asymptotic expansion of solutions to the drift diffusion equation with large initial data abstract we consider the large time behavior of the solution to the initial value problem for the nernst planck type drift diffusion equation in whole spaces in the l p framework the global existence and the decay of the solution were shown moreover the second order asymptotic expansion of the solution as t was derived we also deduce the higher order asymptotic expansion of the solution especially we discuss the contrast between the odd dimensional case and the even dimensional case

spectral variation bounds in hyperbolic geometry abstract we derive new estimates for distances between optimal matchings of eigenvalues of non normal matrices in terms of the norm of their difference we introduce and estimate a hyperbolic metric analogue of the classical spectral variation distance the result yields a qualitatively new and simple characterization of the localization of eigenvalues our bound improves on the best classical spectral variation bounds due to krause if the distance of matrices is sufficiently small and is sharp for asymptotically large matrices our approach is based on the theory of model operators which provides us with strong resolvent estimates the latter naturally lead to a chebyshev type interpolation problem with finite blaschke products which can be solved explicitly and gives stronger bounds than the classical chebyshev interpolation with polynomials as compared to the classical approach our method does not rely on hadamard s inequality and immediately generalizes to algebraic operators on hilbert space

ten methods to bound multiple roots of polynomials given a univariate polynomial p with a k fold multiple root or a k fold root cluster near some z we discuss nine different methods to compute a disc near z which either contains exactly or contains at least k roots of p many of the presented methods are known some are new we are especially interested in the behaviour of methods when implemented in a rigorous way that is when taking into account all possible effects of rounding errors in other words every result shall be mathematically correct we display extensive test sets comparing the methods under different circumstances based on the results we present a tenth a hybrid method combining five of the previous methods which for given z i detects the number k of roots near z and ii computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster therefore the resulting discs are numerically nearly optimal

diffusion approximation for self similarity of stochastic advection in burgers equation self similarity of burgers equation with stochastic advection is studied in self similar variables a stationary solution is constructed which establishes the existence of a stochastically self similar solution for the stochastic burgers equation the analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self similar variables furthermore by a diffusion approximation the long time convergence to the self similar solution is proved in the sense of distribution

bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix we derive some new bounds for the distance between the roots of two polynomials in terms of their coefficients and for the distance between the eigenvalues of two matrices in terms of the norm of their difference

convection diffusion equation with absorption and non decaying initial data in this paper we give the precise description of the large time behavior of the solution u of the cauchy problem tu u a u u in rn 0 u x 0 x 0in rn by using the ordinary differential equation and a linear parabolic equation here n 1 a rn 1 1 0 and is a bounded continuous function such that lp rn for some 1 p furthermore we study the large time behavior of the hot spots for the solution u

resolvent bounds and spectral variation abstract a new spectral variation bound is given for linear operators acting on normed linear spaces this bound is derived using the cayley hamilton theorem and chebychev polynomials

the hierarchical basis multigrid method we derive and analyze the hierarchical basis multigrid method for solving discretizations of self adjoint elliptic boundary value problems using piecewise linear triangular finite elements the method is analyzed as a block symmetric gau seidel iteration with inner iterations but it is strongly related to 2 level methods to the standard multigridv cycle and to earlier jacobi like hierarchical basis methods the method is very robust and has a nearly optimal convergence rate and work estimate it is especially well suited to difficult problems with rough solutions discretized using highly nonuniform adaptively refined meshes

parallel hybrid algebraic multilevel iterative methods abstract for the large scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self adjoint elliptic boundary value problem by making use of the blocked multilevel iteration idea we construct preconditioning matrices for the coefficient matrix and set up a class of parallel hybrid algebraic multilevel iterative methods for solving this kind of system of linear equations theoretical analysis shows that not only do these new methods lend themselves to parallel computation but also their convergence rates are independent of both the sizes and the level numbers of the grids and their computational work loads are also bounded by linear functions of the step sizes of the finest grids

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

parallel multilevel iterative methods abstract for large scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self adjoint elliptic boundary value problem by making use of blocked multilevel iteration we construct preconditioning matrices for the coefficient matrix and set up a class of parallel multilevel iterative methods for solving such system theoretical analysis shows that besides lending themselves to strongly parallel computation these new methods have convergence rates independent of both the sizes and the level numbers of the grids and their computational work loads are also bounded by linear functions about the step sizes of the finest grids

existence stability and smoothness of a bounded solution for nonlinear time varying thermoelastic plate equations in this paper we study the existence stability and the smoothness of a bounded solution of the following nonlinear time varying thermoelastic plate equation with homogeneous dirichlet boundary conditions utt 2u f1 t u t 0 x t ut f2 t u t 0 x u u 0 t 0 x where 0 0 is a sufficiently regular bounded domain in rn n 1 and fe1 fe2 r l2 2 l2 define by fe t u x f t u x x x are continuous and locally lipschitz functions first we prove that the linear system f1 f2 0 generates an analytic strongly continuous semigroup which decays exponentially to zero second under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable and for a large class of functions f1 f2 this bounded solution is almost periodic finally we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution

exact boundary controllability for the linear korteweg de vries equation on the half line this paper is concerned with the controllability of the linear korteweg de vries equation on the domain o 0 infty the control being applied at the left endpoint x 0 it is shown that the em exact boundary controllability holds true in l 2 0 infty provided that the solutions are not required to be in l infty 0 t l 2 0 infty the proof rests on a carleman s estimate and an approximation theorem a similar result is obtained for the heat equation and for the schrodinger equation

a class of new hybrid algebraic multilevel preconditioning methods abstract a class of new hybrid algebraic multilevel preconditioning methods is presented for solving the large sparse systems of linear equations with symmetric positive definite coefficient matrices resulting from the discretization of many second order elliptic boundary value problems by the finite element method the new preconditioners are shown to be of optimal orders of complexities for two dimensional and three dimensional problem domains and their relative condition numbers are estimated to be bounded uniformly independent of the numbers of both the levels and the nodes

on some energy inequalities and supnorm estimates for advection diffusion equations in rn abstract we show how some suitable standard energy inequalities can be used for a simple short derivation of the fundamental supnorm estimate u t l r n k n p u 0 l p r n t n 2 p t 0 for solutions of advection diffusion equations of the form u t div f u u and some generalizations in the whole space r n where n 1 is arbitrary our approach can be used for other parabolic equations as well including degenerate cases like porous medium type equations and p laplacian evolution equations some open problems and related results of interest are also given

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

generalized burgers equations and euler painleve transcendents ii initial value problems for the generalized burgers equation gbe ut u betaux lambdaualpha delta 2 uxx are discussed for the single hump type of initial data both continuous and discontinuous the numerical solution is carried to the self similar intermediate asymptotic regime when the solution is given analytically by the self similar form the nonlinear transformed ordinary differential equations ode s describing the self similar form are generalizations of a class discussed by euler and painleva and quoted by kamke these ode s are new and it is postulated that they characterize gbe s in the same manner as the painlev equations categorize the kortweg de vries kdv type a connection problem for some related ode s satisfying proper asymptotic conditions at x infinity is solved the range of amplitude parameter is found for which the solution of the connection problem exists the other solutions of the above gbe which display several interesting features such as peaking breaking and a long shelf on the left for negative values of the damping coefficient lambda are also discussed the results are compared with those holding for the modified kdv equation with damping journal of mathematical physics is copyrighted by the american institute of physics

null controllability for a semilinear parabolic equation with gradient quadratic growth abstract this article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term the technique in this paper is a combination of cole hopf transformation and some methods from a y khapalov controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term a qualitative approach siam j control optim 41 2003 1886 1900

large time behavior of the vorticity of two dimensional viscous flow and its application to vortex formation we consider the cauchy problem for the two dimensional vorticity equation we show that the solution behaves like a constant multiple of the gauss kernel having the same total vorticity as time tends to infinity no particular structure of initial data 0 x 0 is assumed except the restriction that the reynolds numberr 0 dx v is small wherev is the kinematic viscosity applying a time dependent scale transformation we show a stability of burgers vortex which physically implies formation of a concentrated vortex

carleman estimates for a class of degenerate parabolic operators given alpha in 0 2 and f in l 2 0 t times 0 1 we derive new carleman estimates for the degenerate parabolic problem w t x alpha w x x f where t x in 0 t times 0 1 associated to the boundary conditions w t 1 0 and w t 0 0 if 0 leq alpha 1 or x alpha w x t 0 0 if 1 leq alpha 2 the proof is based on the choice of suitable weighted functions and hardy type inequalities as a consequence for all 0 leq alpha 2 and omega subset subset 0 1 we deduce null controllability results for the degenerate one dimensional heat equation u t x alpha u x x h chi omega with the same boundary conditions as above

pointwise estimates for solutions of fractal burgers equation abstract in this paper we provide two sided estimates and uniform asymptotics for the solution of d dimensional critical fractal burgers equation u t 2 u b u u q 0 1 2 b r d for q 1 d and u 0 l 1 r d we consider also q 1 d under additional condition u 0 l r d in both cases we assume u 0 0 which implies that the solution is non negative the estimates are given in the terms of the function p t u 0 where p t denotes the semigroup for the operator t 2

asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to n waves abstract the goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws general convex flux f u 0 is considered with an assumption lim u 0 uf u f u 1 we show that under suitable conditions on the initial value its solution converges to an n wave in l1 norm with the optimal convergence order of o 1 t the technique we use in this article is to enclose the solution with two rarefaction waves we also show a uniform convergence order in the sense of graphs a numerical example of this phenomenon is included

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

unique continuation for some evolution equations this work is concerned with unique continuation results for some evolution equations essentially of parabolic type to be more precise let l be an evolution operator acting on functions defined on some connected open set b of r r x r l is said to have the unique continuation property if every solution u of lu 0 which vanishes one some nonempty open set u i of fl vanishes in the horizontal component of o in 0 i e in x t e 0 3x x t e co the first general result of this type was established by mizohata 6 in the case where l is a second order parabolic operator with smooth coefficients however mizohata s proof is somewhat delicate it is based in particular on an extended class of calderon zygmund operators and does not lead obviously to a weakening of the smoothness of the coefficients on the other hand it is important for the applications to deal with nonsmooth coefficients since l is often obtained as the linearization of some nonlinear operator at a nonnecessarily smooth solution we refer to lions 4 and kernevez and lions 3 for applications to control theory and to 1 l for applications to transversality techniques let us now describe the content of this work in the first section we prove a unique continuation theorem when l is a second order parabolic equation our proof is simple and based on the derivation of a carleman estimate which is reminiscent of the classical carleman estimates for second order elliptic operators see 121 this carleman inequality allows the weakening of the 118 0022 0396187 3 00

existence and nonexistence of global solutions to the cauchy problem of a class of nonlocal convective reaction diffusion equations we study the existence and nonexistence of global solutions to the cauchy problem where u t x is a scalar function a rn a 0 p 0 q r 1 is a gradient operator the results obtained generalize the results of aguirre and escobedo j aguirre and m escobedo 1993 proc r soc edin a 123 433 60 which do not consider the nonlocal factor in the reaction term of the equation and also generalize the results of wang et al m x wang s wang and c h xie 1999 j partial diff eqs 12 201 11 which do not include the nonlinear convection term in the equation

asymptotic estimates of solutions ofut 12δu u in r rd d 2 abstract we express the solution u of u t 1 2 u u in r r d d 2 u 0 through a functional of a d dimensional bessel process where is invariant by rotations and profiled this allows us to estimate the asymptotic behaviour of u t 1 and u t as t goes to infinity

lipschitz properties of structure preserving matrix perturbations abstract it is proved that under small perturbations that preserve the jordan structure canonical transformation matrices have the lipschitz property analogous result is proved for matrices under additional structure such as hamiltonian skew hamiltonian and symplectic both real and complex cases are considered

variation of grassman powers and spectra the operator norm of the derivative of the map which takes a finite dimensional linear operator to its kth grassman power the kth compound is evaluated this leads to a bound for the distance between the grassman powers of two operators as an important application a bound for the distance between the eigenvalues of two operators is obtained

uniform decay rated for parabolic conservations laws on etablit des taux de decroissance uniformes optimaux pour des solutions du probleme de cauchy avec donnees grandes pour des lois de conservation paraboliques scalaires a plusieurs dimensions d espace

on the variation of the spectra of matrices abstract several distances between the spectra of two matrices are discussed and compared optimal bounds are given which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two the results of bhatia and mukherjea and of bhatia and friedland on the eigenvalue variation are derived in an elementary way using results of henrici on the spectral variation

convergence to diffusion waves of solutions for viscous conservation laws we study the large time behavior of solutions of viscous conservation laws it is shown that solutions tend to diffusion waves which are constructed based on the heat equation and burgers equation the convergence is in thelp 1 p sense and is obtained as a consequence of thel2 decay of the difference between the solution and its asymptotic state of diffusion waves

improving spectral variation bounds with chebyshev polynomials a new spectral variation bound is established for linear operators a and b on finite dimensional vector spaces this bound is v a b c a b 1n a b n 1 n with c 8 this result affirmatively answers a conjecture of friedland

an asymptotic supnorm estimate for solutions of 1 d systems of convection diffusion equations abstract we show how to use the l p l q approach to obtain fundamental estimates for the spatial supnorm values of solutions u x t u 1 x t u m x t to general systems of convection diffusion equations of the form u i t x f i x t u 1 u m x g i t u i i t 2 u i x 2 1 i m with initial data u 0 l p 0 r l r for some 1 p 0 where i t 0 given arbitrary f f 1 f m g g 1 g m such that f x t u b t u for all x r t 0 u r m where b c 0 0

approximate controllability for fractional semilinear parabolic equations abstract in this paper we deal with the control systems described by a large class of fractional semilinear parabolic equations firstly we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate banach space secondly we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator nonlinear analysis method and fixed point theorem then the approximate controllability of the fractional semilinear parabolic equations is formulated and proved at the end of the paper an example is given to illustrate our main results

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

approximate controllability of the semilinear heat equation this article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain when the control acts on any open and nonempty subset of or on a part of the boundary in the case of both an internal and a boundary control the approximate controllability in l p for 1 p l in the case of the interior control we also prove approximate controllability in c 0 the proof combines a variational approach to the controllability problem for linear equations and a fixed point method we also prove that the control can be taken to be of quasi bang bang form

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

some results on controllability of a nonlinear degenerate parabolic system by bilinear control in this paper we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control based on the shrinking property of the solutions we prove that the system is not globally approximately controllable furthermore we give an approximate null controllability result we also prove that the system is not globally exactly null controllable by a comparison principle

asymptotic behaviors of solutions to evolution equations in the presence of translation and scaling invariance abstract there are wide classes of nonlinear evolution equations which possess invariant properties with respect to a scaling and translations if a solution is invariant under the scaling then it is called a self similar solution which is a candidate for the asymptotic profile of general solutions at large time in this paper we establish an abstract framework to find more precise asymptotic profiles by shifting self similar solutions suitably

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

an optimal bound for the spectral variation of two matrices abstract we establish a bound for the spectral variation of two complex n n matrices a b in terms of a b and a b here denotes the spectral norm it is always better than a bound previously given by bhatia and friedland and it is optimal we describe the set of pairs a b for which the bound is attained

the block preconditioned lsqr and gl lsqr algorithms for the block partitioned matrices in this paper a new block preconditioner is proposed for the block partitioned matrices this preconditioner is based on the block c orthogonalization where c is a symmetric positive definite matrix the block preconditioned least squares bpls and block preconditioned global least squares bpgls algorithms are presented to solve the linear system of equations with block partioned coefficient matrix and multiple linear system of equations respectively the bpls algorithm is applied to solve the complex linear system of equations and also bpls and classical preconditioned least squares pls algorithms are compared finally some numerical experiments are given to show the efficiency of the new block preconditioner

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

approximate controllability of a semilinear heat equation in bbb r n we prove the approximate controllability of the semilinear heat equation in bbb r n we introduce the weighted sobolev spaces of escobedo and kavian and in that functional setting we adapt the technique introduced by fabre puel and zuazua for the problem in bounded domains that is we first prove the approximate controllability of the linear equation and by a fixed point method obtain the main result

on uniform and decay estimates for unbounded solutions of partial differential equations abstract we prove that if a function u satisfies certain integral estimates of energy type then it verifies estimates of the type u t l r c u 0 l r 0 0 t 1 t 0 where r and r 0 are exponents that appear in these integral estimates and 1 and 0 are positive constants that can be expressed in terms of these constants we will see that in some cases r r 0 supercontractive estimates and hence a regularizing effect on u appears while in other cases this improvement of regularity does not appear since r r 0 in any case we prove that the l r norm of u decays in time for t large since it verifies the previous estimate we show how to apply this result to obtain new estimates of this kind for the solutions of many nonlinear parabolic equations this new method allows also to explain the reason of the similar behavior of the solutions of very different parabolic problems finally we study sufficient conditions for extinction in finite time

time optimal control of semilinear parabolic equations via bilinear controls abstract we establish the existence of the time optimal control for semilinear parabolic equations with gradient quadratic growth via bilinear controls it is worth pointing out that there is no restriction on the growth of the nonlinearity f s with respect to the variable s in the equation which is a remarkable difference compared to the semilinear parabolic system with locally distributed controls the technique used in this paper is the combination of the hopf cole transformation the a prior estimates on solutions of parabolic equations and the strategy of the stepwise control

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

variational problems related to self similar solutions of the heat equation on demontre la compacite de certains plongements de sobolev sous de bonnes conditions sur les poids k x exp x c 2 r n r on considere des methodes variationnelles pour la resolution d equations elliptiques

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

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

on the lack of null controllability of the heat equation on the half line we consider the linear heat equation on the half line with a dirichlet boundary control we analyze the null controllability problem more precisely we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the l2 boundary control we rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems next the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials eit j in which the usual summability condition on the inverses of the eigenvalues does not hold roughly speaking we prove that controllable data have fourier coefficients that grow exponentially for large frequencies this result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a sobolev space of negative order may be driven to zero in an arbitrarily small time

on a class of non linear parabolic control systems with memory effects this work establishes the local exact null controllability of non linear parabolic integrodifferential equations with non linearities under the integral sign in a bounded domain and with homogeneous dirichlet boundary conditions the control is assumed to be distributed along a subdomain of using a carleman type inequality we first prove the global exact null controllability of a linearized equation then by a fixed point method we obtain the main result for the non linear case this result asserts that when the non linearity is of class c 1 and globally lipschitz the system is null controllable

asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces this paper considers the existence and large time behavior of solutions to the convection diffusion equation ut u b x u u q 1 f x t in n 0 where f x t is slowly decaying and q 1 1 n or in some particular cases q 1 the initial condition u0 is supposed to be in an appropriate lp space uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term

an explicit solution of burgers equation with stationary point source existence uniqueness and regularity of the global weak solution to the burgers equation with a reaction term is shown when the reaction term is given as a time independent point source and produces heat constantly an explicit solution is obtained and used to show the long time asymptotic convergence of the solution to a steady state for the heat equation case without any convection the solution diverges everywhere as time increases and hence it is the first order convection term that gives the compactness of the solution trajectory of the burgers equation with reaction

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

entropy solutions for diffusion convection equations with partial diffusivity we consider the cauchy problem for the following scalar conservation law with partial viscosity ut a u 0 f u x y e r n t o the existence of solutions is proved by the vanishing viscosity method by introducing a suitable entropy condition we prove uniqueness of solutions this entropy condition is inspired by the entropy criterion introduced by kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion introduction we present a theory of existence and uniqueness for suitable entropy solutions of a nonlinear equation which describes the combined effects of diffusion and convection of matter it has the form 0 1 ut axu ayf u posed in the space rn n 2 denoted by the variable z x y x e rn y e r the main characteristic of this equation is that it has mixed parabolichyperboolic type due to the directional separation of the diffusion and convection effects while matter is convected along the y axis it is simultaneously diffused along all orthogonal directions our interest in the model is not merely academic we have encountered it while studying in evz the asymptotic behavior of a standard nonlinear diffusion convection model of the form 0 2 ut azu ayf u in the case f u ululq l for 1 q n 1 n notice that azu axu o2u in fact certain self similar solutions of equation 0 1 will give the large time behaviour of the nonnegative solutions of 0 2 having finite mass the unique characterization of solutions of 0 1 is essential in establishing the asymptotic behaviour of 0 2 the comparison of both equations suggests the name reduced equation for 0 1 the existence of solutions of equation 0 1 can be obtained by the classical method of adding a vanishing artificial viscosity in other terms a diffusion in received by the editors december 21 1992 1991 mathematics subject classification primary 35l65 35k65 35a05

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance