applicable condition for the x approximate controllability and the x observability is established under the assumptions that the control is a finite dimensional one and the system of generalized eigenspaces is projectively complete the condition is stated in terms of eigenvectors and controllers

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

boundary controllability of sobolev type abstract nonlinear integrodifferential systems sufficient conditions are established for boundary controllability of various classes of sobolev type nonlinear systems including integrodifferential systems in banach spaces the results are obtained using the strongly continuous semigroup of operators and the banach contraction principle examples are provided to illustrate the theory

equivalence and normal forms of bilinear forms abstract we present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms beginning from basic results developed by riehm our solution to this problem hinges on the classification of indecomposable forms and in how uniquely they fit together to produce all other forms we emphasize the use of split forms i e those bilinear forms such that the minimal polynomial of the asymmetry of their non degenerate part splits over ground field rather than restricting the field to be algebraically closed in order to obtain the most explicit results without resorting to the classification of hermitian symmetric and quadratic forms we merely require that the underlying field be quadratically closed

structure of isometry group of bilinear spaces abstract we describe the structure of the isometry group g of a finite dimensional bilinear space over an algebraically closed field of characteristic not two if the space has no indecomposable degenerate orthogonal summands of odd dimension it admits a canonical orthogonal decomposition into primary components and g is isomorphic to the direct product of the isometry groups of the primary components each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the lie algebra of that group in the general case the description of g is more complicated we show that g is a semidirect product of a normal unipotent subgroup k with another subgroup which in its turn is a direct product of a group of the type described in the previous paragraph and another group h which we can describe explicitly the group h has a levi decomposition whose levi factor is a direct product of several general linear groups of various degrees we obtain simple formulae for the dimensions of h and k

boundary hemivariational inequality of parabolic type abstract in this paper we consider a parabolic hemivariational inequality with a nonmonotone multivalued boundary condition and with a time dependent pseudomonotone operator we establish the existence of solutions for such inequality our proof is based on a known surjectivity result for operators of pseudomonotone type we present an example of semipermeability heat conduction problem to which the result applies

dynamical behaviors for 1d compressible navier stokes equations with density dependent viscosity abstract the dynamical behaviors of vacuum states for one dimensional compressible navier stokes equations with density dependent viscosity coefficient are considered it is first shown that a unique strong solution to the free boundary value problem exists globally in time the free boundary expands outwards at an algebraic rate in time and the density is strictly positive in any finite time but decays pointwise to zero time asymptotically then it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum the solution is piecewise regular and contains a piece of continuous vacuum before the time t 0 which is compressed at an algebraic rate and vanishes at the time t meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially

boundary controllability of differential equations with nonlocal condition for a semilinear boundary control system with a nonlocal condition using a banach fixed point theorem we prove the boundary controllable and we give an application

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

abstract degenerate cauchy problems in locally convex spaces abstract this paper is concerned with the abstract degenerate cauchy problem dcp d dt bu t au t t 0 bu 0 bu 0 where a and b are closed linear operators in a sequentially complete locally convex space a c propagation family for dcp is introduced c is an operator leading to a general c wellposedness result about dcp moreover conditions are given ensuring the existence of c propagation families for those dcp with differential operators on various function spaces with frechet topologies as coefficient operators these results are new even in the case of banach spaces

the number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field abstract generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over gf q2 and for the number of equivalence or congruence classes of nondegenerate bilinear forms of rank n over gf q

global well posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible navier stokes equations we establish the global existence and uniqueness of classical solutions to the cauchy problem for the isentropic compressible navier stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum the initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large in particular the initial density can even have compact support these results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states 2012 wiley periodicals inc

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

global classical solutions to the 1 d vacuum free boundary problem for full compressible navier stokes equations with large data the vacuum free boundary problem of one dimensional non isentropic compressible navier stokes equations with large initial data is investigated in this paper the fluid is initially assumed to occupy a finite interval and connect to the vacuum continuously at the free boundary which is often considered in the gas vacuum interface problem using the method of lagrangian particle path we derive some point wise estimates and weighted spatial and time energy estimates for the classical solutions then the global existence and uniqueness of classical solutions are shown and the expanding speed for the free boundary is proved to be finite the main difficulty of this problem is the degeneracy of the system near the free boundary previous results are only for the solutions with low regularity cf g q chen and m kratka commun partial differ equations 27 907 943 2002

global well posedness of the cauchy problem of two dimensional compressible navier stokes equations in weighted spaces abstract in this paper we study the global well posedness of classical solution to 2d cauchy problem of the compressible navier stokes equations with large initial data and vacuum it is proved that if the shear viscosity is a positive constant and the bulk viscosity is the power function of the density that is with 3 then the 2d cauchy problem of the compressible navier stokes equations on the whole space r 2 admits a unique global classical solution u which may contain vacuums in an open set of r 2 note that the initial data can be arbitrarily large to contain vacuum states various weighted estimates of the density and velocity are obtained in this paper and these self contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow

controllability of neutral functional differential systems in abstract space in this paper by using fractional power of operators and sadovskii fixed point theorem we study the controllability and the local controllability of abstract neutral functional differential systems with unbounded delay as application an example is provided to illustrate the obtained results

controllability of second order semilinear neutral functional differential systems in banach spaces abstract sufficient conditions for controllability of semilinear second order neutral functional differential systems in banach spaces are established using the theory of strongly continuous cosine families the results are obtained by using the leray schauder alternative

an elementary proof of gabriel s theorem on degenerate bilinear forms and its generalization abstract a theorem due to p gabriel shows that every bilinear space decomposes in a unique manner up to equivalence as the orthogonal direct sum of indecomposable degenerate spaces and a nondegenerate space this has been generalized by c riehm and m a shrader frechette to sesquilinear modules over a semisimple ring with an anti automorphism we give an elementary proof of these facts which avoids the use of kronecker modules

congruence and conjunctivity of matrices to their adjoints abstract every square matrix over a field f is involutorily congruent over f to its transpose and hence each such matrix is the product of a symmetric matrix and an involutory matrix over f in the usual complex case every matrix which is conjunctive with its adjoint conjugate transpose is involutorily conjunctive with its adjoint and hence is the product of a hermitian matrix and an involutory matrix furthermore every such matrix is conjunctive with a real matrix these three conditions on a matrix 1 being conjunctive with its adjoint 2 being involutorily conjunctive with its adjoint and 3 being conjunctive with a real matrix are studied in the more general context of a field f with involution and it is shown in general that 3 implies 2 that 2 implies 3 if char f 2 a 2 2 counterexample exists for each f with char f 2 and that 1 does not in general imply 2 a 2 2 counterexample in the complexification of the rational field is presented the problem of deciding which matrices satisfy 2 is equivalent even in this general context to the problem of deciding which pairs of self adjoint hermitian matrices are involutorily conjunctive for the general 2 2 case the three conditions are characterized in terms of norms

global strong solutions to radial symmetric compressible navier stokes equations with free boundary in this paper we consider the two dimensional barotropic compressible navier stokes equations with stress free boundary condition imposed on the free surface as the viscosity coefficients satisfies 2 1 we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary in particular we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive and the free boundary propagates along the particle path and expand outwards at an algebraic rate

on recent developments in the theory of abstract differential equations with fractional derivatives abstract this note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives we show that the existence results in agarwal et al 2009 1 belmekki and benchohra 2010 2 darwish et al 2009 3 hu et al 2009 4 mophou and n guerekata 2009 6 7 mophou 2010 8 9 muslim 2009 10 pandey et al 2009 11 rashid and el qaderi 2009 12 and tai and wang 2009 13 are incorrect since the considered variation of constant formulas is not appropriate in this note we also consider a different approach to treat a general class of abstract fractional differential equations

lie algebras and bilinear forms in characteristic 2 abstract all bilinear forms defined on a finite dimensional vector space over an algebraically closed field of characteristic 2 whose associated lie algebra is reductive are determined

lagrange structure and dynamics for solutions to the spherically symmetric compressible navier stokes equations the compressible navier stokes system cns with density dependent viscosity coefficients is considered in multi dimension the prototype of the system is the viscous saint venat model for the motion of shallow water a spherically symmetric weak solution to the free boundary value problem for cns with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path which is continuous away from the symmetry center detailed regularity and lagrangian structure of this solution have been obtained in particular it is shown that the particle path is uniquely defined starting from any non vacuum region away from the symmetry center along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved starting from any non vacuum point at a later on time a particle path is also uniquely defined backward in time which either reaches at some initial non vacuum point or stops at a small middle time and connects continuously with vacuum in addition the free boundary is shown to expand outward at an algebraic rate in time and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up this finally leads to the formation of vacuum state almost everywhere as the time goes to infinity

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

existence of s asymptotically ω periodic solutions for fractional order functional integro differential equations with infinite delay abstract we study s asymptotically periodic solutions of the abstract fractional equation u 1 a u f t u t 1 2 where a is a linear operator of sectorial type 0

controllability of neutral functional integrodifferential systems in banach spaces abstract sufficient conditions for controllability of neutral functional integrodifferential systems in a banach space are established the results are obtained by using the schaefer fixed point theorem an example is provided to illustrate the theory

controllability of semilinear boundary problems with nonlocal initial conditions the main purpose of this paper is the existence of solutions and controllability for semilinear boundary problems with nonlocal initial conditions we show that the solutions are given by a variation of constants formula which allows us to study the exact controllability for this kind of problems with control and nonlinear terms at the boundary the included application to a size structured population equation provides a motivation for abstract results

controllability of functional semilinear integrodifferential systems in banach spaces sufficient conditions for controllability of functional semilinear integrodifferential systems in a banach space are established the results are obtained by using the schaefer fixed point theorem

existence of classical solutions to nonautonomous nonlocal parabolic problems abstract we obtain a new existence theorem for classical solutions to nonautonomous equations with nonlocal initial conditions u t a t u t f t u t t s t u s g u u 0 in a banach space x where t s 0 f g are given x valued functions and a t is a sectorial operator not necessarily densely defined in x for each t 0 t both banach s contraction principle and schauder s fixed point theorem as well as the theory of evolution families and interpolation spaces are employed in our approach a concrete example is shown to illustrate the existence theorem

semilinear integrodifferential equations with nonlocal initial conditions abstract of concern is the cauchy problem for semilinear integrodifferential equations with nonlocal initial conditions under general and natural hypotheses we establish some new theorems about the existence and uniqueness of solutions for the cauchy problem as a consequence we unify and extend the corresponding theorems given previously for the cauchy problem for differential equations or integrodifferential equations with nonlocal initial conditions moreover we present two examples one of which comes from heat conduction in materials with memory to show that the existing results are not applicable to them in contrast with ours

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

the failure on continuous dependence on initial data for the navier stokes equations of compressible flow it is shown that physical solutions of the navier stokes equations for one dimensional compressible flow need not depend continuously on their initial data at least when vacuum states are allowed specifically two fluid regions initially separated by a third region of very low density 6 are considered it is shown that as delta to 0 the unique solutions corresponding to delta 0 do not in fact converge to a physical solution but rather to a nonphysical weak solution in which the two fluids cannot collide independent of their initial velocities and whose separate momenta need not be conserved a particular consequence is that solutions of the cavity problem delta 0 are not unique

approximate controllability for a class of semilinear abstract equations in this paper a class of control systems governed by the semilinear abstract equation dot y ay f y bv is considered a sufficient condition for the approximate controllability is obtained which improves j henry s results on a nonlinear parabolic control systems with some more serious restrictions it is suitable not only to the infinite dimensional case but also to the finite dimensional case two examples are given to explain the applications of the theory

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

controllability of evolution inclusions with nonlocal conditions in this paper we examine controllability problems of evolution inclusions with nonlocal conditions using kakutani s fixed point theorem and schauder s fixed point theorem we establish sufficient conditions for the controllability under convex and nonconvex orientor fields respectively

asymptotic behavior of compressible navier stokes equations with density dependent viscosity and vacuum in this paper we study the one dimensional navier stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density precisely when the viscosity coefficient is proportional to with 0 where is the density we give the asymptotic behavior and the decay rate of the density function x t furthermore the behavior of the density function x t near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied the analysis is based on some new mathematical techniques and some new useful estimates this fills a final gap on studying navier stokes equations with the viscosity coefficient dependent on the density

boundary control for the heat equation with non linear boundary condition soit d un domaine borne de r n avec n 1 a frontiere lisse d et soit une normale a d pointant vers l exterieur soit u x t une fonction de x dans d et t l evolution de u est decrite par le probleme aux valeurs limites et initiales suivant u t u dans d 0 t u g u f dans d 0 t u 0 u 0 dans d sous certaines conditions on demontre l existence d une solution faible unique

controllability of sobolev type integrodifferential systems in banach spaces sufficient conditions for controllability of sobolev type integrodifferential systems in banach spaces are established the results are obtained using compact semigroups and the schauder fixed point theorem as an example is provided to illustrate the results

the fractional fourier transform and quadratic field magnetic resonance imaging the fractional fourier transform frft is revisited in the framework of strongly continuous periodic semigroups to restate known results and to explore new properties of the frft we then show how the frft can be used to reconstruct magnetic resonance mr images acquired under the presence of quadratic field inhomogeneity particularly we prove that the order of the frft is a measure of the distortion in the reconstructed signal moreover we give a dynamic interpretation to the order as time evolution of a function we also introduce the notion of r a space as an extension of the fourier or k space in mr and we use it to study the distortions introduced in two common mr acquisition strategies we formulate the reconstruction problem in the context of the frft and show how the semigroup theory allows us to find new reconstruction formulas for discrete sampled signals finally the results are supplemented with numerical examples that show how it performs in a standard 1d mr signal reconstruction

a partial functional differential equation of sobolev type where r t t 0 is the semigroup of bounded linear operators generated by ab equations of type 1 1 have been considered by several authors 24 g 101 in particular our motivation comes from the recent article of brill 2 the method employed here is different from brill s however our results reduce to his when r 0 the article 111 contains results on partial functional differential equations when b i equation 1 1 can be considered as an abstraction of the partial functional differential equation

semigroups and some nonlinear fractional differential equations equations of the formd au t dt a au t f t b 1 t u t b r t u t are considered where 0 0 is a family of linear closed operators defined on dense sets in e into e and f is a given abstract nonlinear function defined on 0 t xe r with values in e t 0 it is assumed that a generates an analytic semigroup under suitable conditions on the family of operators b i t i 1 r t 0 and on f we study the existence and uniqueness of the solution of the cauchy problem for the considered equation some properties concerning the stability of solutions are obtained we also give an application for nonlinear partial differential equations of fractional orders

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

controllability of abstract neutral functional differential systems with unbounded delay in this paper by using sadovskii fixed point theorem we study the controllability and the local controllability of abstract neutral functional differential systems with unbounded delay as application an example is provided to illustrate the obtained results

analysis and semidiscrete galerkin approximation of a class of nonlinear parabolic optimal control problems abstract this paper presents an optimization problem involving a system governed by a nonlinear partial differential equation with control acting on the domain in the boundary conditions or in the initial condition we describe the galerkin approximation and prove its convergence to the solutions of the initial optimization problem

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

existence of solutions for a class of abstract differential equations with nonlocal conditions abstract we discuss the existence of mild classical and strict solutions for a class of abstract differential equations with nonlocal conditions our technical approach allows the study of partial differential equations with nonlocal conditions involving partial derivatives or nonlinear expressions of the solution some concrete applications to partial differential equations are considered

existence of mild solutions for fractional integrodifferential equations of sobolev type with nonlocal conditions abstract this paper is concerned with the fractional integrodifferential equations of sobolev type with nonlocal condition in a separable banach space with the help of the theory of propagation family as well as the theory of the measures of noncompactness and the condensing maps we obtain the existence result of mild solutions for above equations two examples are given to illustrate the existence theorem

a priori estimates of local solutions to compressible navier stokes equations with free boundary abstract in this paper we consider the vacuum free boundary problem of the compressible navier stokes system in two dimension under the lagrangian variables we present a priori estimates of local in time solutions in a proper sobolev space

controllability of semilinear control systems dominated by the linear part while various equivalent conditions for controllability have been obtained in the case of linear control systems controllability problems of semilinear control systems usually require some complicated and limited assumptions in this paper we show the approximate controllability of an abstract semilinear control system under the assumption which has a simple form and can be easily checked in many examples

an abstract semilinear hyperbolic volterra integrodifferential equation abstract we consider semilinear integrodifferential equations of the form u t a t u t 0 t g t s u s ds f t u 0 u 0 for each t 0 the operator a t is assumed to be the negative generator of a strongly continuous semigroup in a banach space x the domain d a t of a t is allowed to vary with t thus our models are volterra integrodifferential equations of hyperbolic type these types of equations arise naturally in the study of viscoelasticity our main results are the proofs of existence uniqueness continuation and continuous dependence of the solutions

existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay we deal in this paper with the mild solution for fractional semilinear differential equations with infinite delay d ax t ax t f t x t bx t email protected 0 t x t email protected t email protected 0 with t 0 and 0 0 on a complex banach space x by means of classical fixed points methods

boundary control problems with nonlinear state equation first order necessary conditions of optimality are obtained for boundary control problems governed by parabolic equations with nonlinear boundary value conditions

global behavior of spherically symmetric navier stokes poisson system with degenerate viscosity coefficients in this paper we study a free boundary problem for compressible spherically symmetric navier stokes poisson equations with degenerate viscosity coefficients and without a solid core under certain assumptions that are imposed on the initial data we obtain the global existence and uniqueness of the weak solution and give some uniform bounds with respect to time of the solution moreover we obtain some stabilization rate estimates of the solution the results show that such a system is stable under small perturbations and could be applied to the astrophysics