existence of positive solutions for some nonlinear parabolic systems in exteriors domains we prove some existence results of positive continuous solutions to the semilinear parabolic system u u t p x t g v v v t q x t f u in an unbounded domain d with compact boundary subject to some dirichlet conditions where and are nonnegative parameters the functions f g are nonnegative continuous monotone on 0 and the potentials p q are nonnegative and satisfy some hypotheses related to the parabolic kato class j d

blow up and global existence for a nonlocal degenerate parabolic system abstract this paper investigates the blow up and global existence of nonnegative solutions of the system u t u m a v p v t v n b u q x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 1 p q a b 0 and dx it is proved that if pq mn there exist both global and blow up nonnegative solutions when pq mn we show that if the domain is sufficiently small every nonnegative solution is global while if the domain is large enough that is if it contains a sufficiently large ball there exists no global solution in particular when p n q m we show that every positive solution exists globally iff x dx 1 ab where x is the unique positive solution of the linear elliptic problem x 1 x x 0 x

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

numerical method of estimating the blow up time and rate of the solution of ordinary differential equations an application to the blow up problems of partial differential equations a numerical method is proposed for estimating the blow up time and the blow up rate of the solution of ordinary differential equation ode when the solution diverges at a finite time that is the blow up time the main idea is to transform the ode into a tractable form by the arc length transformation technique s moriguti c okuno r suekane m iri k takeuchi ikiteiru suugaku suuri kougaku no hatten in japanese baifukan tokyo 1979 and to generate a linearly convergent sequence to the blow up time the sequence is then accelerated by the aitken 2 method the present method is applied to the blow up problems of partial differential equations pdes by discretising the equations in space and integrating the resulting odes by an ode solver that is the method of lines approach numerical experiments on the three pdes the semi linear reaction diffusion equation the heat equation with a nonlinear boundary condition and the semi linear reaction diffusion system show the validity of the present method

global existence of solutions for a weakly coupled system of semilinear damped wave equations abstract in this paper we consider the cauchy problem for a weakly coupled system of semilinear damped wave equations we prove the global existence of solutions for small data in the supercritical case for any space dimension we also give estimates of the weighted energy of solutions and in a special case we prove an almost optimal estimate moreover in the subcritical case we give an almost optimal estimate of the lifespan from both above and below

critical blowup and global existence numbers for a weakly coupled system of reaction diffusion equations let d rn be either all of rn or else a cone in rn whose vertex we may take to be at the origin without loss of generality let pi qj i 1 2 be nonnegative with 0 p1 q1 p2 q2 we consider the long time behavior of nonnegative solutions of the system u t delta u u p 1 v q 1 v t delta v u p 2 v q 2 s in d 0 with u0 v0 0 on d u v t x 0 0 0 t x u0 0 0 u0 0 e l d we obtain fujita type global existence global nonexistence theorems for s analogous to the classical result of fujita and others for the initial value problem for ut u up u x 0 u0 x 0 the principal result in the case d rn and p2q1 0 is that when p1 1 the system behaves like the single equation ut u up1vq1 with respect to fujita type blowup theorems whereas if p1 1 the behavior of the system is more complicated some of the results extend those of escobedo herrero when d rn and of levine when d is a cone these authors considered s in the case of p1 q2 0 an example of nonuniqueness is also given

nonexistence of global solutions of a nonlinear hyperbolic system consider the initial value problem utt au ivip vtt av u q x e r t t o u x 0 f x v x 0 h x ut x 0 g x vt x 0 x with 1 0 we show that there exists a bound b n oo such that if 1 pq b n all nontrivial solutions with compact support blow up in finite time

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

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

weighted lp estimates and fujita exponent for a nonlocal equation abstract we investigate a nonlocal equation t u r n j x y u y t d y j l 1 u x t a x t u p in r n where a is unbounded and j belongs to a weighted space crucial weighted l p and interpolation estimates for the green operator are established by a new method based on the sharp young s inequality the asymptotic behavior of a regular varying coefficients exponential series and the properties of auxiliary functions 1 x 2 b 2 that j and b 2 x b b 2 blow up behaviors are investigated by employing test functions r r instead of principal eigenfunctions global well posedness in weighted l p spaces for the cauchy problem is proved when a x the fujita exponent is shown to be 1 2 n our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

pbw deformations of koszul algebras over a nonsemisimple ring let b be a generalized koszul algebra over a finite dimensional algebra s we construct a bimodule koszul resolution of b when the projective dimension of s b equals two using this we prove a poincare birkhoff witt pbw type theorem for a deformation of a generalized koszul algebra when the projective dimension of s b is greater than two we construct bimodule koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and koszul algebras and then prove the pbw type theorem the results obtained can be applied to standard koszul artin schelter gorenstein algebras in the sense of minamoto and mori adv math 226 4061 4095 2011

homological properties of balanced cohen macaulay algebras a balanced cohen macaulay algebra is a connected algebra a having a balanced dualizing complex a d in the sense of yekutieli 1992 for some integer d and some graded a a bimodule a we study some homological properties of a balanced cohen macaulay algebra in particular we will prove the following theorem theorem 0 1 let a be a noetherian balanced cohen macaulay algebra and m a nonzero finitely generated graded left a module then 1 m has a finite resolution of the form 0 a l mj a l 1j h m 0 where h is a finitely generated maximal cohen macaulay graded left a module 2 m has finite injective dimension if and only if m has a finite resolution of the form 0 a l mj a l 1j a l 0j m 0 as a corollary we will have the following characterizations of as gorenstein algebras and as regular algebras corollary 0 2 let a be a noetherian balanced cohen macaulay algebra 1 a is as gorenstein if and only if a has finite projective dimension as a graded left a module 2 a is as regular if and only if every finitely generated maximal cohen macaulay graded left a module is free

critical exponent for the cauchy problem to the weakly coupled damped wave system abstract in this paper we consider a system of weakly coupled semilinear damped wave equations we determine the critical exponent for any space dimensions our proof of the global existence of solutions for supercritical nonlinearities is based on a weighted energy method whose multiplier is appropriately modified in the case where one of the exponent of the nonlinear term is less than the so called fujita s critical exponent we also give estimates of the lifespan of solutions from above for subcritical nonlinearities

blow up for a system ofheat equations with nonlocal sources and absorptions in this paper we establish the local existence of the nonnegative solution and the finite time blow up results for the following parabolic system u t du wv pdx au r x w t 0 v t dv wv qdx bv s x w t 0 subject to homogeneous dirichlet conditions and nonnegative initial data where p q r s 1 and a b 0 in the situation when the nonnegative solution u v of the above problem blows up in finite time we show that the blow up is global and this differs from the local source case moreover it is proved that for the special case r s 1 if u v blows up then the limits lim t t t t p 1 p q 1 u x t p 1 1 p q 1 q 1 p p q 1 x pq 1 w p 1 p q 1 lim t t t t q 1 p q 1 v x t p 1 q p q 1 q 1 1 p q 1 x pq 1 w q 1 p q 1 converge uniformly on compact subsets of w where t is the blow up time

a fast solver for the ornstein zernike equations in this paper we report on the design and analysis of a multilevel method for the solution of the ornstein zernike equations and related systems of integro algebraic equations our approach is based on an extension of the atkinson brakhage method with newton gmres used as the coarse mesh solver we report on several numerical experiments to illustrate the effectiveness of the method the problems chosen are related to simple short ranged fluids with continuous potentials speedups over traditional methods for a given accuracy are reported the new multilevel method is roughly six times faster than newton gmres and 40 times faster than picard

global existence and finite time blow up for a degenerate reaction diffusion system abstract this paper investigates the blow up and global existence of solutions of the degenerate reaction diffusion system u t u m u v p v t v n u q v x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 0 and p q 0 it is proved that if m n and pq m n every nonnegative solution is global whereas if m or n or pq m n there exist both global and blow up nonnegative solutions when m n and pq m n we show that there exists 1 which depends on the parameters p q m n such that all positive solutions are global if 1 while if 1 1 all positive solutions blow up in finite time where 1 is the first dirichlet eigenvalue for the laplacian on

global and blow up solutions for nonlinear degenerate parabolic systems with crosswise diffusion abstract in this paper the nonlinear degenerate parabolic system u t v 1 u xx au v t u 2 v xx bv with dirichlet boundary condition is studied the regularization method and upper lower solutions technique are employed to show the local existence of a solution for the nonlinear degenerate parabolic system the global existence of a solution is discussed the finite time blow up result together with an estimate of the blow up time are found the blow up set with positive measure is analyzed in some detail

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

existence and nonexistence of the global solution for the semilinear parabolic system on riemannian manifold abstract in this paper we study the global existence and nonexistence of solutions to the following semilinear parabolic system u t u v p in m n 0 v t v u q in m n 0 u x 0 u 0 x in m n v x 0 v 0 x in m n where m n n 3 is a non compact complete riemannian manifold is the laplace beltrami operator we assume that both u 0 x and v 0 x are nonnegative smooth and bounded functions constants p q 1

fujita type phenomenon of nonlinear coupled nonlocal diffusion system abstract this paper studies a cauchy problem for a nonlinear coupled nonlocal diffusion system u t j u u v p v t j v v u q we first deal with the critical fujita exponent and then determine the second critical exponent to characterize the critical space decay rate of initial data in the co existence region of global and non global solutions it turns out that the results are coincident with the classical heat system we not only generalize the results in the scalar case obtained by j garcia melian and f quiros 2010 6 to the case of a system but also delete the monotonicity condition on the kernel function j the key techniques are modifying the rescaled test function method with a new test function rather than the eigenfunction of the nonlocal operator to deal with blow up of the solution and selecting a suitable banach space via the estimates obtained by j terra and n wolanski 2011 13 to obtain the global solutions

the form of blow up for nonlinear parabolic equations on considere des equations paraboliques semi lineaires de la forme u t 2 u sf u ou f est positive et 0 ds f s est finie on etudie des problemes a 1 dimension dans la region semi infinie x o et on trouve le temps t

homological identities for noncommutative rings abstract the auslander buchsbaum formula and bass s theorem are proved for a class of nonlocal noncommutative algebras including certain affine pi hopf algebras we also show that each filtered noetherian affine pi hopf algebra has finite injective dimension which partially answers a question of brown

a remark on the radial and nonradial solutions for an elliptic system with a quadratic gradient term recently the authors yang and zhang in y yang x zhang entire blow up solutions of semilinear elliptic system with quadratic gradient term abstract and applied analysis 2012 have been obtained sufficiently conditions for the existence and non existence of radially symmetric solutions for the system of equations du i u i 2 a i x f i u 1 u d on r n n 3 with i 1 d and d 1 2 3 4 our aim in this paper is to give a necessary and a sufficient condition but for the existence of large or explosive solutions for the same class of systems moreover we intend to obtain a non radial solution for the considered problem we also point out that our framework in the obtained results improve and complete some of the results obtained by yang and zhang

global existence and finite time blow up for a parabolic system on hyperbolic space in this paper we study the global existence and finite time blow up of positive solutions for a parabolic system on hyperbolic space using the heat semigroup and constructing subsolutions and supersolutions we obtain the fujita type results in the case of a critical exponent the critical exponent is not a blow up exponent

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

finite regularity and koszul algebras we determine the positively graded commutative algebras over which the residue field modulo the homogeneous maximal ideal has finite castelnuovo mumford regularity they are the polynomial rings in finitely many indeterminates over koszul algebras this proves a conjecture of avramov and eisenbud we also show that if the residue field of a finitely generated graded algebras has finite regularity then so do all finitely generated graded modules

blow up for a semilinear non local diffusion system abstract in this paper we study the blow up phenomena for the system u t j x y u y d y u u v p x t 0 t v t j x y v y d y v u q v x t 0 t under certain hypothesis we prove that the solution blows up infinite time and we classify in terms of the parameters when both components blow up together simultaneous blow up or when only one component blows up non simultaneous blow up we also prove the blow up rate for a particular case

a nonlocal in time parabolic system whose fujita critical exponent is not given by scaling abstract we consider the nonlocal coupled parabolic system u t u 0 t t s 1 v p 1 v s d s v t v 0 t t s 2 u q 1 u s d s in 0 t when either r n or r n is a bounded domain with smooth boundary we assume that 0 1 2 1 p q 1 and the initial data u 0 v 0 c 0 we obtain the fujita critical exponent for the system above which is not given by the scaling argument

a critical exponent of fujita type for a nonlinear reaction diffusion system on a riemannian manifold abstract in this paper we study the global existence and nonexistence of positive solutions to the following nonlinear reaction diffusion system u t u w x v p s x in m n 0 v t v f x u d g x in m n 0 u x 0 u 0 x in m n v x 0 v 0 x in m n where m n n 3 is a non compact complete riemannian manifold is the laplace beltrami operator and s x g x are non negative l l o c 1 functions we assume that both u 0 x and v 0 x are non negative smooth and bounded functions and constants p d 1 when p d there is an exponent p which is critical in the following sense when p 1 p the above problem has no global positive solution for any non negative constants s x g x not identically zero when p p the problem has a global positive solution for some s x g x 0 and u 0 x v 0 x 0

global existence and nonexistence of solutions for a system of nonlinear damped wave equations abstract we consider the cauchy problem for the system of semilinear damped wave equations with small initial data t 2 u 1 u 1 t u 1 u k p 1 t 0 x r n t 2 u 2 u 2 t u 2 u 1 p 2 t 0 x r n t 2 u k u k t u k u k 1 p k t 0 x r n u j 0 x a j x t u j 0 x b j x x r n 1 j k we show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data the proof of the global existence is based on the l p l q estimates of fundamental solutions for linear damped wave equations k nishihara l p l q estimates of solutions to the damped wave equation in 3 dimensional space and their application math z 244 2003 631 649 k marcati p nishihara the l p l q estimates of solutions to one dimensional damped wave equations and their application to compressible flow through porous media j differential equations 191 2003 445 469 t hosono t ogawa large time behavior and l p l q estimate of 2 dimensional nonlinear damped wave equations j differential equations 203 2004 82 118 t narazaki l p l q estimates for damped wave equations and their applications to semilinear problem j math soc japan 56 2004 585 626 and the blow up is shown by the fujita kaplan zhang method q zhang a blow up result for a nonlinear wave equation with damping the critical case c r acad sci paris 333 2001 109 114 f sun m wang existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping nonlinear anal 66 12 2007 2889 2910 t ogawa h takeda non existence of weak solutions to nonlinear damped wave equations in exterior domains nonlinear anal 70 10 2009 3696 3701

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

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

regularity of modules over a koszul algebra abstract we prove that every module over a commutative homogeneous koszul algebra has regularity bounded by its regularity over a polynomial ring of which the koszul algebra is a homomorphic image from this we derive a result conjectured by george kempf to the effect that a sufficiently high truncation of any module over a homogeneous koszul algebra has a linear free resolution

global nonexistence for the cauchy problem of some nonlinear reaction diffusion systems we first consider the parabolic equationut 2um a x uq f t x u p w t x t 0 x rn where 2 is the 2 fractional power of the laplacian which for 0 2 stands for impurities and f t x and w t x are given nonnegative functions and find its critical exponent then we consider the criticality for five systems of strongly coupled parabolic equations two of them with nonlinear convective terms our results answer positively some open problems raised recently by bandle deng levine and zhang

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

algorithms for the computation of solutions of the ornstein zernike equation we introduce a robust and efficient methodology to solve the ornstein zernike integral equation using the pseudoarc length pal continuation method that reformulates the integral equation in an equivalent but nonstandard form this enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so called branch points prevents newton s method from converging we illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations namely the evaluation of the so called no solution line of the ornstein zernike hypernetted chain hnc integral equation for the lennard jones potential we are able to use the pal algorithm to solve the integral equation along this line and to connect physical and nonphysical solution branches both isotherms and isochores where appropriate we also show that pal continuation can compute solutions within the no solution region that cannot be computed when newton and picard methods are applied directly to the integral equation while many solutions that we find are new some correspond to states with negative compressibility and consequently are not physical

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

blow up phenomena for a nonlinear reaction diffusion system with time dependent coefficients abstract we investigate the blow up phenomena for the solution to a nonlinear reaction diffusion system with time dependent coefficients subject to null dirichlet boundary conditions by virtue of kaplan s method method of subsolutions and supersolutions and modified differential inequality technique we establish the blow up criteria for the solution moreover lower and upper bounds for the blow up time are derived

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

blow up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions abstract in this paper we consider the following reaction diffusion systems with gradient nonlinearity under nonlinear boundary condition u t u u p v q u x t 0 t v t v v r u s v x t 0 t u g u v h v x t 0 t u x 0 u 0 x v x 0 v 0 x x where r n n 1 is a bounded region with smooth boundary p q r s 0 1 t is a possible blow up time when blow up occurs by constructing an appropriate auxiliary functions and by means of payne weinberger or scott s method a lower bound on blow up time when blow up occurs is derived

on the stability of equilibria to weakly coupled parabolic systems in unbounded domains abstract we investigate weakly coupled semilinear parabolic systems in unbounded domains of r 2 or r 3 with general nonlinearities we present several sufficient conditions on the nonlinearities which ensure the stability of the zero solution with respect to h 2 perturbations in addition various examples are discussed to illustrate the scope of application of our results

coupled diffusion systems with localized nonlinear reactions this paper deals with the blowup rate and profile near the blowup time for the system of diffusion equations uit ui upii 1 x0 t i 1 k uk 1 u1 in 0 t with boundary conditions ui 0 on 0 t we show that the solution has a global blowup the exact rate of the blowup is obtained and we also derive the estimate of the boundary layer and on the asymptotic behavior of the solution in the boundary layer

properties of the fixed ring of a preprojective algebra abstract for a finite group acting on a polynomial ring the chevalley shephard todd theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo reflections in recent years progress was made in work of kirkman kuzmanovich zhang and others to extend this result to artin schelter regular algebras by expanding pseudo reflections to quasi reflections naturally the question arises if the theory generalizes further to non connected noncommutative algebras our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended dynkin diagrams of type a d or e this work answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible on our way we will point out additional difficulties in establishing quasi reflections using the trace and reveal situations which do not occur for artin schelter regular algebras

critical exponents and lower bounds of blow up rate for a reaction diffusion system abstract this paper is concerned with a reaction diffusion system with absorption terms under dirichlet boundary conditions modelling the cooperative interaction of two diffusion biological species by constructing blow up sub solutions and bounded super solutions we obtain the optimal conditions on the exponent of reaction and absorption terms for the existence or nonexistence of global solutions moreover for a special case we derive the lower bound estimates of blow up rate

non commutative castelnuovo mumford regularity and as regular algebras abstract let a be a connected graded k algebra with a balanced dualizing complex we prove that a is a koszul as regular algebra if and only if that the castelnuovo mumford regularity and the ext regularity coincide for all finitely generated a modules this can be viewed as a non commutative version of t romer on the regularity over positively graded algebras j algebra 319 2008 1 15 theorem 1 3 by using castelnuovo mumford regularity we prove that any koszul standard as gorenstein algebra is as regular as a preparation to prove the main result we also prove that the following statements are equivalent 1 a is as gorenstein 2 a has finite left injective dimension 3 the dualizing complex has finite left projective dimension this generalizes i mori homological properties of balanced cohen macaulay algebras trans amer math soc 355 2002 1025 1042 corollary 5 9

global existence and blow up of solutions of the cauchy problem for a time fractional diffusion system abstract we prove sharp blow up and global existence results for the cauchy problem of a time fractional diffusion system when initial values u 0 0 and v 0 0 the critical exponent of this problem is determined

pointwise bounds and blow up for systems of nonlinear fractional parabolic inequalities abstract we investigate nonnegative nonlocal solutions u x t and v x t of the nonlinear system of inequalities 0 t u v 0 t v u in r n r n 1 satisfying the initial conditions u v 0 in r n 0 where and are positive constants specifically using the definition of the fractional heat operator t given in taliaferro 2020 we obtain when they exist optimal pointwise upper bounds on r n 0 for nonnegative nonlocal solutions u and v of this initial value problem with particular emphasis on these bounds as t 0 and as t when 1 we compare our results for nonlocal solutions to those of escobedo and herrero 1991 for classical pointwise solutions

blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions abstract this paper concerns with blow up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space we establish the rate estimates for blow up solutions and prove that the blow up set is r n under proper conditions on initial data furthermore for n 1 more complete conclusions about such two topics are given

global and nonglobal weak solutions to a degenerate parabolic system this paper deals with a degenerate parabolic system coupled via general reaction terms of power type global weak solutions are obtained by means of energy estimates and the de giorgi s technique in particular the criterion for global nonexistence of weak solutions is proved by introducing suitable weak sub solutions together with a weak comparison principle in summary the critical exponent for weak solutions of the degenerate parabolic system is determined

note on critical exponents for a system of heat equations coupled in the boundary conditions this note establishes the blow up estimates near the blow up time for a system of heat equations coupled in the boundary conditions under certain assumptions the exact rate of blow up is established we also prove that the only solution with vanishing initial values whenpq 1 is the trivial one

nonexistence of positive solutions to a quasilinear elliptic system and blow up estimates for a quasilinear reaction diffusion system the prior estimate and decay property of positive solutions are derived for a system of quasilinear elliptic differential equations first then the nonexistence result for radially nonincreasing positive solutions of the system is implied by using this nonexistence result blow up estimates for a class of quasilinear reaction diffusion systems non newtonian filtration systems are established to extend the result for semilinear reaction diffusion systems fujita type

a reaction diffusion system with mixed type coupling this paper deals with asymptotic behavior of solutions to a reaction diffusion system coupled via localized and local sources u t du v p x t t v t dv u q both the initial boundary problem with null dirichlet boundary condition and the cauchy problem are considered to study the interaction between the two kinds of sources for the initial boundary problem we prove that the nonglobal solutions blow up everywhere in the bounded domain with uniform blow up profiles in addition it is interesting to observe that the cauchy problem admits an infinity fujita exponent namely the solutions blow up under any nontrivial and nonnegative initial data whenever pq 1 all these imply that the blow up behavior of solutions is governed by the localized source for the two problems with mixed type coupling

critical exponent for parabolic system with time weighted sources in bounded domain abstract this paper considers the time weighted parabolic system u t u e t v p v t v e t u q in bounded domain with r and p q 0 subject to null dirichlet boundary condition the critical fujita curve is determined as p q c 1 max p q 0 1 where 1 is the first eigenvalue of the laplacian as an extension it is observed for another coupled system u t u m u v p v t v n v u q with p q 1 that there is the fujita critical coefficient max m n 1 namely any nontrivial solution blows up in finite time if and only if max m n 1 the studies of critical curves for coupled systems in the current literature are all heavenly relying upon jensen s inequality and the kaplan method for which one has to deal with complicated discussions on the exponents p q being greater or less than 1 differently in the present framework the heat semigroup is introduced to study critical curves for coupled systems where various superlinear and sublinear cases can be treated uniformly by estimates involved this greatly simplifies the arguments for establishing fujita type theorems

boundedness and blow up for a semilinear reaction diffusion system abstract we consider the semilinear parabolic system s u t u p t u q where x rnn 1 t 0 and p q are positive real numbers at t 0 nonnegative continuous and bounded initial values u0 v0 x are prescribed the corresponding cauchy problem then has a nonnegative classical and bounded solution u t x v t x in some strip st 0 t r n 0 t sup t 0 u v remain bounded in s t we show in this paper that if 0 t so that solutions can be continued for all positive times when pq 1 and 1 pq 1 n 2 with max p q one has t for every nontrivial solution u v t is then called the blow up time of the solution under consideration finally if 1 pq 1 n 2 both situations coexist since some nontrivial solutions remain bounded in any strip s while others exhibit finite blow up times

on the solutions of the hypernetted chain equation inside the gas liquid coexistence region we present a detailed study of the solutions of the hypernetted chain integral equation inside the gas liquid coexistence region for simple lennard jones fluids the study is performed by means of a hybrid newton raphson algorithm extended to cope with complex solutions in this way we have unequivocally confirmed that the origin of the well known hnc singular behavior inside the coexistence curve is linked to the onset of complex solutions as density is increased starting from the vapor phase along isotherms inside the coexistence region another singularity is encountered very likely linked with the existence of a complex multiple solution point and correlations start to diverge therefore with the numerical approach here presented it is not feasible to join the liquid and vapor phases through an analytically continuous path of real and complex solutions finally a study of the transition from the mean spherical approximation behavior characterized by the presence of a spinodal divergence to the peculiar hypernetted chain sort of singularity is also presented

on the critical exponent for some semilinear reaction diffusion systems on general domains abstract in this study we consider the parabolic systems u t u f t v p v t v f t u q and u t u f t u r v p v t v f t u q v s in 0 t with homogeneous dirichlet boundary condition p q r s 1 and f c 0 the initial data are considered in the space u 0 v 0 c 0 2 u 0 v 0 0 where is a general domain bounded or unbounded with a smooth boundary we find the conditions that guarantee the global existence or the blow up in finite time of nonnegative solutions these conditions are given in terms of the asymptotic behavior of the solution of the homogeneous linear problem u t u 0 in 0

blow up results for evolution problems with inhomogeneous nonlocal diffusion abstract we provide a sufficient condition on the existence and nonexistence of global positive solutions to the cauchy problem for an inhomogeneous nonlocal diffusion u t j u u u p f x where j is a nonnegative function p 0 and f 0 0 meanwhile the case of an inhomogeneous nonlocal diffusion system is considered it turns out that the inhomogeneous terms substantially contribute to the blow up exponent which coincides with the classical one when the diffusion is given by the laplacian

global existence and blow up for a nonlinear reaction diffusion system abstract we consider the nonlinear reaction diffusion system u t u u m 1 v n 1 v t v u m 2 v n 2 subject to dirichlet boundary conditions andm2 m1 1 n1 n2 1 we prove that ifm1 1 n2 1 andm2n1 1 m1 1 n2 all nonnegative solutions are global while ifm1 1 orn2 1 orm2n1 1 m1 1 n2 both global existence and finite time blow up coexist

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

the critical exponent of doubly singular parabolic equations abstract in this paper we study the cauchy problem of doubly singular parabolic equations u t div u u m t s x u p with non negative initial data here 1 m max 0 1 2 n satisfying 0 m 1 p 1 and s 0 we prove that if max 2 1 s n 1 m 2 then p c m m 1 s 2 1 s n 1 is the critical exponent i e if 1 p p c then every non trivial solution blows up in finite time but for p p c a positive global solution exists

existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping abstract this paper concerns with the cauchy problem for the damped nonlinear hyperbolic system u t t u u t v p t x r 1 r n v t t v v t u q t x r 1 r n u 0 x u 0 x u t 0 x u 1 x x r n v 0 x v 0 x v t 0 x v 1 x x r n where p q 1 and satisfy p q 1 we shall prove that if max 1 p p q 1 1 q p q 1 n 2 for n 1 or 3 then the solution with small initial data exists globally in time and satisfies the decay estimates u t c 1 t and v t c 1 t for some positive constants and while if max 1 p p q 1 1 q p q 1 n 2 for n 1 then every solution with initial data having positive average value does not exist globally

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

critical exponents of fujita type for certain evolution equations and systems with spatio temporal fractional derivatives this paper is concerned with establishing necessary or sufficient conditions for the existence of solutions to evolution equations with fractional derivatives in space and time the fujita exponent is determined then these results are extended to systems of reaction diffusion equations our new results shed lights on important practical questions

iterative approximation of solutions for semilinear parabolic equations system in this paper we present the convergence analysis of iterative schemes for solving semilinear systems resulting from multidimensional semilinear second order parabolic partial differential equations pdes defined in a domain n x and subject to cauchy boundary conditions on n using the method of integral successive iterative approximation and the principle of decay estimation we derive the bounded nonnegative local solutions and global solutions blowing up results of the solutions are also presented

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

large time behavior of ode type solutions to a nonlinear parabolic system abstract let u v be a solution to the cauchy problem for a nonlinear parabolic system p t u u v in r n 0 t v v u in r n 0 u x 0 x 0 in r n v x 0 x 0 in r n where 0 with 1 0 and b c r n l r r n with 1 r and inf x r n x inf x r n x then the solution u v to problem p behaves like a positive solution to ode s and in 0 and both of u t l r n and v t l r n diverge as t in this paper we obtain the precise description of the large time behavior of the solution u v

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

homological properties of graded noetherian pi rings abstract let r be a connected graded noetherian pi ring if injdim r n r is auslander gorenstein and cohen macaulay with gelfand kirillov dimension equal to n if gldim r n r is a domain finitely generated as a module over its centre and a maximal order in its quotient division ring similar results hold if r is assumed to be local rather than connected graded alternatively suppose that r is a noetherian pi ring with gldim r r m 1 hd r m 2 for any two maximal ideals m i in the same clique then r is a direct sum of prime rings is integral over its centre and is auslander gorenstein if r is a prime ring then the centre z r of r is a krull domain and r equals its trace ring tr moreover hd r m height m for every maximal ideal m of r