on the existence of maximal solution for generalized algebraic riccati equations arising in stochastic control abstract existence of maximal solution is proved for a generalized version of the well known standard algebraic riccati equations which arise in certain stochastic optimal control problems

on some iterations for optimal control of jump linear equations we consider different iterative methods for computing hermitian solutions of the coupled riccati equations of the optimal control problem for jump linear systems we have constructed a sequence of perturbed lyapunov algebraic equations whose solutions define matrix sequences with special properties proved under proper initial conditions several numerical examples are included to illustrate the effectiveness of the considered iterations

improved conditions for single point blow up in reaction diffusion systems abstract we study positive blowing up solutions of the system u t u v p v t v u q as well as of some more general systems for any p q 1 we prove single point blow up for any radially decreasing positive and classical solution in a ball this improves on previously known results in 3 directions i no type i blow up assumption is made and it is known that this property may fail ii no equidiffusivity is assumed i e any 0 is allowed iii a large class of nonlinearities f u v g u v can be handled which need not follow a precise power behavior as a side result we also obtain lower pointwise estimates for the final blow up profiles

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

supersolutions for a class of nonlinear parabolic systems abstract in this paper by using scalar nonlinear parabolic equations we construct supersolutions for a class of nonlinear parabolic systems including t u u v p x t 0 t v v u q x t 0 u v 0 x t 0 u x 0 v x 0 u 0 x v 0 x x where p 0 q 0 is a possibly unbounded smooth domain in r n and both u 0 and v 0 are nonnegative and locally integrable functions in the supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow up rate of the solutions

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

two iterative algorithms for stochastic algebraic riccati matrix equations abstract in this paper two iterative algorithms are proposed to solve stochastic algebraic riccati matrix equations arising in the linear quadratic optimal control problem of linear stochastic systems with state dependent noise in the first algorithm a standard riccati matrix equation needs to be solved at each iteration step and in the second algorithm a standard lyapunov matrix equation needs to be solved at each iteration step in the proposed algorithms a weighted average of the estimates in the last and the previous steps is used to update the estimate of the unknown variable at each iteration step some properties of the sequences generated by these algorithms under appropriate initial conditions are presented and the convergence properties of the proposed algorithms are analyzed finally two numerical examples are employed to show the effectiveness of the proposed algorithms

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

a particle system with explosions law of large numbers for the density of particles and the blow up time consider a system of independent random walks in the discrete torus with creation annihilation of particles and possible explosion of the total number of particles in finite time rescaling space and rates for diffusion creation annihilation of particles we obtain a strong law of large numbers for the density of particles in the supremum norm the limiting object is a classical solution to the semilinear heat equation t u xx u f u if f u u p 1 p 3 we also obtain a law of large numbers for the explosion time

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

asymptotic and limiting profiles of blowup solutions of the nonlinear schrodinger equation with critical power this paper is a sequel to previous ones 38 39 41 we continue the study of the blowup problem for the nonlinear schrodinger equation with critical power nonlinearity nsc we introduce a new idea to prove the existence of a blowup solution in h1 n without any weight condition and reduce the problem to a kind of variational problem our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions for a certain class of initial data the blowup solution behaves like a finite superposition of zero energy h1 bounded global in time solutions of nsc these singularities stay in a bounded region in n and one can see that the so called shoulder emerges outside these singularities as suggested by some numerical computations see e g 26 we investigate the asymptotic behavior of zero energy global in time solutions of nsc and find that such a solution behaves like a multisoliton however it is not an assemblage of free particles the solitons interact with each other 1999 john wiley sons inc

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

convergence for degenerate parabolic equations abstract we prove that any bounded global solution to a degenerate parabolic problem in one spatial dimension converges to a unique stationary state

critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation ii abstract the blowup of solutions of the cauchy problem formula is studied let k be the set of functions on r which change sign k times it is shown that for p k 1 2 k 1 k 0 1 2 any solution with u 0 k blows up in finite time if 1 p p k whereas a global solution with u 0 k exists if p p k this is an extension of our previous result 17 in which a fast decay condition was imposed on initial data it is also shown in this paper that if u0 decays more slowly than x 2 p 1 as x then the solution blows up in finite time regardless of the number of sign changes

comparisons of spectral radii and the theorem of stein rosenberg abstract the theorem of stein rosenberg is generalized to the case of two m splittings a m 1 n 1 m 2 n 2 where the m i are m matrices and n 1 n 2 0 n 1 n 2 n 2 0

blow up of solutions with sign changes for a semilinear diffusion equation abstract this paper is concerned with the initial boundary value problem formula with the dirichlet neumann or periodic boundary condition here 0 is a parameter and f is an odd function of u satisfying f 0 0 and some convexity condition let z u be the number of times of sign changes for u c 0 1 it is shown that there exists an increasing sequence of positive numbers k k 0 1 2 such that any solution with z u 0 k blows up in finite time if k and there exists a global solution with z u 0 k if k

decentralized h_ 2 control for multi channel stochastic systems this technical note investigates a decentralized h 2 control problem for multi channel linear time invariant stochastic systems governed by ito s differential equation after using a stochastic algebraic riccati equation sare to establish the necessary conditions for the existence of the optimal strategy set that minimizes an h 2 norm we show that the necessary conditions can also be expressed using a linear matrix inequality lmi the equivalence between the solvability of the sare and the feasibility of the lmi is proved for the first time using a karush kuhn tucker kkt condition furthermore the static output feedback solution was also studied a numerical example is given to demonstrate the usefulness of the obtained features

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

existence and stability of time periodic solutions in a model for spherical flames with time periodic heat losses the problem under investigation is the study of the effect of periodicity when time periodic heat losses are added in a model for spherical flames introduced by g joulin in the present context the flame either quenches in finite or infinite time or its radius converges to an upper stable time periodic solution or to a lower unstable time periodic solution

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

blowup solutions for a reaction diffusion system with exponential nonlinearities abstract we consider the following parabolic system whose nonlinearity has no gradient structure t u u e p v t v v e q u u 0 u 0 v 0 v 0 p q 0 in the whole space r n we show the existence of a stable blowup solution and obtain a complete description of its singularity formation the construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude in particular our analysis uses neither the maximum principle nor the classical methods based on energy type estimates which are not supported in this system the stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem

blow up set of type i blowing up solutions for nonlinear parabolic systems we consider the blow up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow up set our criterion enables us to obtain sufficient conditions for the boundedness of the blow up set and no boundary blow up for type i blowing up solutions

global blow up of a nonlinear heat equation solutions to semilinear parabolic equations of the form u t u f u x in which blow up at some finite time t are investigated for slowly growing functions f for nonlinearities such as f s 2 s ln 2 s 1 b with 0 u becomes infinite throughout as t t it is alsofound that for marginally more quickly growing functions e g f s 2 s ln 2 s 2 u is unbounded on some subset of which has positive measure and is unbounded throughout if is a small enough region

a symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions abstract in this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for u t u xx u p in a bounded interval 0 1 with dirichlet boundary conditions we focus in the behaviour of blowing up solutions we find that the blow up rate for the numerical scheme is the same as for the continuous problem also we find the blow up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow up set of the continuous problem when the mesh parameter is small enough

properties of stein lyapunov iterations for solving a general riccati equation abstract we consider different iterative methods for computing a hermitian or maximal hermitian solution of two types of rational riccati equations arising in stochastic control the classical newton procedure and its modification applied to equations are very expensive new less expensive iterations for these equations are introduced and some convergence properties of new iterations are proved

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

generalized stein rosenberg theorems for the regular splittings and convergence of some generalized iterative methods we study the convergence properties of the aor and gaor iterative methods for the solution of the linear system ax b and we give more general forms of the stein rosenberg theorem under the assumptions that 1 l u 0 in a i b i l u are not necessarily strictly lower and strictly upper triangular matrices respectively and 2 l 1 we derive new results regarding the spectral radii of the gaor or aor and the gj or jacobi iteration matrices and give in the plane sufficient conditions domains for the gaor and aor method to converge

convergence to periodic solutions in autonomous reaction diffusion systems in one space dimension we consider the autonomous reaction diffusion system ddt uv uv f uv uv vu t 0 0 x 1 with dirichlet boundary conditions where f c1 r2 r and is positive we show a poincare bendixson theorem which means that all solutions converge either to zero or to a periodic orbit furthermore we determine stability properties of the periodic orbits

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

on a simple maximum principle technique applied to equations on the circle abstract for a class of general quasilinear equations on s 1 we show that by a very simple maximum principle technique as long as the solution stays finite all of its derivatives also remain finite some specific examples are given under suitable assumptions we also derive exponential decay of the derivatives of the solution

convergence of solutions to the equation of quasi static approximation of viscoelasticity with capillarity the long time behavior of solutions to the quasi steady approximation of the equation of viscoelasticity with capillarity is investigated we consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one dimensional we prove that any solution converges to an equilibrium point we admit some kind of initial data with infinite energy we use tools based on the assumption that the nonlinear term is real analytic

symmetry and convergence properties for non negative solutions of nonautonomous reaction diffusion problems nonautonomous parabolic equations of the form u t u f u t on a symmetric domain are considered using the moving hyperplane method it is proved that any bounded nonnegative solution symmetrises as t this is then used to show that for nonlinearities periodic in t any non negative bounded solution approaches a periodic solution

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

technical communique solution of the state dependent noise optimal control problem in terms of lyapunov iterations in this paper we present the solution to the algebraic riccati type equation of the state dependent noise linear quadratic optimal control problem in terms of algebraic lyapunov iterations by properly initializing the sequence of algebraic lyapunov iterations we got monotonic convergence from above to the positive definite stabilizing solution of the algebraic riccati type equation the solution that represents the optimal solution to the corresponding optimal control problem the proposed algorithm requires much less computational efforts than those previously used to solve the same problem

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

analytic solutions of some nonlinear diffusion equations on considere u t u u 2 x n u 0 avec des fonctions initiales non constantes est un domaine borne de r n a frontiere lisse

brief paper soft constrained stochastic nash games for weakly coupled large scale systems in this paper we discuss infinite horizon soft constrained stochastic nash games involving state dependent noise in weakly coupled large scale systems first we formulate linear quadratic differential games in which robustness is attained against model uncertainty it is noteworthy that this is the first time conditions for the existence of robust equilibria have been derived based on the solutions of sets of cross coupled stochastic algebraic riccati equations csares after establishing an asymptotic structure with positive definiteness for csares solutions we derive a recursive algorithm by means of newton s method so that it can be used to obtain solutions for csares as another important feature we propose a high order approximate nash strategy based on iterative solutions finally we provide a numerical example to verify the efficiency of the proposed algorithms

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

single point blow up for parabolic systems with exponential nonlinearities and unequal diffusivities abstract we study positive blowing up solutions of systems of the form u t 1 u e p v v t 2 v e q u with 1 2 0 and p q 0 we prove single point blow up for large classes of radially decreasing solutions this answers a question left open in a paper of friedman and giga 1987 where the result was obtained only for the equidiffusive case 1 2 and the proof depended crucially on this assumption

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

static large deviations for a reaction diffusion model we consider the superposition of a symmetric simple exclusion dynamics speeded up in time with a spin flip dynamics in a one dimensional interval with periodic boundary conditions we prove the large deviations principle for the empirical measure under the stationary state we deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable