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

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

motion estimation using the correlation transform the zero mean normalized cross correlation is shown to improve the accuracy of optical flow but its analytical form is quite complicated for the variational framework this paper addresses this issue and presents a new direct approach to this matching measure our approach uses the correlation transform to define very discriminative descriptors that are pre computed and that have to be matched in the target frame it is equivalent to the computation of the optical flow for the correlation transforms of the images the smoothness energy is non local and uses a robust penalty in order to preserve motion discontinuities the model is associated with a fast and parallelizable minimization procedure based on the projected proximal point algorithm the experiments confirm the strength of this model and implicitly demonstrate the correctness of our solution the results demonstrate that the involved data term is very robust with respect to changes in illumination especially where large illumination exists

some existence and nonexistence theorems for solutions of degenerate parabolic equations abstract the initial and boundary value problem for the degenerate parabolic equation v t v f v in the cylinder 0 r n bounded for a certain class of point functions satisfying v 0 e g v v m sign v is considered in the case that f v sign v c 1 v the equation has a global time solution the same is true for 1 provided the measure of is sufficiently small in the case that f v v is nondecreasing a condition is given on the initial state v x 0 which implies that the solution must blow up in finite time the existence of such initial states is discussed

universal bounds for global solutions of a forced porous medium equation we show that in a smooth bounded domain rn n 2 all global nonnegative solutions of ut um up with zero boundary data are uniformly bounded in by a constant depending on p and but not on u0 provided that 1 m p n 1 n 1 m furthermore we prove an a priori bound in l 0 depending on u0 l under the optimal condition 1 m p n 2 n 2 m

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

on a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium the main result of the paper is the uniqueness of nonnegative solutions of the cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut um xx un x where m 1 and n 0 in particular n is allowed to be smaller than one the proof is based on a refined test function argument the condition that u be nonnegative is crucial but the restriction to one space variable is not

additive maps on standard operator algebras preserving parts of the spectrum let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y respectively and let be a unital additive surjection from a onto b we introduce thirteen parts of the spectrum for elements in a and b and prove that if preserves any one of these parts of the spectrum then it is either an isomorphism or an anti isomorphism

on determinant preserver problems abstract let m n and t n be the vector spaces of n n matrices and upper triangular matrices over a field f with some cardinality and characteristic restrictions respectively we characterise transformations on these two spaces separately which satisfy one of the following conditions 1 det a b det a b for all a b and 2 is surjective and det a b det a b for all a b and two specific 3 is additive and preserves determinant

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

boundedness of global positive solutions of a porous medium equation with a moving localized source abstract in this paper a porous medium equation with a moving localized source u t u r u a f u x 0 t t is considered it is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data moreover in one space dimension case it is also shown that all global positive solutions of the above equation are uniformly bounded and this differs from that of a porous medium equation with a local source

maps on matrix algebras preserving idempotents abstract let m n be the algebra of all n n complex matrices and p n the set of all idempotents in m n suppose m n m n is a surjective map satisfying a b p n if and only if a b p n a b m n c then either is of the form a tat 1 a m n or is of the form a ta t t 1 a m n where t m n is a nonsingular matrix

determinant preserving maps on matrix algebras abstract let m n be the algebra of all n n complex matrices if m n m n is a surjective mapping satisfying det a b det a b a b m n c then either is of the form a man a m n or is of the form a ma t n a m n where m n m n are nonsingular matrices with det mn 1

a class of non linear non classical parabolic equations abstract in this paper we deal with a class of non classical parabolic equations which arise from some practical problems such as the identification of unknown parameters free boundary problems etc one of the characteristics of this kind of equation is that the maximum principle is no longer valid we will first study the local solvability for a general equation of this kind via an auxiliary problem and the schauder fixed point principle then some a priori estimates are derived to establish the global existence of the solution moreover uniqueness and continuous dependence of the solution are proved finally some examples are given to illustrate the applications of the theory

unital and multiplicatively spectrum preserving surjections between semi simple commutative banach algebras are linear and multiplicative abstract let t be a surjective map from a unital semi simple commutative banach algebra a onto a unital commutative banach algebra b suppose that t preserves the unit element and the spectrum f g of the product of any two elements f and g in a coincides with the spectrum t f t g then b is semi simple and t is an isomorphism the condition that t is surjective is essential an example of a non linear and non multiplicative unital map from a commutative c algebra into itself such that t f t g f g holds for every f g are given we also show an example of a surjective unital map from a commutative c algebra onto itself which is neither linear nor multiplicative such that t f t g f g holds for every f g

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

boundedness of global solutions for a porous medium system with moving localized sources abstract this paper deals with a class of porous medium systems with moving localized sources u t u r 1 u a f v x 0 t t v t v r 2 v b g u x 0 t t with homogeneous dirichlet boundary conditions it is shown that under certain conditions solutions of the above system blow up in finite time for large a and b or large initial data while there exist global positive solutions to the above system for small a and b or small initial data moreover in the one dimensional space case it is also shown that all global positive solutions of the above problem are uniformly bounded

boundedness of global solutions of a porous medium equation with a localized source abstract in this paper a localized porous medium equation u t u r u af u x 0 t is considered it is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data moreover it is also shown that all global positive solutions of the above equation are uniformly bounded and this differs from that of a porous medium equation with a local source

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

hua s fundamental theorems of the geometry of matrices and related results abstract we briefly survey some recent improvements of hua s fundamental theorem of the geometry of rectangular matrices then we discuss possible further generalizations as well as some related open problems in the theory of preservers we solve one such open problem using ovchinnikov s characterization of automorphisms of the poset of idempotent matrices using ovchinnikov s result we obtain a short proof of the fundamental theorem of the geometry of square matrices

boundedness of global solutions for a nonlinear degenerate parabolic porous medium system with localized sources abstract this paper deals with nonlinear degenerate parabolic porous medium system with localized sources it is shown that under certain conditions solutions of the equation blow up in finite time for large a and b or large initial data while there exist global positive solutions for small a and b or small initial data moreover it is also shown that all global positive solutions of the above equation are uniformly bounded these extend a recent work of chen et al in y chen q liu h gao boundedness of global solutions of a porous medium equation with a localized source nonlinear anal 64 2006 2168 2182 which considered the single equation with localized source