a global approach to fully nonlinear parabolic problems we consider the general initial boundary value problem 1 u t f x t u du du f x t x t qt 0 t 2 g x t u du g x t x t st 0 t 3 u x 0 h x x where is a bounded open set in rn with sufficiently smooth boundary the problem 1 3 is first reduced to the analogous problem in the space w 4 0 p qt with zero initial condition and f w 2 0 p qt g w 3 1 p 0 p st the resulting problem is then reduced to the problem au 0 where the operator a w 4 0 p qt w 4 0 p qt satisfies condition s this reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in sobolev spaces the local and global solvability of the operator equation au 0 are achieved via topological methods developed by i v skrypnik further applications are also given involving relevant coercive problems as well as galerkin approximations

global solutions to semilinear parabolic systems for small data abstract we consider semilinear parabolic systems u t au f u g u 0 u 0 a being the generator of an analytic semigroup with spectrum a in the right half plane for appropriate small data g u 0 global solutions are obtained if 0 belongs to the resolvent set and local existence is proved if 0 a the inverse function theorem is used in a class of functions which are holder continuous with respect to time t the asymptotic behaviour of the solution is given too as a first application we prove the existence of global respectively local solutions to the navier stokes equations in l spaces in bounded respectively unbounded domains second we consider the case where a is an elliptic operator of order 2 m and f has bounded growth third a quasilinear example where the nonlinearity involves derivatives of the same order as a is studied then an example of a nonautonomous nonlinearity is given and finally the convergence of solutions of the time dependent system to solutions of the stationary semilinear elliptic one is derived for 0 a

holder classes with boundary conditions as interpolation spaces on etudie la caracterisation de certains espaces d interpolation reels entre le domaine d un operateur differentiel elliptique avec des conditions aux limites generales et l espace de banach de fonctions continues dans lequel le domaine est plonge

parabolic evolution equations in interpolation and extrapolation spaces on etudie le probleme de cauchy pour des equations d evolution lineaires a t u f t o t t de type parabolique dans un espace de banach x 0 on considere des resultats d existence d unicite et de continuite pour a t u u f t u o t t

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

tightness of products of i i d random matrices ii in this paper we prove a necessary and sufficient condition for tightness of products of i i d finite dimensional random real matrices we give an example illustrating the use of our theorem and treat in more details the case of 2 x 2 real matrices

a rigorous reduction of the l2 stability of the solutions to a nonlinear binary reaction diffusion system of pde s to the stability of the solutions to a linear binary system of ode s abstract a basic peculiar lyapunov functional v is introduced for the dynamical systems generated by a pair of nonlinear reaction diffusion pde s with nonconstant coefficients the sign of v and of its derivative along the solutions is linked through an immediate simple relation to the eigenvalues by using v and the l 2 norm the non linear l 2 stability instability is rigorously reduced to the stability instability of the solutions to a linear binary system of ode s

predicting response bounds for friction damped gas turbine blades with uncertain friction coupling abstract friction dampers are often used to reduce high amplitude vibration within gas turbines they are a robust solution that are able to withstand extreme operating environments although the turbine blades are manufactured to tight tolerances there can be significant variability in the overall response of the assembly uncertainties associated with the frictional contact properties are a major factor contributing to this variability this paper applies a recently developed method for predicting response bounds to friction damped gas turbines when the characteristics of the friction dampers are unknown including uncertainty regarding the underlying functional form of the friction law the approach taken is to represent the frictional contact using a describing function and formulate an optimisation problem to seek upper and lower bounds on a chosen response metric such as displacement amplitude constraints are chosen that describe known properties of the frictional nonlinearity without needing to specify a particular constitutive law the method was validated by comparison with numerical and experimental results from an idealised test system the experimental test rig consisted of an array of eight beams coupled by pin contact friction dampers a modal description of this test rig formed the basis of a numerical model which uses the harmonic balance method hbm for nonlinear simulations a set of monte carlo tests was carried out numerically and experimentally for both a two beam sub assembly as well as for the full eight beam assembly comparisons with numerical results showed excellent agreement providing confident verification of the implementation and comparisons with experimental results revealed that the bounds became less conservative as the system complexity increased overall the results are promising upper and lower response bounds for an array of friction damped systems can be computed at similar cost to a single hbm simulation giving reliable bounds that are valid for both parametric and model uncertainties associated with the friction couplings

dynamic theory of quasilinear parabolic equations i abstracts evolution equations on etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

blow up problems for the heat equation with a local nonlinear neumann boundary condition abstract this paper estimates the blow up time for the heat equation u t u with a local nonlinear neumann boundary condition the normal derivative u n u q on 1 one piece of the boundary while on the rest part of the boundary u n 0 the motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects we show the finite time blow up of the solution and estimate both upper and lower bounds of the blow up time in terms of the area of 1 in many other work they need the convexity of the domain and only consider the problem with 1 in this paper we remove the convexity condition and only require to be c 2 in addition we deal with the local nonlinearity namely 1 can be just part of

response bounds for complex systems with a localised and uncertain nonlinearity predicting the vibration response of complex nonlinear structures is a significant challenge the response may involve many modes of the structure nonlinearity precludes the use of efficient techniques developed for linear systems and there is often uncertainty associated with the nonlinear law even to the extent that its functional form is not always known this paper builds on a recently developed method for handling this class of problem in a novel way the method exploits the fact that nonlinearities are often spatially localised and seeks the best and worst case system response with respect to a chosen metric by regarding the internal nonlinear force as an independent excitation to the underlying linear system constraints are used to capture what is thought to be known about the nonlinearity without needing to specify a particular law this paper focuses on the case of systems with a single point nonlinearity but with arbitrarily complex underlying linear dynamics driven by a sinusoidal force excitation semi analytic upper and lower bounds are proposed for root mean square response metrics subject to constraints which specify that the nonlinearity should be a combination of a passive b displacement limited and or c force saturating the concept of equivalent linear bounds is also introduced for cases where the response metric is thought to be dominated by the same frequency as the input the bounds corresponding to a passive and displacement limited nonlinearity are compared with monte carlo experimental and numerical results from an impacting beam test rig the bounds corresponding to a passive and force saturating nonlinearity are compared with numerical results for a friction damped beam the global upper and lower bounds are satisfied for all input frequencies but are generally found to be rather conservative the equivalent linear bounds show remarkably good agreement for predicting the range of root mean square velocity responses finally the principle of maximum entropy is used to estimate the response distributions which was found to give surprisingly good agreement with experimental and numerical data

recurrent random walks in nonnegative matrices ii in this paper we continue the study undertaken in our earlier paper m1 one of the main results here can be described as follows letx0 x1 be a sequence of iid random affine maps from r d into itself let us write w n x n x n 1 x0 andz n x0x1 x n where composition of maps is the rule of multiplication by the attractora u u r d we mean the setau y r d p wnu n i o 0 for every openn containingy it is shown that the attractora u under mild conditions is the support of a stationary probability measure when the random walk z n has at least one recurrent state

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

well posedness and stability results for a quasilinear periodic muskat problem abstract we study the muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two dimensional geometry we first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown secondly we show that this evolution equation has a quasilinear structure which is at a formal level not obvious and we also disclose the parabolic character of the equation exploiting these aspects we establish the local well posedness of the problem for arbitrary initial data in h s s with s 3 2 2 determine a new criterion for the global existence of solutions and uncover a parabolic smoothing property besides we prove that the zero steady state solution is exponentially stable

blow up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions abstract by constructing different auxiliary functions and using hopf s maximum principle the sufficient conditions for the blow up and global solutions are presented for nonlinear parabolic equation u t a u b x c t u f x u q t with different kinds of boundary conditions the upper bounds of the blow up time and the upper estimates of global solutions are provided finally some examples are presented as the application of the obtained results

