laplacian and diffusional growth a unified theoretical description for symmetrical and parity broken patterns abstract nowadays laplacian growth patterns such as the saffman taylor instability are well understood mainly because of exact results derived by conformal mapping techniques this is not the case for crystal growth processes except for the free needle crystal normally it is thought that the crystal growth process can be understood by an analogy with the equivalent steady laplacian process the results that follow from this analogy however are in contradiction with numerical and experimental results even at very low velocities we present here a way to mimic the diffusion instabilities in a simple way which allow to describe the formation and the dynamics of new patterns we show that the problem of crystal growth in a channel is closely related to the saffman taylor problem in a sector by using this analogy we obtain the family of zero surface tension solutions for small peclet numbers including an asymmetric degree of freedom we study analytically the selection problem and confirm that parity broken solutions originate from the symmetrical solutions through a bifurcation in agreement with recent numerical investigations we obtain scaling laws for the parity broken solutions for a wide range of parameters we extend these results to the infinite geometry and show that parity broken double fingers can exist even in the range of small undercooling with fully isotropic surface tension the scaling law for the selected velocity of double fingering structures is very different from the scaling law of a free dendrite with anisotropy

chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

an attraction repulsion chemotaxis system with nonlinear productions abstract this paper studies the semilinear attraction repulsion chemotaxis system with nonlinear productions and logistic source u t u u v u w f u 0 v u k v 0 w u l w in bounded domain r n n 1 subject to the non flux boundary conditions where the nonlinear productions for the attraction and repulsion chemicals are described via u k and u l respectively and the logistic source f c 2 0 satisfying f u u a b u s with s 0 f 0 0 it is proved that if one of the random diffusion logistic source and repulsion mechanisms dominates the attraction with max l s 2 n k the solutions would be globally bounded furthermore under the three balance situations namely k s l k l s or k s l the boundedness of solutions depends on the sizes of the coefficients involved this extends the global boundedness criteria established by zhang and li 2016 20 for the attraction repulsion chemotaxis system with linear productions and logistic source

how far does small chemotactic interaction perturb the fisher kpp dynamics abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the fully parabolic chemotaxis growth system u e t u e e u e v e u e 1 u e x t 0 v e t v e v e u e x t 0 with positive small parameter e 0 in a bounded convex domain r n n 1 with smooth boundary the solutions converge to the solution u to the fisher kpp equation as e 0 it is shown that for all 0 and any suitably regular nonnegative initial data u i n i t v i n i t there are some constants e 0 0 and c 0 such that sup t 0 u e t u t l c e for all e 0 e 0

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

blow up in a higher dimensional chemotaxis system despite logistic growth restriction abstract we study radially symmetric solutions of a class of chemotaxis systems generalizing the prototype u t u u v u u x t 0 0 v m t u x t 0 in a ball r n with parameters 0 0 0 and 1 and m t 1 u x t d x it is shown that when n 5 and 3 2 1 2 n 2 then there exist initial data such that the smooth local in time solution of blows up in finite time this indicates that even superlinear growth restrictions may be insufficient to rule out a chemotactic collapse as is known to occur in the corresponding system without any proliferation

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

prevention of blow up by fast diffusion in chemotaxis abstract in this paper we study a strongly coupled parabolic system with cross diffusion term which models chemotaxis the diffusion coefficient goes to infinity when cell density tends to an allowable maximum value such fast diffusion leads to global existence of solutions in bounded domains for any given initial data irrespective of the spatial dimension which is usually the goal of many modifications to the classical keller segel model the key estimates that make this possible have been obtained by a technique that uses ideas from moser s iterations

boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source abstract this paper deals with a parabolic elliptic chemotaxis system with nonlinear sensitivity and logistic source u t u u v f u x t 0 0 v v g u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 1 where 0 the function u is the chemotactic sensitivity g u is the production rate of the chemoattractant and f u is the logistic source under some suitable assumptions on the nonlinearities u g u and logistic source f u we study the global boundedness of solutions for the problem

tight wavelet frames for subdivision in this paper we construct multivariate tight wavelet frame decompositions for scalar and vector subdivision schemes with nonnegative masks the constructed frame generators have one vanishing moment and are obtained by factorizing certain positive semi definite matrices the construction is local and allows us to obtain framelets even in the vicinity of irregular vertices constructing tight frames instead of wavelet bases we avoid extra computations of the dual masks in addition the frame decomposition algorithm is stable as the discrete frame transform is an isometry on 2 if the data are properly normalized

global strong solution to the semi linear keller segel system of parabolic parabolic type with small data in scale invariant spaces abstract we shall show existence of global strong solution to the semi linear keller segel system in r n n 3 of parabolic parabolic type with small initial data u 0 h n r 2 r r n and v 0 h n r r r n for max 1 n 4 r n 2 our method is based on the perturbation of linealization together with the l p l q estimates of the heat semigroup and the fractional powers of the laplace operator as a by product of our method we shall prove the decay property of solutions as the time goes to infinity

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

triangular 3 subdivision schemes the regular case the paper deals with the investigation of triangular 3 subdivision schemes in the stationary shift invariant setting in section 2 we collect the available theory on refinable functions subdivision surfaces with emphasis on their sobolev and holder smoothness families of interpolatory and approximating 3 subdivision schemes are investigated in section 3 some dual 3 subdivision schemes which are related to vector valued refinable functions are also analyzed for this purpose we have developed matlab routines for numerically investigating properties of vector subdivision schemes

wavelet bi frames with uniform symmetry for curve multiresolution processing this paper is about the construction of univariate wavelet bi frames with each framelet being symmetric as bivariate filter banks are used for surface multiresolution processing it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices for open surfaces special multiresolution algorithms are designed to process boundary vertices when the multiresolution algorithms derived from univariate wavelet bi frames are used as the boundary algorithms it is desired that not only the scaling functions but also all framelets be symmetric in addition the algorithms for curve surface multiresolution processing should be given by templates so that they can be easily implemented in this paper first by appropriately associating the lowpass and highpass outputs to the nodes of z we show that both biorthogonal wavelet multiresolution algorithms and bi frame multiresolution algorithms can be represented by templates then using the idea of the lifting scheme we provide frame algorithms given by several iterative steps with each step represented by a symmetric template finally with the given templates of algorithms we obtain the corresponding filter banks and construct bi frames based on their smoothness and vanishing moments two types of symmetric bi frames are studied in this paper in order to provide a clearer picture on the template based procedure for bi frame construction in this paper we also consider the template based construction of biorthogonal wavelets the approach of the template based bi frame construction introduced in this paper can be extended easily to the construction of bivariate bi frames with high symmetry for surface multiresolution processing

bi frames with 4 fold axial symmetry for quadrilateral surface multiresolution processing when bivariate filter banks and wavelets are used for surface multiresolution processing it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry this symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices recently lifting scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing if biorthogonal wavelets have certain smoothness then the analysis or synthesis scaling function or both have big supports in general in particular when the synthesis low pass filter is a commonly used scheme such as loop s scheme or catmull clark s scheme the corresponding analysis low pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness big supports of scaling functions or in other words big templates of multiresolution algorithms are undesirable for surface processing on the other hand a frame provides flexibility for the construction of basis systems this paper concerns the construction of wavelet or affine bi frames with high symmetry in this paper we study the construction of wavelet bi frames with 4 fold symmetry for quadrilateral surface multiresolution processing with both the dyadic and 2 refinements considered the constructed bi frames have 4 framelets or frame generators for the dyadic refinement and 2 framelets for the 2 refinement namely with either the dyadic or 2 refinement a frame system constructed in this paper has only one more generator than a wavelet system the constructed bi frames have better smoothness and smaller supports than biorthogonal wavelets furthermore all the frame algorithms considered in this paper are given by templates so that one can easily implement them

quincunx fundamental refinable functions and quincunx biorthogonal wavelets we analyze the approximation and smoothness properties of quincunx fundamental refinable functions in particular we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in r2 their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order in addition these examples are minimally supported with symmetry for two special families of such quincunx interpolatory masks we prove that their symbols are nonnegative finally a general way of constructing quincunx biorthogonal wavelets is presented several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed

boundedness and global existence in the higher dimensional parabolic parabolic chemotaxis system with without growth source abstract in this paper we are concerned with a general class of quasilinear parabolic parabolic chemotaxis systems with without growth source under homogeneous neumann boundary conditions in a smooth bounded domain r n with n 2 it is recently known that blowup is possible even in the presence of superlinear growth restrictions here we derive new and interesting characterizations on the growth versus the boundedness we show that the hard task of proving the l boundedness of the cell density can be reduced to proving its l r boundedness in other words we show that the l r boundedness of the cell density can successfully guarantee its l boundedness and hence its global boundedness where r n or n 2 depending on whether the growth restriction is essentially linear including no growth or superlinear hence a blowup solution also blows up in l p norm for any suitably large p more detailed information on how the growth source affects the boundedness of the solution is derived these results reveal deep understandings of blowup mechanism for chemotaxis models then we use these criteria to establish uniform boundedness and hence global existence of the underlying models logistic source in 2 d cubic source as initially proposed by mimura and tsujikawa in 3 d n 1 st source in n d with n 4 as a consequence in a chemotaxis growth model blowup is impossible if the growth effect is suitably strong finally we underline that our results remove the commonly assumed convexity on the domain

nonlocal nonlinear reaction preventing blow up in supercritical case of chemotaxis system abstract this paper is devoted to the analysis of non negative solutions for the chemotaxis model with nonlocal nonlinear source in bounded domain the qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction when the growth factor is stronger than the dampening effect with the help of the nonlocal nonlinear term in the reaction for appropriately chosen exponents and arbitrary initial data the model admits a classical solution which is uniformly bounded moreover when the growth factor has the same order with the dampening effect the nonlocal nonlinear exponents can prevent the chemotactic collapse

void electromigration as a moving free boundary value problem abstract the void electromigration process in the strip geometry is investigated analytically the void is assumed to travel steadily along the axis of symmetry of the metal strip when its length is large compared to the width of the metal ribbon a strong similarity with the classical saffman taylor viscous fingering is pointed out it allows us to adapt the hodograph method to predict both the width and the growth velocity at fixed applied current in the absence of surface tension the void shape is unique and has a relative width close to 2 3 a result in strong contradiction with the viscous fingering in the same geometry the propagation of symmetric bubbles is also investigated and shows that the set of solutions is less rich than the flow problem the addition of surface tension appears non standard in the area of pattern formation it does not allow large void propagation suppressing the unique infinite solution as for the bubbles the numerically computed shapes with surface tension appear rather circular this paper shows that the modification of a single boundary condition the dirichlet boundary condition is enough to strongly modify our intuition about pattern formation

on a quasilinear parabolic elliptic chemotaxis system with logistic source abstract this paper deals with a quasilinear parabolic elliptic chemotaxis system with logistic source under homogeneous neumann boundary conditions in a smooth bounded domain for the case of positive diffusion function it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded moreover if the diffusion function is zero at some point or a positive diffusion function and the logistic damping effect is rather mild we proved that the weak solutions are global existence finally it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source

shape selection of saffman taylor fingers among all problems of pattern selection the one posed by the saffman taylor finger is yet unsolved although numerics and experiments indicate the very simple property that the relative width of the finger tends to 1 2 in the low surface tension limit we explain this by performing an expansion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions in particular we predict that the relative width of the finger tends to 1 2 as the surface tension to the power 2 3 pacs numbers 47 10 g 03 40 gc 68 10 m

spatial pattern formation in a chemotaxis diffusion growth model abstract mimura and one of the authors 1996 proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis for this model tello and winkler 2007 22 obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution while kurata et al 2008 numerically showed several spatio temporal patterns in a rectangle motivated by their work we consider some qualitative behaviors of stationary solutions from global and local bifurcation viewpoints in the present paper first we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle for this case the directions of the branches near the bifurcation points are also obtained finally we exhibit several numerical results for the stationary and oscillating patterns

compactly supported tight affine spline frames in l 2 r d the theory of fiberization is applied to yield compactly supported tight affine frames wavelets in l 2 r d from box splines the wavelets obtained are smooth piecewise polynomials on a simple mesh furthermore they exhibit a wealth of symmetries and have a relatively small support the number of mother wavelets however increases with the increase of the required smoothness two bivariate constructions of potential practical value are highlighted in both the wavelets are derived from four direction mesh box splines that are refinable with respect to the dilation matrix 1 1 1 1

triangular 7 and quadrilateral 5 subdivision schemes regular case abstract this paper is devoted to the study of triangular 7 and quadrilateral 5 surface subdivisions both approximation and interpolatory subdivision schemes are considered with illustrative examples of both scalar valued and matrix valued 7 and 5 subdivision masks that satisfy the sum rule of sufficiently high orders in particular optimal sobolev smoothness is determined and holder smoothness estimates are presented

a quasilinear attraction repulsion chemotaxis system of parabolic elliptic type with logistic source abstract this study considers the following quasilinear attraction repulsion chemotaxis system of parabolic elliptic type with logistic source u t d u u u v u w f u x t 0 0 v u v x t 0 0 w u w x t 0 under homogeneous neumann boundary conditions in a bounded domain r n n 2 with smooth boundary where d u c d u m 1 with m 1 0 and c d 0 and f u a b u with a 0 b 0 and 1 in the case of non degenerate diffusion i e 0 we show that the system admits a unique global bounded classical solution provided that the repulsion prevails over the attraction in the sense that 0 or the logistic dampening is sufficiently strong or the diffusion is sufficiently strong while in the case of degenerate diffusion i e 0 we show that the system admits a global bounded weak solution at least under the same assumptions finally we obtain the large time behavior of solutions for a specific logistic source

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

a moving boundary model motivated by electric breakdown ii initial value problem abstract an interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a laplacian growth model regularized by a kinetic undercooling boundary condition using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle where they decay an initially analytic interface stays analytic for all finite times but singularities from outside the physical region approach the interface for t which results in some anomalous relaxation at the back of the circle for the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations but larger perturbations may lead to branching we also present results for more general initial shapes which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time

global existence of solutions to a parabolic parabolic system for chemotaxis with weak degradation abstract we study the global existence of solutions to a parabolic parabolic system for chemotaxis with a logistic source in a two dimensional domain where the degradation order of the logistic source is weaker than quadratic we introduce nonlinear production of a chemoattractant and show the global existence of solutions under certain relations between the degradation and production orders

nonseparable two and three dimensional wavelets we present two and three dimensional nonseparable wavelets they are obtained from discrete time bases by iterating filter banks we consider three sampling lattices quincunx separable by two in two dimensions and fco the design methods are based either on cascade structures or on the mcclellan transformation in the quincunx case we give a few design examples in particular the first example of an orthonormal 2 d wavelet basis with symmetries is constructed

global existence and decay properties for a degenerate keller segel model with a power factor in drift term abstract the following degenerate parabolic system modelling chemotaxis is considered ks u t u m u q 1 v x r n t 0 v t v v u x r n t 0 u x 0 u 0 x v x 0 v 0 x x r n where m 1 q 2 0 or 1 and n 1 the aim of this paper is to prove the existence of a time global weak solution u v of ks with the l 0 l r n bound such a global bound is obtained in the case of i m q 2 n for large initial data and ii 1 m q 2 n for small initial data in the case of ii the decay properties of the solution u v are also discussed

non existence of finite time self similar solutions of the keller segel system in the scaling invariant class let us consider the problem whether there does exist a finite time self similar solution of the backward type to the semilinear keller segel system in the case of parabolic elliptic type for n 3 we show that there is no such a solution with a finite mass in the scaling invariant class on the other hand in the case of parabolic parabolic type for n 2 non existence of finite time self similar solutions is proved in a larger class of a finite mass with some local bounds

a moving boundary problem motivated by electric breakdown i spectrum of linear perturbations abstract an interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a laplacian growth model regularized by a kinetic undercooling boundary condition we study the linear stability of uniformly translating circles that solve the problem in two dimensions in a space of smooth perturbations of the circular shape the stability operator is found to have a pure point spectrum except for the eigenvalue 0 0 for infinitesimal translations all eigenvalues are shown to have negative real part therefore perturbations decay exponentially in time we calculate the spectrum through a combination of asymptotic and series evaluation in the limit of vanishing regularization parameter all eigenvalues are found to approach zero in a singular fashion and this asymptotic behavior is worked out in detail a consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations both the linear and the nonlinear initial value problem are considered in a second paper

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k

symmetry property and construction of wavelets with a general dilation matrix in this note we are interested in the symmetry property of a refinable function with a general dilation matrix we investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has a certain kind of symmetry given two dilation matrices which produce the same lattice we demonstrate that if a mask has a certain kind of symmetry then its associated refinable functions with respect to the two dilation matrices are the same therefore the two corresponding derived wavelet systems are essentially the same finally we illustrate that for any dilation matrix orthogonal masks as well as interpolatory masks having nonnegative symbols can be easily constructed with any preassigned order of sum rules by employing a linear transform without solving any equations the method in this note on constructing masks with certain desirable properties is simple painless and general examples of quincunx wavelets and wavelets with respect to the checkerboard lattice are presented to illustrate the general theory

periodic traveling waves in a reaction diffusion model with chemotaxis and nonlocal delay effect abstract in this paper we study the existence of periodic traveling wave solutions with large wave speed for a reaction diffusion model with chemotaxis and nonlocal delay effect by applying the perturbation method the proof relies on an abstract formulation of the wave profile as a solution of an operator equation in a certain banach space coupled with the lyapunov schmidt reduction and the implicit function theorem