critical exponents for semilinear equations of mixed elliptic hyperbolic and degenerate types for semilinear gellerstedt equations with tricomi goursat or dirichlet boundary conditions we prove poho aev type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations a critical exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic hyperbolic or degenerate settings where the power is 1 less than the critical exponent in a relevant sobolev embedding c 2003 wiley periodicals inc

algebras of generalized quaternion type abstract we introduce and study the algebras of generalized quaternion type being natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups we prove that all these algebras with 2 regular gabriel quivers are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles the main result of the paper forms an important step towards the morita equivalence classification of all periodic symmetric tame algebras of non polynomial growth applying the main result we establish existence of wild periodic algebras of period 4 with arbitrary large number at least 4 of pairwise non isomorphic simple modules these wild periodic algebras arise as stable endomorphism rings of cluster tilting cohen macaulay modules over one dimensional hypersurface singularities

conservation laws for equations of mixed elliptic hyperbolic and degenerate types for partial differential equations of mixed elliptic hyperbolic and degenerate types which are the euler lagrange equations for an associated lagrangian invariance with respect to changes in independent and dependent variables is investigated as are results in the classification of continuous one parameter symmetry groups for the variational and divergence symmetries conservation laws are derived via the method of multipliers the conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems

blocks with trivial intersection defect groups we show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups or is morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non abelian simple groups in particular we classify all blocks of automorphism groups of non abelian simple groups whose defect groups are non cyclic and intersect pairwise trivially a consequence is that donovans conjecture holds for blocks whose defect groups intersect pairwise trivially

periodicity of self injective algebras of polynomial growth abstract let a be an indecomposable representation infinite tame finite dimensional algebra of polynomial growth over an algebraically closed field we prove that a is a periodic algebra with respect to action of the bimodule syzygy operator if and only if a is morita equivalent to a socle deformation of an orbit algebra b g where b is the repetitive category of a tubular algebra b and g is an admissible infinite cyclic automorphism group of b the main contribution in the paper is to prove the sufficiency part of this equivalence it is known that every orbit algebra b g of a tubular algebra b admits a presentation as an orbit algebra t b r h of an r fold trivial extension algebra t b r of b with respect to free action of a finite cyclic automorphism group h of t b r a significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self injective algebras of polynomial growth and showing that all of them are periodic algebras then we deduce the periodicity of all algebras socle equivalent to the orbit algebras b g t b r h of tubular algebras b from the periodicity of these ten exceptional algebras using invariance of periodicity for finite galois coverings and derived equivalences of algebras

global existence for the one dimensional second order semilinear hyperbolic equations abstract the aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one dimensional semilinear equation appearing in the boundary value problems of gas dynamics we investigate the cauchy problem for such equation in the domain where the operator is weakly hyperbolic we obtain the necessary condition for the existence of the self similar solutions for the semilinear gellerstedt type equation the approach used in the paper is based on the fundamental solution of the linear gellerstedt operator and the l p l q estimates

the self similar solutions of the one dimensional semilinear tricomi type equations abstract in this article we investigate the issue of existence of global in time solutions of semilinear tricomi type equations we give conditions that relate the nonlinearity the speed of propagation and the order of singularities of initial data these conditions guarantee existence of global in time solutions in particular we prove existence of solutions invariant under dilation by solving the cauchy problem with initial data which are homogeneous functions

spatial boundary problem with the dirichlet neumann condition for a singular elliptic equation abstract in the present work we find an explicit solution of a boundary problem with the dirichlet neumann condition for elliptic equation with singular coefficients in a quarter of ball for this aim the method of green s function has been used since found green s function contains a hypergeometric function of appell we had to deal with decomposition formulas formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem

fundamental solutions of the wave equation in robertson walker spaces abstract in this article we construct the fundamental solutions for the wave equation in the robertson walker spaces arising in the de sitter model of the universe we then use these fundamental solutions to represent solutions of the cauchy problem for the equation with and without a source term

derived equivalence classification of algebras of dihedral semidihedral and quaternion type abstract algebras of dihedral semidihedral and quaternion type were introduced by k erdmann as a generalization of blocks of finite groups of tame representation type in a series of papers erdmann gave a description of the morita equivalence classes of these algebras the aim of the present article is to present a classification of algebras of dihedral semidihedral and quaternion type up to derived equivalence as an application we can show that all basic algebras in erdmann s list are actually of tame representation type

price s law on nonstationary space times abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate price s law price 1972 54 for linear waves as a corollary we also prove price s law for certain small perturbations of the kerr metric this result was previously established by the second author in tataru 65 on stationary backgrounds the present work was motivated by the problem of nonlinear stability of the kerr schwarzschild solutions for the vacuum einstein equations which seems to require a more robust approach to proving linear decay estimates

on the existence of low regularity solutions to semilinear generalized tricomi equations in mixed type domains abstract in 20 21 we have established the existence and singularity structures of low regularity solutions to the semilinear generalized tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations respectively in the present paper we shall be concerned with the low regularity solution problem for the semilinear mixed type equation t 2 u t 2 l 1 u f t x u with an initial data u 0 x x h s r n 0 s n 2 where t x r r n n 2 l n f t x u is c 1 smooth in its arguments and has compact support with respect to the variable x under the assumption of the subcritical growth of f t x u on u we will show the existence and regularity of the considered solution in the mixed type domain t 0 t 0 r n for some fixed constant t 0 0

interior regularity of the dirichlet problem for the tricomi equation for the tricomi equation with dirichlet boundary conditions we study the relationship between singularites at the boundary and singularities in the interior of a bounded planar region with smooth non characteristic boundary necessary and sufficient conditions for interior smoothness are stated in terms of microlocal regularity at the boundary and are proven via known microlocal propagation of singularities results along the generalized bicharacteristic flow in particular a trapped gliding ray phenomenon at parabolic boundary points is demonstrated under a sharp geometric hypothesis which provides a microlocal explanation for the possibility of having only isolated singularities at the boundary which is a question left open in the work of morawetz

a note on the fundamental solution for the tricomi type equation in the hyperbolic domain abstract barros neto and gelfand duke math j 98 3 1999 465 duke math j 117 2 2003 561 constructed for the tricomi operator y x 2 y 2 on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the tricomi operator we give for the tricomi type operator t 2 t m x a fundamental solution relative to an arbitrary point of r n 1 with the support in the region t 0 where the operator is hyperbolic our key observation is that the fundamental solution for the tricomi type operator can be written like an integral of the distributions generated by the fundamental solution of the cauchy problem for the wave equation the application of that fundamental solution to the l p l q estimate for the forced tricomi type equation is given as well

fundamental solutions for the tricomi operator in this paper we explicitly calculate fundamental solutions for the tricomi operator relative to an arbitrary point in the plane and show that all such fundamental solutions originate from the hypergeometric function f 1 6 1 6 1 that is obtained when we look for homogeneous solutions to the reduced hyperbolic tricomi equation

a new class of analytic solutions of the two state problem a new class of solutions of the time dependent schrodinger equation is found for the two state problem often encountered in quantum optics magnetic resonance and atomic collisions the authors use the riemann papperitz differential equation to find exact solutions in terms of hypergeometric functions they consider only cases in which the final occupation probabilities are elementary functions of the parameters of the model

expansions for some confluent hypergeometric functions power series for the parabolic cylinder function dv z are derived from known addition theorems thus enabling a more compact and efficient expression of the function one of the expansions is applied to find an asymptotic approximation for the function as mod v mod to infinity with mod arg v mod pi next a large argument z asymptotic addition theorem is derived from an integral representation of dv z before extending this type of summation theorem to the two general confluent hypergeometric functions motivating mathematical applications to three integration problems are given as examples of the practical usefulness of the formulae

nonlinear waves on 3d hyperbolic space in this article global in time dispersive estimates and strichartz estimates are explored for the wave equation on three dimensional hyperbolic space due to the negative curvature extra dispersion is noted as compared to the euclidean case and a wider range of strichartz estimates is proved using these small data global existence to semilinear wave equations is shown for a range of powers that is broader than that known for euclidean space

bessel integrals and fundamental solutions for a generalized tricomi operator abstract the method of partial fourier transform is used to find explicit formulas for two remarkable fundamental solutions for a generalized tricomi operator these fundamental solutions reflect clearly the mixed type of the tricomi operator in proving these results we establish explicit formulas for fourier transforms of some functions involving bessel functions

on semilinear tricomi equations with critical exponents or in two space dimensions abstract this paper is a complement of our recent works on the semilinear tricomi equations in 9 and 10 for the semilinear tricomi equation t 2 u t u u p with initial data u 0 t u 0 u 0 u 1 where t 0 x r n n 3 p 1 and u i c 0 r n i 0 1 we have shown in 9 and 10 that there exists a critical exponent p crit n 1 such that the solution u in general blows up in finite time when 1 p p crit n and there is a global small solution for p p crit n in the present paper firstly we prove that the solution of t 2 u t u u p will generally blow up for the critical exponent p p crit n and n 2 secondly we establish the global existence of small data solution to t 2 u t u u p for p p c r i t n and n 2 thus we have given a systematic study on the blowup or global existence of small data solution u to the equation t 2 u t u u p for n 2

multidomain spectral method for the helically reduced wave equation we consider the 2 1 and 3 1 scalar wave equations reduced via a helical killing field respectively referred to as the 2 dimensional and 3 dimensional helically reduced wave equation hrwe the hrwe serves as the fundamental model for the mixed type pde arising in the periodic standing wave psw approximation to binary inspiral we present a method for solving the equation based on domain decomposition and spectral approximation beyond describing such a numerical method for solving strictly linear hrwe we also present results for a nonlinear scalar model of binary inspiral the psw approximation has already been theoretically and numerically studied in the context of the post minkowskian gravitational field with numerical simulations carried out via the eigenspectral method despite its name the eigenspectral technique does feature a finite difference component and is lower order accurate we intend to apply the numerical method described here to the theoretically well developed post minkowski psw formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation

a competition between fujita and strauss type exponents for blow up of semi linear wave equations with scale invariant damping and mass abstract we obtain a blow up result for solutions to a semi linear wave equation with scale invariant dissipation and mass and power non linearity in the case in which the model has a wave like behavior we perform a change of variables that transforms our starting equation in a strictly hyperbolic semi linear wave equation with time dependent speed of propagation applying kato s lemma we prove a blow up result for solutions to the transformed equation under some assumptions on the initial data the limit case that is when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow up needs special considerations in this critical case an explicit integral representation formula for solutions of the corresponding linear cauchy problem in 1d is derived finally carrying out the inverse change of variables we get a non existence result for global in time solutions to the original model

derived equivalent tame blocks abstract in this article we provide derived equivalences between several blocks of tame representation type based on erdmann s work on morita equivalence classes of tame blocks we obtain a classification of tame blocks up to derived equivalence as in erdmann s work in the case of two simple modules our classification is only complete up to scalars occurring in certain relations

on fundamental solutions for multidimensional helmholtz equation with three singular coefficients abstract the main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients which could be expressed in terms of a confluent hypergeometric function of four variables in addition the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found

standard lagrangians for wave laplace and tricomi like equations with variable coefficients general methods to derive lagrangians for first and second order linear partial differential equations with variable coefficients are developed the methods are used to obtain standard lagrangians for different physical problems that are described by wave laplace and tricomi like equations

the wave equation on hyperbolic spaces abstract we study the dispersive properties of the wave equation associated with the shifted laplace beltrami operator on real hyperbolic spaces and deduce new strichartz estimates for a large family of admissible pairs as an application we obtain local well posedness results for the nonlinear wave equation

exactly soluble two state quantum models with linear couplings a class of exact analytic solutions of the time dependent schrequation is presented for a two state quantum system coherently driven by a nonresonant external field the coupling is a linear function of time with a finite duration and the detuning is constant four special models are considered in detail namely the shark double shark tent and zigzag models the exact solution is derived by rotation of the landau zener propagator at an angle of 4 and is expressed in terms of weber s parabolic cylinder function approximations for the transition probabilities are derived for all four models by using the asymptotics of the weber function these approximations demonstrate various effects of physical interest for each model

an l p a priori estimate for the tricomi equation in the upper half space we establish an lp a priori estimate for the tricomi equation our main tool is mihlin s multiplier theorem combined with well known estimates of the newtonian potential 0 introduction the purpose of this paper is to establish an lp a priori estimate for the tricomi equation in the upper half space the tricomi equation arises in transonic gas dynamics and is a typical model equation of changing type it has been extensively investigated from the various viewpoints the tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary which is our viewpoint in this paper we can formulate the dirichlet boundary value problem in the upper half space as follows 02 yacuz f in r 0 2 u 0 x x on mt where r y x y 0 x e rn and x is the laplacian in the x variable an l2 a priori estimate can be obtained very easily following the presentation in 8 we suppose that u e co rn and multiply the equation 0 1 by 92u then we integrate over r r to obtain 0 3 ir dy2x 2jj v dydx rn 1 o 9y2 rn 1 o a y 2irn1l vq dx j 1 f 3 dydx where vx stands for the gradient in x e r n this yields 0 4 2u f 1 yx 4y21 l2 rn dy l2 rn l2 rn m f 1 l2 rn 11vx01il2 rn 1 7 received by the editors december 30 1996 and in revised form february 10 1998 1991 mathematics subject classification primary 35j70 35b45

dispersion for the wave equation inside strictly convex domains i the friedlander model case we consider a model case for a strictly convex domain r d of dimension d 2 with smooth boundary 6 and we describe dispersion for the wave equation with dirichlet boundary conditions more specically we obtain the optimal xed time decay rate for the smoothed out green function a t 1 4 loss occurs with respect to the boundary less case due to repeated occurrences of swallowtail type singularities in the wave front set

derived equivalences for tame weakly symmetric algebras having only periodic modules abstract recently a classification up to morita equivalence of all weakly symmetric algebras which have tame representation type simply connected galois coverings and all of whose modules are periodic equivalently the stable auslander reiten quiver consist only of tubes has been completed in j bialkowski a skowronski on tame weakly symmetric algebras having only periodic modules arch math to appear we classify in this paper these algebras up to derived equivalence and give a set of representatives for the derived equivalence classes

sonic supersonic solutions for the two dimensional steady full euler equations this paper focuses on the structure of classical sonic supersonic solutions near sonic curves for the two dimensional full euler equations in gas dynamics in order to deal with the parabolic degeneracy near the sonic curve a novel set of dependent and independent variables are introduced to transform the euler equations into a new system of governing equations which displays a clear regularity singularity structure with the help of technical characteristic decompositions the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables this is the first construction of a classical sonic supersonic solution near a sonic curve for the full euler equations

degenerate operators of tricomi type in l p spaces and in spaces of continuous functions we study elliptic operators l with dirichlet boundary conditions on a bounded domain whose diffusion coefficients degenerate linearly at in tangential directions we compute the domain of l and establish existence uniqueness and maximal regularity of the elliptic and parabolic problems for l in lp spaces and in spaces of continuous functions moreover the analytic semigroups generated by l are consistent positive compact and exponentially stable

one dimensional degenerate operators in lp spaces abstract we comprehensively study a one dimensional elliptic operator degenerating of first order at the boundary in an l p setting here the coefficient of the drift term determines the regularity and the possible boundary conditions of the problem

spectral bounds for tricomi problems and application to semilinear existence and existence with uniqueness results abstract for the linear tricomi problem it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum exploiting the linear solvability theory and spectral information standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems in particular using the leray schauder principle existence of generalized solutions with sublinear nonlinearities is established for sublinear or asymptotically linear nonlinearities that satisfy a lipschitz condition the contraction mapping principle is employed to give results on existence with uniqueness the lipschitz constant depends on lower bounds for the spectrum of the linear problem for certain superlinear problems maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence

analyticity of solutions of semi linear equations with double characteristics we prove the analyticity and gevrey regularity of solutions of elliptic degenerate semi linear differential equations principle part of which is a linear operator with double characteristics considered first by gilioli and treves a new elementary proof for hypoellipticity in the weak sense is given

on closed boundary value problems for equations of mixed elliptic hyperbolic type for partial differential equations of mixed elliptic hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of dirichlet and mixed dirichlet conormal types such problems are of interest for applications to transonicflow and are overdetermined for solutions with classical regularity the method employed consists in variants of the a b c integral method of friedrichs in sobolev spaces with suitable weights particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function in addition interior regularity results are also given in the important special case of the tricomi equation c

integral transform approach to generalized tricomi equations abstract in this article we introduce the integral transform that can be used to obtain solutions of the generalized tricomi equation via solutions of a simpler equation we employed a particular version of this transform in 13 14 41 42 43 44 45 46 to investigate in a unified way several equations such as the linear and semilinear tricomi equations gellerstedt equation the wave equation in the einstein de sitter spacetime the wave and the klein gordon equations in the de sitter and anti de sitter spacetimes

the weighted w2 p estimate on the solution of the gellerstedt equation in the upper half space abstract in this note we are concerned with the weighted w 2 p estimate for the gellerstedt equation which is degenerate on the boundary r 2 based on the estimates of the generalized airy functions and mihlin s multiplier theorem we establish a weighted w 2 p a priori estimate

hochschild cohomology of tame blocks abstract we study the additive structure of hochschild cohomology for blocks of modular group algebras of tame representation type we give explicit formulae for the dimensions of the cohomology groups and compute the poincare series of the hochschild cohomology ring for all tame blocks having one or three simple modules