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given

convergence in distribution of products of random matrices a semigroup approach on considere le probleme de la convergence faible de la suite des puissances de convolution d une mesure de probabilite dans le contexte general d un semigroupe non compact

blow up solution for the degeneratequasilinear parabolic system with nonlinear boundary condition in this paper we discuss the global existence and blow up of the solution to the quasilinear parabolic system with nonlinear boundary condition we also discuss the blow up set of the blow up solution and obtain the blow up rate of the blow up solution and prove the existence of the self similar blow up solution

global existence of solutions of the liquid crystal flow for the oseen frank model in r2 abstract in the first part of this paper we establish the global existence of solutions of the liquid crystal gradient flow for the well known oseen frank model the liquid crystal flow is a prototype of equations from the ericksen leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2 sphere the ericksen leslie system is a system of the navier stokes equations coupled with the liquid crystal flow in the second part of this paper we also prove the global existence of solutions of the ericksen leslie system for a general oseen frank model in r 2

bounds on the maximum modulus of solutions to quasilinear parabolic equations by means of a modified lyapunov method on obtient des bornes l aux solutions des equations paraboliques d ordre 2 de la forme u t x t u u x t u x t q ou toutes les quantites sont reelles q 0 t un ouvert borne de r n 0 t et ou l application non lineaire peut avoir une nature non locale

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

blow up solutions and global solutions for nonlinear parabolic problems abstract this paper deals with the blow up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions we obtain the conditions under which the solutions may exist globally or blow up in a finite time by a new approach moreover upper estimates of the blow up time blow up rate and global solutions are obtained also

the principle of linearized stability for quasilinear parabolic evolution equations on donne des conditions pour la stabilite des points d equilibre des equations paraboliques quasilineaires avec des conditions aux limites non lineaires u a t u u u f t u 7b b t u u 0 u t 0 u 0 t 0 t

improvements on lower bounds for the blow up time under local nonlinear neumann conditions abstract this paper studies the heat equation u t u in a bounded domain r n n 2 with positive initial data and a local nonlinear neumann boundary condition the normal derivative u n u q on partial boundary 1 for some q 1 while u n 0 on the other part we investigate the lower bound of the blow up time t of u in several aspects first t is proved to be at least of order q 1 1 as q 1 since the existing upper bound is of order q 1 1 this result is sharp secondly if is convex and 1 denotes the surface area of 1 then t is shown to be at least of order 1 1 n 1 for n 3 and 1 1 ln 1 1 for n 2 as 1 0 while the previous result is 1 for any 1 n 1 finally we generalize the results for convex domains to the domains with only local convexity near 1

blow up and global solutions for nonlinear reaction diffusion equations with nonlinear boundary condition abstract this paper deals with the blow up of positive solutions for a nonlinear reaction diffusion equation subject to nonlinear boundary conditions we obtain the conditions under which the solutions may exist globally or blow up in finite time moreover an upper bound of the blow up time an upper estimate of the blow up rate and an upper estimate of the global solutions are given at last we give two examples to which the theorems obtained in the paper may be applied

semigroups and nonlinear evolution equations abstract in the main part of this paper we survey some recent results on a semigroup approach to parabolic equations under nonlinear boundary conditions in particular we give a detailed motivation and a heuristic derivation of a generalized variation of constant formula which is the basis for the whole method in the appendix we prove some new results about semigroups in interpolation and extrapolation spaces

on the solutions of the first boundary value problem for the linear parabolic equations on etudie le premier probleme aux valeurs limites pour une equation parabolique lineaire d ordre 2 sous l hypothese que le terme inhomogene est continu en espace et en temps et continu de holder seulement par rapport aux variables d espace

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

on quasilinear parabolic systems on demontre certains resultats sur l existence locale des solutions continument differentiables u u 1 u n de systemes paraboliques quasilineaires sous des conditions aux limites non lineaires generales

quasilinear reaction diffusion systems with nonlinear boundary conditions this paper deals with the existence and nonexistence of global positive solutions of quasilinear reaction diffusion systems with nonlinear boundary conditions necessary and sufficient conditions on the global existence of all positive solutions are obtained

doubly nonlinear degenerate parabolic systems with coupled nonlinear boundary conditions in this paper we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions we first prove a local existence result by the regularization method next we construct a weak comparison principle then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions necessary and sufficient conditions on the global existence of all positive weak solutions are obtained

uniform blow up profile for a degenerate parabolic system with nonlocal source in this paper we establish the local existence of the solution and the finite time blow up result for the following system ut ur1 u a updx forx ohm t 0 ut ur2 u b uqdx forx ohm t 0 where p q 1 and 0 r1 r2 1 moreover it is proved that the solution has global blow up and limt t t t p 1 r2 yu x t 1 r1 11 r1 a a y p 1 r2 y q 1 r1 py xa p 1 r2 b1 r2 y limt t t t q 1 r1 yu x t 1 r2 11 r2 b a y q 1 r1 y p 1 r2 qy xb q 1 r1 a1 r1 y uniformly on compact subsets of where pq 1 r1 1 r2 and t is the blow up time

quasilinear evolution equations and parabolic systems it is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values provided the boundary conditions are time independent moreover it is shown that in the autonomous case these equations generate local semi flows on appropriate sobolev spaces our results apply in particular to the case of prescribed boundary values dirichlet boundary conditions

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

holder regularity in variational parabolic non homogeneous equations abstract we prove optimal holder regularity results for the solutions to second order variational parabolic equations with holder continuous coefficients and non homogeneous boundary data using analytic semigroups methods

blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition abstract in this paper we study the large time behavior of positive solutions of the heat equation under the nonlinear boundary condition u f u where is the outward normal and f is nondecreasing with f u 0 for u 0 we show that if b r and 1 f is integrable at infinity there is finite time blow up for any initial datum in the two dimensional case we show that this is true for any smooth simply connected domain in the radially symmetric case if f c2 is convex and satisfies the properties above we show that blow up occurs only at the boundary

multifidelity importance sampling estimating statistics of model outputs with the monte carlo method often requires a large number of model evaluations this leads to long runtimes if the model is expensive to evaluate importance sampling is one approach that can lead to a reduction in the number of model evaluations importance sampling uses a biasing distribution to sample the model more efficiently but generating such a biasing distribution can be difficult and usually also requires model evaluations a different strategy to speed up monte carlo sampling is to replace the computationally expensive high fidelity model with a computationally cheap surrogate model however because the surrogate model outputs are only approximations of the high fidelity model outputs the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high fidelity model we introduce a multifidelity importance sampling mfis method which combines evaluations of both the high fidelity and a surrogate model it uses a surrogate model to facilitate the construction of the biasing distribution but relies on a small number of evaluations of the high fidelity model to derive an unbiased estimate of the statistics of interest we prove that the mfis estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself the mfis method can be used with any type of surrogate model such as projection based reduced order models and data fit models furthermore the mfis method is applicable to black box models i e where only inputs and the corresponding outputs of the high fidelity and the surrogate model are available but not the details of the models themselves we demonstrate on nonlinear and time dependent problems that our mfis method achieves speedups of up to several orders of magnitude compared to monte carlo with importance sampling that uses the high fidelity model only

a parabolic system model for the formation of porous silicon existence uniqueness and stability in this paper a nonlinear weakly coupled parabolic system is constructed to model porous silicon formation the model considers physical and chemical factors such as charge carriers transport ions transport electrochemical reactions and material balance in order to prove the existence of a unique solution for the proposed parabolic model with zero neumann boundary we use a monotone method based on the upper and lower solutions the asymptotic stability of the steady state solution is also demonstrated and the behavior of this model is shown for specific cases where the analytical stationary solution is known finally a discussion of the results is carried out and we conclude that the proposed model reproduces important qualitative features which are observed experimentally

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

group actions on strongly monotone dynamical systems si l action du groupe g est monotone alors la symetrie est incluse dans le comportement non chaotique

numerical and experimental validation of variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea approach to vibration modeling a complex system is represented as an assembly of coupled subsystems and the method leads to the prediction of the vibrational energy level of each subsystem the averaging procedures implicit in the technique imply that the predicted energy is the mean value taken over an ensemble of random structures such as a set of vehicles leaving a production line recently a new method has been developed to allow the ensemble variance in addition to the mean to be predicted within the context of sea and the present paper concerns further extension and validation of this work the theoretical extension concerns the variance of the energy density at a single point in any of the subsystems and the validation includes both simulation and experimental studies the simulation results concern plate assemblies while experimental results are presented both for a single plate and for a cylinder plate structure in each case an ensemble of random structures has been generated by adding small point masses at random locations on the structure in general good agreement between the predictions and the validation results is observed

regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in lp abstract the equation lu x u on b 0 b bounded smooth domain in r n with nonlinear boundary conditions u v g x u on b 0 is studied l being the uniformly parabolic operator with time independent coefficients under suitable conditions on the nonlinearities that do not involve monotonicity global existence uniqueness compactness of the orbits and certain regularizing effects of the semigroup are established in the case that l is in divergence form it is shown that under generic conditions orbits tend as t to some equilibrium and that the stable equilibria attract essentially baire category the whole space l2 b

