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

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

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

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

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

nonexistence of global solutions of a nonlinear hyperbolic system consider the initial value problem utt au ivip vtt av u q x e r t t o u x 0 f x v x 0 h x ut x 0 g x vt x 0 x with 1 0 we show that there exists a bound b n oo such that if 1 pq b n all nontrivial solutions with compact support blow up in finite time

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

global existence for the dissipative wave equations with space time dependent potential abstract this research paper aims at studying the long time behavior of solutions for wave equations with focusing nonlinearity we consider p c r n p n 2 n 2 and separable potential a t x x t with x 1 x and t 1 t satisfying conditions 3 4 which are given below the main result is that if the exponent p lies between p c r 1 4 1 2 n 1 2 p n 2 n 2 the solutions are global for sufficiently small data also the precise decay estimates for the energy l 2 and l p 1 norms of solutions in the region of global solutions have been found

blow up and critical exponents for nonlinear hyperbolic equations we prove nonexistence results for the cauchy problem for the abstract hyperbolic equation in a banach space x utt f u t 0 u 0 u0 ut 0 u1 where f x r is a c1 function several applications to the second and higher order hyperbolic equations with local and nonlocal nonlinearities are presented we also describe an approach to kato s and john s critical exponents for the semilinear equations ut u b x t u p p 1 which are responsible for phenomena of stability unstability blow up and asymptotic behaviour

lower central series and free resolutions of hyperplane arrangements if m is the complement of a hyperplane arrangement and a h m k is the cohomology ring of m over a field of characteristic 0 then the ranks k of the lower central series quotients of 1 m can be computed from the betti numbers bii dimtor a k k i of the linear strand in a minimal free resolution of k over a we use the cartan eilenberg change of rings spectral sequence to relate these numbers to the graded betti numbers b dimtor e a k j of a minimal resolution of a over the exterior algebra e from this analysis we recover a formula of falk for 3 and obtain a new formula for 4 the exact sequence of low degree terms in the spectral sequence allows us to answer a question of falk on graphic arrangements and also shows that for these arrangements the algebra a is koszul if and only if the arrangement is supersolvable we also give combinatorial lower bounds on the betti numbers b 1 of the linear strand of the free resolution of a over e if the lower bound is attained for i 2 then it is attained for all i 2 for such arrangements we compute the entire linear strand of the resolution and we prove that all components of the first resonance variety of a are local for graphic arrangements which do not attain the lower bound unless they have no braid sub arrangements we show that b 1 is determined by the number of triangles and k4 subgraphs in the graph

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

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

on the solvability of one multidimensional version of the first darboux problem for some nonlinear wave equations in the present paper for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first darboux problem in the conic domain

global nonexistence results for a class of hyperbolic systems abstract our concern in this paper is to prove blow up results to the non autonomous nonlinear system of wave equations u t t u a t x v p v t t v b t x u q t 0 x r n in any space dimension we show that a curve f p q 0 depending on the space dimension on the exponents p q and on the behavior of the functions a t x and b t x exists such that all nontrivial solutions to the above system blow up in a finite time whenever f p q 0 our method of proof uses some estimates developed by galaktionov and pohozaev in 11 for a single non autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived our result generalizes some important results such as the ones in del santo et al 1996 12 and galaktionov and pohozaev 2003 11 the advantage here is that our result applies to a wide variety of problems

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

a dimension spectrum for sle boundary collisions we consider chordal sle curves for where the intersection of the curve with the boundary is a random fractal of almost sure hausdorff dimension we study the random sets of points at which the c

blow up theorem for semilinear wave equations with non zero initial position abstract one of the features of solutions of semilinear wave equations can be found in blow up results for non compactly supported data in spite of finite propagation speed of the linear wave we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak this was first observed by asakura 1986 2 finding out a critical decay to ensure the global existence of the solution but the blow up result is available only for zero initial position having positive speed in this paper the blow up theorem for non zero initial position by uesaka 2009 22 is extended to higher dimensional case and the assumption on the nonlinear term is relaxed to include an example u p 1 u moreover the critical decay of the initial position is clarified by example

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

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

critical criteria of fujita type for a system of inhomogeneous wave inequalities in exterior domains abstract we consider blow up results for a system of inhomogeneous wave inequalities in exterior domains we will handle three type boundary conditions dirichlet type neumann type and mixed boundary conditions we use a unified approach to show the optimal criteria of fujita type for each case our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation we provide many new results and close some open questions

critical blowup for systems of semilinear wave equations in low space dimensions abstract we consider the cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data del santo et al geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 140 birkhauser boston 1997 have studied the existence and nonexistence of global classical solutions of the cauchy problem except for the critical case in this paper we study the critical case and we show the nonexistence of global classical solutions and also give the upper bounds of the life span

on singularity of a nonlinear variational sine gordon equation abstract we study a sine gordon type of nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude this equation arises naturally from long waves on a dipole chain in the continuum limit which provides a crude model for some polymers using characteristic methods we describe a blow up result for the one dimensional nonlinear variational sine gordon equation which shows that smooth solutions breakdown in finite time

life span of small solutions to a system of wave equations abstract we study the cauchy problem with small initial data for a system of semilinear wave equations u v q v t u p in n dimensional space when n 2 we prove that blow up can occur for arbitrarily small data if p q lies below a curve in the p q plane on the other hand we show a global existence result for n 3 which asserts that a portion of the curve is in fact the borderline between global in time existence and finite time blow up we also estimate the maximal existence time and get its upper bound which is sharp at least for n p q 2 2 2 and 3 2 2

the first darboux problem for wave equations with a nonlinear positive source term abstract we consider the first darboux problem for nonlinear wave equations with positive power nonlinearity source term depending on the power of nonlinearity we investigate the problem on a global existence and blow up of solutions of the first darboux problem the question of local solvability of the problem is also considered