focusing blow up for quasilinear parabolic equations we study the behaviour of the non negative blowing up solutions to the quasilinear parabolic equation with a typical reaction diffusion right hand side and with a singularity in the space variable which takes the form where m 1 p 1 are arbitrary constants in the critical exponent case q p 1 m 0 we impose zero dirichlet boundary conditions at the singular point x 0 and at x 1 and take large initial data for a class of concave initial functions we prove focusing at the origin of the solutions as t approaches the blow up time t in the sense that x 0 belongs to the blow up set the proof is based on an application of the intersection comparison method with an explicit separable solution which has the same blow up time as u the method has a natural generalisation to the case of more general nonlinearities in the equation a description of different fine structures of blow up patterns in the semilinear case m 1 and in the quasilinear one m 1 is also presented a numerical study of the semilinear equation is also made using an adaptive collocation method this is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour

an adaptive splitting approach for the quenching solution of reaction diffusion equations over nonuniform grids the numerical solution of a nonlinear degenerate reaction diffusion equation of the quenching type is investigated while spatial derivatives are discretized over symmetric nonuniform meshes a peaceman rachford splitting method is employed to advance solutions of the semidiscretized system the temporal step is determined adaptively through a suitable arc length monitor function a criterion is derived to ensure that the numerical solution acquired preserves correctly the positivity and monotonicity of the analytical solution weak stability is proven in a von neumann sense via the norm computational examples are presented to illustrate our results

a splitting moving mesh method for reaction diffusion equations of quenching type this paper studies the numerical solution of multi dimensional nonlinear degenerate reaction diffusion differential equations with a singular force term over a rectangular domain the equations may generate strong quenching singularities our work focuses on a variable temporal step peaceman rachford splitting method with an adaptive moving mesh in space the temporal and spatial adaptation is implemented based on arc length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs the multi dimensional problem is split into a few one dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced the physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching as stability analysis may not be valid when it is very close to the quenching thus an exact linear problem is introduced to justify the stability near the quenching time finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities we will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi cpu

dilation method and smoothing effects of solutions to the benjamin ono equation in this paper we study smoothing effects of solutions to the benjamin ono equation where h is the hilbcrt transform defined by we prove that if h 2 and x x 4 then the solution u of bo belongs to where

blow up for a degenerate parabolic equation with a nonlocal source abstract in this paper existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained for the degenerate parabolic problem x q u t u xx 0 a u m x t d x ku n x t in 0 a 0 t where 0 t a 0 m n 1 k 0 are constants

complete blow up for degenerate semilinear parabolic equations abstract let t a q and x0 be constants with a 0 q 0 and 0 x q u t u xx f u x 0 t 0 0 u x 0 u 0 x 0 0 x a u 0 t u a t 0 0 where f and u0 are given functions we show that under certain conditions u blows up in a finite time and the set of blow up points is the entire interval 0 a

single point blow up for a degenerate parabolic problem with a nonlinear source of local and nonlocal features let q m and t be any real numbers such that q 0 m 1 and t 0 this article studies the following degenerate semilinear parabolic first initial boundary value problem x qu t x t u x x x t a m q email protected x b f u x t u m t for0 0 f u and f u are positive for u 0 and j is nontrivial nonnegative and continuous such that j 0 email protected 1 and j a m q email protected x b f j x u m 0 0 for 0

rank one twists of a certain elliptic curve the purpose of this note is to give the first known example for a conjecture of goldfeld on the number of rank one curves appearing in a family of quadratic twists we show unconditionally that the curve x0 19 has the property that a positive proportion of its quadratic twists have analytic rank one this amounts to a strong nonvanishing statement for the derivatives of certain l functions when the sign of the functional equation is 1 we simultaneously obtain the fact that a positive proportion of twists have analytic rank zero when the sign of the functional equation is 1 let e be a modular elliptic curve over q with corresponding newform f z anq n s2 0 n let l s f l s e ann s be the usual l series then l s f satisfies a functional equation under s 2 s with sign 1 e let d be the fundamental discriminant of a quadratic field k k d and write d for the associated quadratic character in this setting we define the twisted l function l s fd d n ann if e is given by the equation y 2 p x then l s fd is the l function attached to the twisted elliptic curve ed given by dy 2 p x therefore l s fd also satisfies a functional equation and if d n 1 then the sign is given by d d n for r 0 1 and a positive real number x define m r f x d d x ords 1l s fd r the following conjecture of goldfeld gol79 is well known

existence uniqueness and quenching properties of solutions for degenerate semilinear parabolic problems with second boundary conditions let q 0 p 0 t d 0 a d 0 a d 0 t and lu xqut uxx this article considers the following degenerate semilinear parabolic initial boundary value problem lu xpf u in u x 0 0on d ux 0 t 0 ux a t for t 0 where f 0 0 f 0 f 0 and limu c f u for some positive constant c existence of a unique classical solution is proved it is shown that if p q then quenching occurs only at the boundary point x a while if p q then the only quenching point is x 0 if p q then the quenching set is d

neumann initial boundary value problem for benjamin ono equation abstract we consider the neumann initial boundary value problem for benjamin ono equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

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

b methods for the numerical solution of evolution problems with blow up solutions part i variation of the constant in the last two decades the field of geometric numerical integration and structure preserving algorithms has focused on the design of numerical methods that preserve properties of hamiltonian systems evolution problems on manifolds and problems with highly oscillatory solutions in this paper we show that a different geometric property namely the blow up of solutions in finite time can also be taken into account in the numerical integrator giving rise to geometric methods we call b methods we give a first systematic approach for deriving such methods for scalar and systems of semi and quasi linear parabolic and hyperbolic partial differential equations we show both analytically and numerically that b methods have substantially better approximation properties than standard numerical integrators as the solution approaches the blow up time

global existence and blow up for the degenerate and singular nonlinear parabolic system with a nonlocal source abstract the existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained for the degenerate and singular parabolic system u t x u x x 0 a u m x t v n x t d x v t x v x x 0 a v p x t u q x t d x in 0 a 0 t where t a 0 are constants and m n p q are positive real numbers furthermore under certain conditions it is proved that the blow up set of the solution is the entire interval 0 a and we also obtain the blow up rate under the condition

impulsive effects on global existence of solutions for degenerate semilinear parabolic equations let q be a nonnegative real number and l and s be positive constants this article studies the following impulsive problem for n 1 2 3 x email protected email protected email protected email protected x 2u email protected u 0 0 the number l is called the critical value if the problem has a unique global solution u for l l for s 1 existence of a unique l and three criteria for the blow up of the solution in a finite time are given respectively it is also shown that there exists a unique t such that u exists globally for t t and u blows up in a finite time for t t

complete blow up of solutions for degenerate semilinear parabolic first initial boundary value problems let q be a nonnegative real number and tb be a positive constant this paper studies integral solutions of the following degenerate semilinear parabolic first initial boundary value problem xqut uxx f u 0 x 1 0 t tb u x 0 u0 x 0 x 1 u 0 t 0 u 1 t 0 t tb where f 0 0 f 0 0 f u 0 for u 0 f u o u3 as u and u0 x c2 0 1 for some constant 0 1 is a nontrivial and nonnegative function such that u0 f u0 0 in 0 1 and u0 0 0 u0 1 the complete blow up of the integral solutions in the interval 0 1 is investigated

beyond quenching for degenerate singular semilinear parabolic equations let q be a nonzero real constant greater than 1 d 0 a w dx 0 email protected and email protected be their respective closures lu x qu t u x x email protected email protected s email protected sbe the characteristic function of the set s beyond quenching is studied for the problem lu f u g u c email protected u x 0 u 0 x email protected u 0 t 0 u a t for0

global existence and blow up for degenerate and singular parabolic system with localized sources this paper deals with the blow up properties of the solution to degenerate and singular parabolic system with localized sources and homogeneous dirichlet boundary conditions the existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally and blows up in finite time are obtained furthermore under certain conditions it is proved that the blow up set of the solution is the whole domain

the benjamin ono equation in weighted sobolev spaces abstract in this paper we continue the study of the initial value problem t u x u 2 2 x u u 0 in the spaces y s h s r l 2 r where 0 s 3 2 and l r 2 r denotes the set of all complex valued measurable functions such that f r 2 r 1 x 2 r f x 2 dx our main result is the proof of well posedness of the aforementioned problem in y 2 0 1 the proof involves parabolic regularization the riesz thorin interpolation theorem and kato s theory of linear evolution equations of hyperbolic type

blow up at the boundary for degenerate semilinear parabolic equations this paper treats a superlinear parabolic equation degenerate in the time derivative it is shown that the solution may blow up in finite time moreover it is proved that for a large class of initial data blow up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic various estimates are obtained which determine the asymptotic behaviour near the blow up the mathematical analysis is then extended to equations with other degeneracies

mixed initial boundary value problem for the benjamin ono equation abstract we consider the mixed initial boundary value problem for the benjamin ono equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

ternary cubic forms having bounded invariants and the existence of a positive proportion of elliptic curves having rank 0 we prove an asymptotic formula for the number of sl3 z equivalence classes of integral ternary cubic forms having bounded invariants we use this result to show that the average size of the 3 selmer group of all elliptic curves when ordered by height is equal to 4 this implies that the average rank of all elliptic curves when ordered by height is less than 1 17 combining our counting techniques with a recent result of dokchitser and dokchitser we prove that a positive proportion of all elliptic curves have rank 0 assuming the niteness of the tate shafarevich group we also show that a positive proportion of elliptic curves have rank 1 finally combining our counting results with the recent work of skinner and urban we show that a positive proportion of elliptic curves have analytic rank 0 i e a positive proportion of elliptic curves have nonvanishing l function at s 1 it follows that a positive proportion of all elliptic curves satisfy bsd

blow up for a degenerate parabolic equation with a nonlocal source abstract in this paper we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation v x v m xx 0 l v p 1 dx kv q 1 with homogeneous dirichlet boundary conditions in the interval 0 l where 0 m 1 we first establish the local existence and uniqueness of its classical solutions then we show that the positive solution blows up in finite time if the initial datum is sufficient large finally we prove that the blow up set is the whole interval and we also obtain the estimates of the blow up rate

binary quartic forms having bounded invariants and the boundedness of the average rank of elliptic curves we prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants this extends to the quartic case the classical results of gauss and davenport in the quadratic and cubic cases respectively our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups we use these counting results to prove that the average rank of elliptic curves over q when ordered by their heights is bounded in particular we show that when elliptic curves are ordered by height the mean size of the 2 selmer group is 3 this implies that the limsup of the average rank of elliptic curves is at most 1 5

smoothing properties and existence of solutions for the generalized benjamin ono equation abstract we study the cauchy problem for the generalized benjamin ono equation tu d d 2 u dv u where d d dx u 0 and v e 1 r r with v 0 v 0 0 using commutator identities and estimates we prove that the equation exhibits smoothing properties similar to those of the generalized korteweg de vries equation gkdv which is the special case 1 of if v satisfies suitable estimates at zero and at infinity we prove that the cauchy problem for the equation with initial data u 0 u0 has a solution u l r l2 l2loc r h loc if u0 l2 and has a solution u l r h l2loc r h2 loc if u0 h finite energy solution in addition we prove that the usual benjamin ono equation 1 2 v u u 2 has a solution u l r h1 l2loc r h 3 2 loc for u0 h1 those results are easily extended to related equations such as the intermediate long wave equation or the smith equation

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

no quenching in infinite time for degenerate singular semilinear parabolic equations let q be any nonnegative real number and t for the degenerate singular semilinear parabolic initial boundary value problemx qu t u x x f u for0

large time asymptotics of solutions to the generalized benjamin ono equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized benjamin ono bo equation ut u 1u x huxx 0 where h is the hilbert transform x t r when the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time in the case 3 critical for the asymptotic behavior i e for the modified benjamin ono equation we prove that the solutions have the same l time decay as in the corresponding linear bo equation also we find the asymptotics for large time of the solutions of the cauchy problem for the bo equation in the critical and noncritical cases for the critical case we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted sobolev spaces these modified scattering states differ from the free scattering states by a rapidly oscillating factor

l series with nonzero central critical value suppose that f zn 1 an f qtn is a cusp form of weight 2k k e n we denote by l f s the l function of f for re s sufficiently large the value of l f s is given by l f s en 1 an and one can show that l f s has analytic continuation to the entire complex plane the value of l f s at s k will be of particular interest to us and we will refer to this value as the central critical value of l f s let xd denote the dirichlet character associated to the extension q d q that is xd n va where ad denotes the discriminant of q d q define the dth quadratic twist of f to be fxd en 1 an f xd n qn for any integer d the l function of fx is the dth quadratic twist of l f s that is l fxd s zn aan f xd n we will be interested in determining how often l fxd s has nonzero central critical value as d varies over all integers since xd 2 xd we will restrict our attention to the square free integers d we expect that as we let d vary over all of the square free integers a positive proportion of the l functions l fxd s will have nonzero central critical value indeed goldfeld 7 conjectures that for newforms f of weight 2 l fxd 1 78 0 for 1 of the square free integers given an elliptic curve e y2 x3 ax2 bx c a b c e z with conductor ne and an integer d we define the dth quadratic twist of e to be the curve ed y2 x3 adx2 bd2x cd3 let l ed s denote the l function associated to ed for square free d coprime to 2ne l ed s is simply the dth quadratic twist of l ei s if f e s2 n is a newform with integer coefecients we know via the theory of eichler and shimura that there is an elliptic curve e over q having conductor n so that l e s l f s thus if d is coprime to 2n then l ed s l fxd s also one knows from the work of kolyvagin 13 as supplemented by the work of murty and murty 17 or that of bump friedberg and hoffstein 3 see also 10 for a shorter proof that if e is a modular elliptic curve and if l e 1 7 0 then the rank of e is 0 thus if f has the property that a positive proportion of the twists of l f s have nonzero central critical value then this implies that a positive density of the quadratic twists ed have rank 0 there have been many papers which have proved results in this direction for example in 2 3 6 10 16 17 19 28 one can find general theorems on

disparity in selmer ranks of quadratic twists of elliptic curves we study the parity of 2 selmer ranks in the family of quadratic twists of an arbitrary elliptic curve e over an arbitrary number eld k we prove that the fraction of twists of a given elliptic curve over a xed number eld having even 2 selmer rank exists as a stable limit over the family of twists and we compute this fraction as an explicit product of local factors we give an example of an elliptic curve e such that as k varies these fractions are dense in 0 1 more generally our results also apply to p selmer ranks of twists of 2 dimensional self dual fp representations of the absolute galois group of k by characters of order p

existence of classical solutions for degenerate semilinear parabolic problems let q be a nonnegative real number and t be a positive real number existence of a unique classical solution is established for the following degenerate semilinear parabolic initial boundary value problem x qu t u x x f u 0