recurrent random walks in nonnegative matrices attractors of certain iterated function systems in this paper the structure of the set of recurrent points for random walks in finite dimensional nonnegative matrices is determined the structural results are then used in understanding attractors of certain not necessarily contractive iterated function systems

a novel method for predicting the response variability of friction damped gas turbine blades abstract predicting the response of gas turbine blades with underplatform friction dampers is challenging due to the combination of frictional nonlinearity and system uncertainty a traditional monte carlo approach to predicting response distributions requires a large number of nonlinear simulations which is computationally expensive this paper presents a new approach based on the principle of maximum entropy that provides an estimate of the response distribution that is approximately two orders of magnitude faster than monte carlo harmonic balance method simulations the premise is to include the concept of computational uncertainty incorporating lack of knowledge of the solution as part of the uncertainty on the basis that there are diminishing returns in computing precise solutions to an uncertain system to achieve this the method uses a describing function approximation of the friction damped part of the system chooses an ignorance prior probability density function for the complex value of the describing function based on coulombs friction law updates the distribution using an estimate of the mean solution the admissible domain of solutions and the principle of maximum entropy then carries out a linear monte carlo simulation to estimate the response distribution the approach is validated by comparison with hbm simulations and experimental tests using an idealised academic system consisting of a periodic array of beams with controllable uncertainty coupled by single point friction dampers comparisons with two and eight blade systems show generally good agreement predicting the response statistics of the maximum blade amplitude reveals specific well understood circumstances when the method is less effective predictions of the overall blade response statistics agree with monte carlo hbm extremely well across a wide range of excitation amplitudes and uncertainty levels critically experimental comparisons reveal the care that is needed in accurately characterising uncertainty in order to obtain agreement of response percentiles the new method allowed fast iteration of uncertainty parameters and correlations to achieve good agreement which would not have been possible using traditional methods

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

analysis of a population model with strong cross diffusion in unbounded domains we study a parabolic population model in the full space and prove the global in time existence of a weak solution this model consists of two strongly coupled diffusion equations describing the population densities of two competing species the system features intrinsic growth inter and intra specific competition of the species as well as diffusion cross diffusion and self diffusion and drift terms related to varying environment quality the cross diffusion terms can be large making the system non parabolic for large initial data for the first time solutions in unbounded domains are studied the method of our proof is a combination of a time semi discretization a transformation of the system guaranteeing the positivity of the solution a special entropy symmetrizing the system and compactness arguments 2000 mathematics subject classification 35k55 primary 92d25 35d05 secondary

global existence for semilinear parabolic systems on etudie des problemes aux valeurs limites et initiales paraboliques semilineaires de la forme u t a t u f t x u du d 2m 1 u dans t 0 b t u 0 sur t 0 u t 0 u 0 sur ou est un domaine de r n

quasilinear parabolic equations and the ricci flow on manifolds with boundary the first part of the paper discusses a second order quasilinear parabolic equation in a vector bundle over a compact manifold m with boundary m we establish a short time existence theorem for this equation the second part of the paper is devoted to the investigation of the ricci flow on m we propose a new boundary condition for the flow and prove two short time existence results

estimates of solutions of impulsive parabolic equations under neumann boundary condition abstract in this paper we prove the relation v t u t x w t where u t x is the solution of an impulsive parabolic equations under neumann boundary condition u t x 0 and v t and w t are solutions of two impulsive ordinary equations we also apply these estimates to investigate the asymptotic behavior of a model in the population dynamics and it is shown that there exists a unique solution of the model which converges to the periodic solution of an impulsive ordinary equation asymptotically

blow up solutions and global solutions for a class of quasilinear parabolic equations with robin boundary conditions the type of problem under consideration isu email protected email protected a u b x u g x t f u indx 0 t email protected email protected x t u 0 email protected dx 0 t u x 0 u 0 x 0 ind where d is a smooth bounded domain of r n by constructing an auxiliary function and using hopf s maximum principles on it existence theorems of blow up solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a b f g s and initial data u 0 x the obtained results are applied to some examples in which a b f g and s are power functions or exponential functions

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse