asymptotic behavior of solutions for the damped wave equation with slowly decaying data abstract we consider the cauchy problem for the damped wave equation u t t u u t u 1 u t x r r n and the heat equation t 1 t x r r n if the data is small and slowly decays likely c 1 1 x k n 0 k 1 then the critical exponent is c k 1 2 k n for the semilinear heat equation in this paper it is shown that in the supercritical case there exists a unique time global solution to the cauchy problem for the semilinear heat equation in any dimensional space r n whose asymptotic profile is given by 0 t x r n e x y 2 4 t 4 t n 2 c 1 1 y 2 k n 2 d y provided that the data 0 satisfies lim x x k n 0 x c 1 0 even in the semilinear damped wave equation in the supercritical case a time global solution u with the data u u t 0 x u 0 u 1 x is shown in low dimensional spaces r n n 1 2 3 to have the same asymptotic profile 0 t x provided that lim x x k n u 0 u 1 x c 1 0 those proofs are given by elementary estimates on the explicit formulas of solutions

the behavior of the life span for solutions tout δu a x upinrd abstract let t denote the life span of the positive bounded solution u x t to the equation u t u a u u p in r d and u x 0 x where 0 a x c r d 0 x c b r d p 1 and 0 is a parameter depending on a p and d it is possible that t for 0 sufficiently small or that t 0 in which case lim 0 t it is always true that lim t 0 in this paper we investigate the asymptotic behavior of t as 0 in the case that t 0 and as in all cases the asymptotic order depends heavily on a p and d in the case that 0 whereas in the case that it depends only on whether there exists an x 0 with a x 0 x 0 0 or whether the supports of a and are separated by a positive distance

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

on superimposed training for mimo channel estimation and symbol detection channel estimation for multiple input multiple output mimo time invariant channels using superimposed training is considered a user specific periodic nonrandom training sequence is arithmetically added superimposed at a low power to each user s information sequence at the transmitter before modulation and transmission two versions of a two step approach are adopted where in the first step we estimate the channel using only the first order statistics of the data using the estimated channel from the first step a linear minimum mean square error mmse equalizer and hard decisions or a viterbi detector are used to estimate the information sequence in the second step of the two step approach a deterministic maximum likelihood dml approach based on a viterbi detector or a linear mmse equalizer based approach is used to iteratively estimate the mimo channel and the information sequences sequentially we also present a performance analysis of the first order statistics based approach to obtain a closed form expression for the channel estimation variance we then address the issue of superimposed training power allocation for complex gaussian random rayleigh channels for mimo systems arising from spatial multiplexing of a single user signal illustrative simulation examples are provided

life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity abstract we present a new upper bound of the life span of positive solutions of a semilinear heat equation for initial data having positive limit inferior at space infinity the upper bound is expressed by the data in limit inferior not in every direction but around a specific direction it is also shown that the minimal time blow up occurs when initial data attains its maximum at space infinity

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

existence and nonexistence of global solutions for a semilinear reaction diffusion system abstract this paper is concerned with the blow up and global existence of nonnegative solutions to the following cauchy problem u t u v p t 0 x r n v t v a x u q t 0 x r n u x 0 u 0 x v x 0 v 0 x x r n where the constants p q 0 and a x 0 is on the order x m as x m r the fujita critical exponent is determined when m 0 and some results of global existence of solution under some assumptions when m 0 are also obtained the results extend those in escobedo and herrero 1991 9 and indicate that m affects the fujita critical exponent

a semilinear pseudo parabolic equation with initial data non rarefied at abstract in this paper we study the cauchy problem for a semilinear pseudo parabolic equation with initial data non rarefied at our interest lies in the discussion of the effect of the non rarefied factors on the bow up phenomenon and the life span of the solution

hair triggered instability of radial steady states spread and extinction in semilinear heat equations we first study the initial value problem for a general semilinear heat equation we prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low n 10 or if the steady state is flat enough at infinity the solution of the heat equation either becomes unbounded as t approaches the lifespan or eventually stays above or below another bounded radial steady state depending on if the initial value is above or below the first steady state moreover the second steady state must be a constant if n 10 using this instability result we then prove that every nonconstant radial steady state of the generalized fisher equation is a hair trigger for two kinds of dynamical behavior extinction and spreading we also prove more criteria on initial values for these types of behavior similar results for a reaction diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained

asymptotically self similar global solutions of a general semilinear heat equation we consider the general nonlinear heat equation partial t u delta u a u p 1 1 u g u u 0 varphi on 0 infty times i r n where a in i r p 1 1 2 n and g satisfies certain growth conditions we prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation we also prove that some of those global solutions are asymptotic for large time to self similar solutions of the single power nonlinear heat equation i e with g equiv 0

self similar subsolutions and blowup for nonlinear parabolic equations abstract for a wide class of nonlinear parabolic equations of the formut u f u u we prove the nonexistence of global solutions for large initial data we also estimate the maximal existence time to do so we use a method of comparison with suitable blowing up self similar subsolutions as a consequence we improve several known results onut u up on generalized burgers equations and on other semilinear equations this method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems both semilinear and quasilinear

remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional exterior domain abstract in this paper we consider the following initial boundary value problem with the power type nonlinearity u p with 1 p 2 in a two dimensional exterior domain 0 1 t 2 u x t u x t e i t u x t u x t p x t 0 t u x t 0 x t 0 t u x 0 e f x x t u x 0 e g x x where is given by x r 2 x 1 2 2 c and 0 1 switches the parabolicity dispersivity and hyperbolicity remark that 2 1 2 n is well known as the fujita exponent if p 2 then there exists a small global in time solution of 0 1 for some initial data small enough see ikehata 11 and if p 2 then global in time solutions cannot exist for any positive initial data see ogawa takeda 22 and lai yin 14 the result is that for given initial data f g h 0 1 l 2 satisfying f g log x l 1 with some requirement the solution blows up at finite time and moreover the upper bound for lifespan of solutions to 0 1 is given as the following double exponential type when p 2 lifespan u exp exp c e 1 the crucial idea is to use test functions which approximates the harmonic function log x satisfying dirichlet boundary condition and the technique modified from 9

existence and nonexistence of global solutions of higher order parabolic problems with slow decay initial data weinberger 1 for a reference of the extensive literature on the extension of the classicalpaper by fujita in several directions we refer the reader to the survey papers by levine 10 and deng and levine 2 and to the monograph by mitidieri and pohozaev 11 the purpose of this paper is to study the problem of the existence and of the nonex istence of global solutions for problem 1 2 for initial data

the critical exponent of a reaction diffusion equation on some lie groups we shall study the global existence and blow up of the following semilinear parabolic cauchy problem 1 1 hu h0u up x i aijxju tu up 0 in rn 0 u x 0 u0 x 0 in rn where aij aij x t are bounded measurable functions and the matrix a aij is uniformly elliptic xj j 1 k are smooth vector fields in rn and x j is the formal adjoint of xj basic assumptions i we assume that xj satisfy the hormander s condition for hypoellipticity i e the rank of the lie algebra generated by x1 xp equals n given two points x y rn d x y will be the x1 xp control distance x denotes the euclidean distance d x d x 0 b x r will represent the metric ball y rn d x y 6 r properties of the distance metric balls and the fundamental solutions of h0 have been studied extensively in nsw fes ks1 2 and sa ii there are positive constants q b c and b such that 1 2 b x r br lim x d x lim d x x

nonexistence of solutions to parabolic differential inequalities with a potential on riemannian manifolds we are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact riemannian manifolds in particular we highlight the interplay between the geometry of the underlying manifold the power nonlinearity and the behavior of the potential at infinity

global and nonglobal existence for a strongly coupled parabolic system on a general domain abstract we consider the parabolic system u t u f t u r v s v t v g t u q v s in 0 t where r n is either an unbounded or bounded domain and f g c 0 we find conditions for the global existence or nonglobal existence which are expressed in terms of the behavior of s t u 0 as t where u t s t u 0 is the solution of the linear problem u t u 0 u 0 u 0

on the stability of the positive radial steady states for a semilinear cauchy problem abstract this paper is a contribution to the analysis of the cauchy problem 0 1 u t u i 1 k c i x l i u p i in r n 0 t u x 0 x in r n with initial function 0 the stability and asymptotic stability of the positive radial steady states which are positive solutions of 0 2 u i 1 k c i x l i u p i 0 have been discussed with in weighted banach spaces

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

well posedness global existence and large time behavior for hardy henon parabolic equations in this paper we study the nonlinear parabolic equation tu u a x u u tu u a x u u t 0 x rn 0 n 1 a r 0 0 min 2 n and with initial value u 0 u 0 we establish local well posedness in lq rn lq rn and in c0 rn c0 rn in particular the value q n 2 q n 2 plays a critical role for 2 n 2 n we show the existence of global self similar solutions with initial values x x x 2 x x x 2 where l rn l rn is homogeneous of degree 00 and is sufficiently small we then prove that if x x x 2 x x x 2 for x x large then the solution is global and is asymptotic in the l l norm to a self similar solution of the nonlinear equation while if x x x x x x for x x large with 2 n 2 n then the solution is global but is asymptotic in the l l norm to et x x et x x the equation with more general potential tu u v x u u v x x l rn is also studied in particular for initial data x x x 2 x x x 2 x x large we show that the large time behavior is linear if vv is compactly supported near the origin while it is nonlinear if vv is compactly supported near infinity

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0