hyperbolic like estimates for higher order equations abstract the main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time dependent coefficients in particular we estimate the energy in the hyperbolic zone of the extended phase space by means of a function f t which depends on the principal part and on the coefficients of the terms of order m 1 then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space we call this class of estimates hyperbolic like since the energy behavior is deeply depending on the hyperbolic structure of the equation in some cases these estimates produce a dissipative effect on the energy

on the congruence 2 n 2 1 mod n there exist infinitely many positive integers n such that 2 2 1 mod n in the monograph 5 i proposed the following problem problem 18 p 138 let a k 1 be fixed positive integers do there exist infinitely many composite n such that a 1 put a 2 k 2 in the above problem since by fermat s theorem 2p l 1 mod p for odd primes p if 2 2 1 mod n and n 2 n must be composite r matuszewski and p rudnicki with the aid of the computer k 202 in warsaw checked that below 4208 such integers do not exist the following theorem holds congruence 2m 1 mod2m 1 it follows that 2m v v m 1 mod2m 1 22 1 3 s 1 mod2m 1 and 2 2 1 modn for n 2m 1 thus 2 2 1 mod n for n 2 1 where 0 4700063497 suppose now that 2 2 1 mod and n 8 let be a primitive factor of the number 2 2 1 a prime factor of 2 1 is said to be primitive if it does not divide any of the numbers 2m 1 for m 1 2 n 1 by a theorem of k zsigmondy 7 such a prime factor exists for any n 6 and is of the form nt 1 now we shall show that nx np is also a solution of the congruence 2 1 2 1 mod we have p 2 n 2 k 1 where k is a positive integer and p 2 3 n and 1 thus

integrals involving airy functions the authors show how to evaluate a large number of integrals involving airy functions the method uses the fact that the wronskian has a very simple form

long time asymptotics for 2 by 2 hyperbolic systems the goal of this paper is to study the behavior of the energy for 2 by 2 strictly hyperbolic systems on the one hand we are interested in generalized energy conservation which excludes blow up and decay of the energy for t on the other hand we present scattering results which take account of terms of order zero

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

an analytic solution to the thomas fermi equation a perturbative procedure due to bender et al here referred to as the bmps procedure j math phys 30 1447 1989 and useful in solving difficult nonlinear problems has been used here to solve the thomas fermi t f equation the present work attempts to balance the ease of the ensuing analysis with the use of an analytic zero order function that already contains a good deal of the nonlinearity of the t f equation the initial slope of the t f potential is computed with 0 35 error in a second order application of the theory

critical exponents for semilinear dissipative wave equations in rn abstract we shall present new critical exponents 1 2m n with m 1 2 to the cauchy problem utt u ut u p 1u with the initial data u 0 u 1 h 1 r n l m r n l 2 r n l m r n that is the small data global existence property can be derived to the cauchy problem above with power 1 2m n

blow up of the energy at infinity for 2 by 2 systems abstract the goal of this paper is to study the long time behavior of the energy of solutions to 2 by 2 linear hyperbolic systems some blow up results at t are given for a large class of initial data of the cauchy problem we shall prove the optimality of these results for a special class of systems we present also an example of generalized energy conservation under a c m stabilization condition

the influence of a nonlinear memory on the damped wave equation abstract in this paper we study the influence of a nonlinear memory f t u 0 t t s u s x p d s 0 1 on the global existence of small data solutions to u t t u u t f t u u 0 x u 0 x u t 0 x u 1 x in space dimension 1 n 5 we prove the global existence for p p n where p n is the critical exponent i e no global weak solution exists for 1 p p n for suitable arbitrarily small data to prove our result we consider small data in some energy space h k h k 1 where k 1 with additional l 1 regularity we also discuss what happens if this latter assumption is dropped i e data are only assumed to be small in h k h k 1 for some k 1

some pseudoprimes and related numbers having special forms we give an example of a pseudoprime which is itself of the form 2n 2 answering a question posed by a rotkiewicz show that lehmer s example of an even pseudoprime having three prime factors is not unique and answer a question of benkoski concerning the solutions of 2n 2 1 mod n

integrals involving products of airy functions their derivatives and bessel functions abstract a new integral representation of the hankel transform type is deduced for the function f n x z z n 1 a i x z a i x z with x r z 0 and n n this formula involves the product of airy functions their derivatives and bessel functions the presence of the latter allows one to perform various transformations with respect to z and obtain new integral formulae of the type of the mellin transform k transform laplace and fourier transform some integrals containing airy functions their derivatives and chebyshev polynomials of the first and second kind are computed explicitly a new representation is given for the function a i z 2 with z c

differential and integral relations involving fractional derivatives of airy functions and applications abstract various differential and integral relations are deduced that involve fractional derivatives of the airy function ai x and the scorer function gi x several new wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the airy functions new fractional derivative conservation laws are derived for equations of the korteweg de vries type

global existence of solutions for semilinear damped wave equations in rn with noncompactly supported initial data abstract we consider the n dimensional cauchy problem in r n for a semilinear damped wave equation with a power type nonlinearity u p for a noncompactly supported initial data which has a small energy we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 2 n p n n 2 this generalizes a previous result due to todorova yordanov j differential equations 174 2001 464 489 which dealt with a solution in the framework of compactly supported initial data

an equation whose fujita critical exponent is not given by scaling in this paper we prove sharp blow up and global existence results for a heat equation with nonlinear memory it turns out that the fujita critical exponent is not the one which would be predicted from the scaling properties of the equation

the lp lq estimates of solutions to one dimensional damped wave equations and their application to the compressible flow through porous media we first obtain the lp lq estimates of solutions to the cauchy problem for one dimensional damped wave equation vtt vxx vt 0 v vt t 0 v0 v1 x x t r r corresponding to that for the parabolic equation t xx 0 t 0 v0 v1 x the estimates are shown by v t e t 2v0 t v0 t 2lp ct 121q 1p 1 v0 v1 lq t 1 etc for 1 q p to show the explicit formula of the damped wave equation will be used to apply the estimates to nonlinear problems is the second aim we will treat the system of a compressible flow through porous media the solution is expected to behave as the diffusion wave which is the solution to the porous media equation due to the darcy law when the initial data has the same constant state at a sharp lp convergence rate for p 2 has been recently obtained by nishihara proc roy soc edinburgh sect a 133a 2003 1 20 by choosing a suitably located diffusion wave we will show the l1 convergence applying

the continuing search for wieferich primes a prime p satisfying the congruence 2 p 1 1 mod p 2 is called a wieferich prime although the number of wieferich primes is believed to be infinite the only ones that have been discovered so far are 1093 and 3511 this paper describes a search for further solutions the search was conducted via a large scale internet based computation the result that there are no new wieferich primes less than 1 25 10 15 is reported

evaluation of an integral involving airy functions the authors evaluate an integral which has occurred in recent studies of the asymptotic behaviour of the gap in the mathieu equation