nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

blow up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities abstract in this work we study the blow up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities we apply an iteration method to study both the subcritical case and the critical case in the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions in the critical case we use the slicing method and a couple of auxiliary functions recently introduced by wakasa yordanov to modify the definition of the functionals with the introduction of weight terms in particular we find as critical curve for the pair p q of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities

lifespan of semilinear wave equation with scale invariant dissipation and mass and sub strauss power nonlinearity abstract in this paper we study the blow up of solutions for semilinear wave equations with scale invariant dissipation and mass in the case in which the model is somehow wave like a strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems two blow up results are obtained for the subcritical case and for the critical case respectively in both cases an upper bound lifespan estimate is given

the lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations abstract the sharp upper bound of the lifespan of the solution to p q systems in high space dimensions has been obtained in the lifespan of solutions to nonlinear systems of high dimensional wave equation preprint for the sub critical case they also proved the lower bound of the lifespan with a small error term the present paper remove this error term and give the sharp lifespan under the assumption of the radial symmetricity of the solution

blow up for semilinear damped wave equations with subcritical exponent in the scattering case abstract it is well known that the critical exponent for semilinear damped wave equations is fujita exponent when the damping is effective lai takamura and wakasa in 2017 have obtained a blow up result not only for super fujita exponent but also for the one closely related to strauss exponent when the damping is scaling invariant and its constant is relatively small which has been recently extended by ikeda and sobajima introducing a multiplier for the time derivative of the spatial integral of unknown functions we succeed in employing the techniques on the analysis for semilinear wave equations and proving a blow up result for semilinear damped wave equations with sub strauss exponent when the damping is in the scattering range

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

an elementary proof of the blow up for semilinear wave equation in high space dimensions abstract this paper concerns the blow up of solutions to u tt u u p in high dimensions for n 4 and 1 p p 0 n where p 0 n is a critical exponent we proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

the equation utt δu u p for the critical value of p the equation u tt u u p is considered in two and three space dimensions smooth cauchy data of compact support are given at t 0 for the case of three space dimensions john has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if glassey has shown the two dimensional case is similar this paper shows that small data solutions blow up in finite time when p is the critical value in three dimensions and in two

on the nonexistence of global solutions for critical semilinear wave equations with damping in the scattering case abstract we consider the cauchy problem for semilinear wave equations with variable coefficients and time dependent scattering damping in r n where n 2 it is expected that the critical exponent will be strauss number p 0 n which is also the one for semilinear wave equations without damping terms lai and takamura 2018 have obtained the blow up part together with the upper bound of lifespan in the sub critical case p p 0 n in this paper we extend their results to the critical case p p 0 n the proof is based on wakasa and yordanov 0000 which concerns the blow up and upper bound of lifespan for critical semilinear wave equations with variable coefficients

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

the sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions the final open part of strauss conjecture on semilinear wave equations was the blow up theorem for the critical case in high dimensions this problem was solved by yordanov and zhang 2006 18 or zhou 2007 21 independently but the estimate for the lifespan the maximal existence time of solutions was not clarified in both papers in this paper we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan as a result with the sharp lower bound by li and zhou 1995 10 the lifespan t e of solutions of utt u u2 in r4 0 with the initial data u x 0 ef x ut x 0 eg x of a small parameter e 0 compactly supported smooth functions f and g has an estimate exp ce 2 t e exp ce 2 where c and c are positive constants depending only on f and g this upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations

critical curve for p q systems of nonlinear wave equations in three space dimensions abstract the existence of the critical curve for p q systems for nonlinear wave equations was already established by d del santo v georgiev and e mitidieri 1997 global existence of the solutions and formation of singularities for a class of hyperbolic systems in geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 139 birkhauser basel except for the critical case our main purpose is to prove a blow up theorem for which the nonlinearity p q is just on the critical curve in three space dimensions moreover the lower and upper bounds of the lifespan of solutions are precisely estimated including the sub critical case

sharp upper bound for lifespan of solutions to some critical semilinear parabolic dispersive and hyperbolic equations via a test function method abstract this paper is concerned with the blowup phenomena for initial boundary value problem 0 1 t 2 u x t u x t a x t u x t u x t p x t c 0 t u x t 0 x t c 0 t u x 0 e f x x c t u x 0 e g x x c where c is a cone like domain in r n n 2 defined as c int r r n r 0 with a connected open set in s n 1 with smooth boundary if n 1 then we only consider two cases c 0 and c r here a x is a non zero coefficient of t u which could be complex valued and space dependent is a fixed constant and e 0 is a small parameter the constants 0 1 switch the parabolicity and hyperbolicity of the problem 0 1 the result proposes an argument for sharp upper lifespan estimates of solutions to 0 1 by a test function method based on mitidieri and pokhozhaev 2001 the crucial idea is to introduce an ordinary differential inequality with a parameter as a variable

blow up for semilinear wave equations with the scale invariant damping and super fujita exponent abstract the blow up for semilinear wave equations with the scale invariant damping has been well studied for sub fujita exponent however for super fujita exponent there is only one blow up result which is obtained in 2014 by wakasugi in the case of non effective damping in this paper we extend his result in two aspects by showing that i the blow up will happen for bigger exponent which is closely related to the strauss exponent the critical number for non damped semilinear wave equations ii such a blow up result is established for a wider range of the constant than the known non effective one in the damping term

the lifespan of solutions to nonlinear systems of a high dimensional wave equation abstract in this work we study the lifespan of solutions to a p q system in the higher dimensional case n 4 a suitable local existence curve in the p q plane is found the curve characterizes the local solutions in sobolev space h s with s 0 further some lower and upper bounds of the lifespan of classical solutions are found too this work is an extension of work geometric optics and related topics progress in nonlinear differential equations and their applications vol 32 birkhauser basel 1997 pp 117 140 where a suitable global existence small data curve is studied in the subcritical case we give almost precise results for the lower bounds of the lifespan by using a suitable weighted strichartz estimate for the higher dimensional wave equation

strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation the aim of this article is twofold first we consider the wave equation in the hyperbolic space hi and obtain the counterparts of the strichartz type estimates in this context next we examine the relationship between semilinear hyperbolic equations in the minkowski space and in the hyperbolic space this leads to a simple proof of the recent result of georgiev lindblad and sogge on global existence for solutions to semilinear hyperbolic problems with small data shifting the space time strichartz estimates from the hyperbolic space to the minkowski space yields weighted strichartz estimates in rn x 1r which extend the ones of georgiev lindblad and sogge