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

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

stability of moving fronts in the ginzburg landau equation we use renormalization group ideas to study stability of moving fronts in the ginzburg landau equation in one spatial dimension in particular we prove stability of the real fronts under complex perturbations this extends the results of aronson and weinberger to situations where the maximum principle is inapplicable and constitutes a step in proving the general marginal stability hypothesis for the ginzburg landau equation

asymptotic behavior near planar transition fronts for the cahn hilliard equation abstract we consider the asymptotic behavior of perturbations of planar wave solutions arising in the cahn hilliard equation in space dimensions d 2 such equations are well known to arise in the study of spinodal decomposition a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into regions in which one component or the other is dominant with these regions separated by steep transition layers a critical feature of the cahn hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability the analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator leading in the case of stability moves into the negative real half plane with cubic scaling 3 where r d 1 denotes a fourier variable associated with spatial components transverse to the planar wave under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from planar wave solutions establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an l 1 norm of the transverse variables

asymptotic behavior of global solutions of a semilinear parabolic problem abstract we study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem u t u u p x r n t 0 u 0 u 0 x r n t 0 it is known that the nonnegative solution u t of this problem blows up in finite time for 1 p 1 2 n moreover if p 1 2 n and the norm of u 0 is small enough the problem admits a global solution in this work we use the entropy method to obtain the decay rate of the global solution u t

asymptotic behavior near transition fronts for equations of generalized cahn hilliard form we consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary pde of generalized cahn hilliard form in one space dimension such equations are well known to arise in the study of spinodal decomposition a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur resolving the mixture into its two components with their concentrations separated by sharp transition layers motivated by work of bricmont kupiainen and taskinen 5 we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions a critical feature of the cahn hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis a feature that is known to complicate the step from spectral to nonlinear stability under the assumption of spectral stability described in terms of an appropriate evans function we develop detailed asymptotics for perturbations from standing wave solutions establishing phase asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate

stability of cahn hilliard fronts we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity we prove stability of the kink solution of the cahn hilliard equation partial derivative t u partial derivative x 2 partial derivative x 2 u u 2 u 3 2 x is an element of r the proof is based on an inductive renormalization group method and we obtain detailed asymptotics of the solution as t infinity c 1999 john wiley sons inc

renormalisation group theory for two problems in linear continuum mechanics abstract we consider the problems of 1 the stress field in an infinite wedge when a moment is applied at the tip and 2 the flow of an inviscid incompressible fluid past an infinite wedge both problems are linear but nevertheless exhibit anomalous dimensions for wedge angles larger than a critical value c in 1 it is the stress field and in 2 it is the velocity potential which can have anomalous power law behaviour in distance from the tip we use these problems as simple examples to illustrate how partial differential equations can be solved using the renormalisation group

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

solutions without phase slip for the ginsburg landau equation we consider the ginsburg landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx if these solutions are not very different then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips i e which does not go through zero amplitude

renormalization group theory and variational calculations for propagating fronts we study the propagation of uniformly translating fronts into a linearly unstable state both analytically and numerically we introduce a perturbative renormalization group rg approach to compute the change in the propagation speed when the fronts are perturbed by structural modification of their governing equations this approach is successful when the fronts are structurally stable and allows us to select uniquely the numerical experimentally observable propagation speed for convenience and completeness the structural stability argument is also briefly described we point out that the solvability condition widely used in studying dynamics of nonequilibrium systems is equivalent to the assumption of physical renormalizability we also implement a variational principle due to hadeler and rothe which provides a very good upper bound and in some cases even exact results on the propagation speeds and which identifies the transition from linear to nonlinear marginal stability as parameters in the governing equation are varied

the non linear stability of front solutions for parabolic partial differential equations for the ginzburg landau equation and similar reaction diffusion equations on the line we show convergence of complex perturbations of front solutions towards the front solutions by exhibiting a coercive functional

almost global existence and transient self similar decay for poiseuille flow at criticality for exponentially long times abstract we consider nonlinear diffusion equations with critical exponent as tu x2u u3 with x r for small initial data in l1 l it is well known that almost all solutions of this system explode in finite time however we make the observation that in terms of the norm of the initial conditions it takes an exponentially long time moreover before explosion the l norm of such solutions becomes exponentially small which makes it almost impossible to observe the instability in experiments as an application we consider the long time transient self similar decay to unstable poiseuille flow at criticality for exponentially long times this together with a subcritical bifurcation and short time transient amplification is a principal obstruction in all attempts to measure the critical reynolds number for this experiment more and more precisely

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

self similar decay of localized perturbations in the integral boundary layer equation abstract the integral boundary layer equation ible arises as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane the trivial solution of the ible is linearly at best marginally stable i e it has essential spectrum at least up to the imaginary axis here we show that in the stable case this trivial solution is in fact nonlinearly stable with a burgers like self similar decay of localized perturbations the proof uses renormalization theory and the fact that in the stable case burgers equation is the amplitude equation for long small amplitude waves in the ible

large time behavior for the fast diffusion equation with critical absorption we study the large time behavior of nonnegative solutions to the cauchy problem for a fast diffusion equation with critical zero order absorption partial t u delta u m u q 0 quad quad hbox in 0 infty times real n with m c n 2 n 1

cumulative incidence association models for bivariate competing risks data association models like frailty and copula models are frequently used to analyze clustered survival data and evaluate within cluster associations the assumption of noninformative censoring is commonly applied to these models though it may not be true in many situations in this paper we consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function cif a nonparametrically identifiable quantity copula models are proposed which relate the bivariate cif to its corresponding univariate cifs similarly to independently right censored data and accommodate frailty models for the bivariate cif two estimating equations are developed to estimate the association parameter permitting the univariate cifs to be estimated either parametrically or nonparametrically goodness of fit tests are presented for formally evaluating the parametric models both estimators perform well with moderate sample sizes in simulation studies the practical use of the methodology is illustrated in an analysis of dementia associations

renormalization group analysis of nonlinear diffusion equations with time dependent coefficients and marginal perturbations in this paper we use a renormalization group rg method to study the long time asymptotics of nonlinear diffusion equations with time dependent diffusion coefficients and nonlinearities which are marginal or critical with respect to the rg operator these equations describe the time evolution of the average concentration of a passive scalar being advected by a random velocity field we prove that besides the expected diffusive behavior there is an extra logarithmic correction which is the imprint of the critical nonlinearity

renormalization theoretic analysis of non equilibrium phase transitions i the becker doring equations with power law rate coefficients we study in detail the application of renormalization theory to models of cluster aggregation and fragmentation of relevance to nucleation and growth processes we investigate the becker d oring equations originally formulated to describe and analyse non equilibrium phase transitions and more recently generalized to describe a wide range of physicochemical problems in this paper we analyse how the systematic coarse graining renormalization of the becker d oring system of equations affects the aggregation and fragmentation rate coefficients we consider the case of power law size dependent cluster rate coefficients which we show lead to only three classes of system that require analysis coagulation dominated systems fragmentation dominated systems and those where coagulation and fragmentation are exactly balanced we analyse the late time asymptotics associated with each class

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

a maximum principle for partial information backward stochastic control problems with applications this paper studies the partial information control problems of backward stochastic systems there are three major contributions made in this paper i first we obtain a new stochastic maximum principle for partial information control problems our method relies on a direct calculation of the derivative of the cost functional ii second we introduce two classes of partial information linear quadratic backward control problems for the first time and then investigate them using the maximum principle complete and explicit solutions are obtained in terms of some forward and backward stochastic differential filtering equations iii last but not least we study a class of full information stochastic pension fund optimization problems which can be viewed as a special case of our general partial information ones applying the aforementioned maximum principle we derive the optimal contribution policy in closed form and present some related economic remarks

phase turbulence in the complex ginzburg landau equation via kuramoto sivashinsky phase dynamics we study the complex ginzburg landau initial value problem for a complex field u c with r we consider the benjamin feir linear instability region we show that for all and for all initial data u 0 sufficiently close to 1 up to a global phase factor e i 0 0 r in the appropriate space there exists a unique spatially periodic solution of space period l 0 these solutions are small even perturbations of the traveling wave solution and s have bounded norms in various l p and sobolev spaces we prove that apart from corrections whenever the initial data satisfy this condition and that in the linear instability range the dynamics is essentially determined by the motion of the phase alone and so exhibits phase turbulence indeed we prove that the phase satisfies the kuramoto sivashinsky equation for times while the amplitude 1 2 s is essentially constant

anomalous scaling for three dimensional cahn hilliard fronts we prove the stability of the one dimensional kink solution of the cahn hilliard equation under d dimensional perturbations for d 3 we also establish a novel scaling behavior of the large time asymptotics of the solution the leading asymptotics of the solution is characterized by a length scale proportional to t1 3 instead of the usual t1 2 scaling typical to parabolic problems 2004 wiley periodicals inc

geometric stability analysis for periodic solutions of the swift hohenberg equation in this paper we describe invariant geometrical structures in the phase space of the swift hohenberg equation in a neighborhood of its periodic stationary states we show that in spite of the fact that these states are only marginally stable i e the linearized problem about these states has continuous spectrum extending all the way up to zero there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long time behavior of solutions near these stationary solutions in particular using this point of view we obtain a new demonstration of schneider s recent proof that these states are nonlinearly stable

asymptotics for nonlinear integral equations with a generalized heat kernel using renormalization group technique in this paper we employ the renormalization group method to study the long time asymptotics of solutions to a class of nonlinear integral equations with a generalized heat kernel the nonlinearities are classified and studied according to its role in the asymptotic behavior here we prove that the behavior in the limit as t goes to infinity remains unchanged when compared with the one in the linear case if the nonlinearities are the ones classified as irrelevant in the renormalization group sense