stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

a solvable one dimensional model of a diffusion inventory system we analyse the optimal ordering policy for impulse control of a one product inventory system subject to a demand modelised by a diffusion process the purpose is to minimize the expected discounted cost that includes a fixed set up cost and linear costs of purchase storage and shortage the optimal cost is explicitly obtained as the smoothest solution of a quasi variational inequality derived from the optimal principle of dynamic programming the optimal s s policy is determined as the unique solution of a system of algebraic equations

global existence for the higher order camassa holm shallow water equation abstract in this paper we investigate the global existence of the higher order camassa holm equation in the case of k 2 we prove the local well posedness of this equation and find a conservation law then a global existence result is obtained

on the blow up phenomena of the periodic dullin gottwald holm equation we first present several new blow up results for the periodic dullin gottwald holm equation by using some new sobolev inequalities we then investigate the blow up rate for all nonglobal solutions and determine the blow up set of blowing up solutions to the equation for a large class of initial data we finally address the lower semicontinuity of the existence time of solutions with smooth initial data

well posedness global existence and blow up phenomena for an integrable multi component camassa holm system abstract this paper is concerned with a new integrable multi component camassa holm system which has been proven to be integrable and has peakon solutions this system includes many one component and two component camassa holm type systems as special cases in this paper we first establish the local well posedness and a continuation criterion for the system then we present several global existence and blow up results for two important integrable two component subsystems our obtained results cover and improve considerably recent results in gui et al 2013 yan et al 2015

on the wave breaking phenomena for the periodic two component dullin gottwald holm system abstract considered herein is the well posedness problem of the periodic two component dullin gottwald holm dgh system on the circle which can be derived from euler s equation with constant vorticity in shallow water waves moving over a linear shear flow the result of blow up solutions for certain initial profiles in a manner which corresponds to wave breaking is established

stability of peakons and linear dispersion limit for the periodic dullin gottwald holm equation the dullin gottwald holm equation can be described as the unidirectional propagation of surface waves in a shallow water regime in this paper we study the orbital stability problem of the peaked solitons peakons for the periodic dullin gottwald holm equation on the line by constructing a function only depending on three important conservative laws we prove that the shapes of peakons are stable under small perturbations we also demonstrate that the solutions of the cauchy problem for this equation converge to those of the corresponding periodic camassa holm equation as the linear dispersive parameter converges to zero

global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation this paper is concerned with the global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation we prove the global existence and uniqueness of solution to the viscous degasperis procesi equation for u0 l2 r the local well posedness for u0 l2 r has been proven based on the energy estimate the strong and weak limit behaviors of solution to the viscous degasperis procesi equation are obtained

revisit to wave breaking phenomena for the periodic dullin gottwald holm equation in this paper we mainly devote to investigate the periodic dullin gottwald holm equation by overcoming the difficulties caused by the complicated mixed nonlinear structure a very useful priori estimate is derived in lemma 2 7 based on h 1 conservation and l estimate of solution some new blow up phenomena are derived for the periodic dullin gottwald holm equation under different initial conditions

on existence and uniqueness of the global weak solution for a shallow water equation abstract a shallow water equation including the camassa holm and degasperis procesi equations as special cases is investigated provided that initial value u 0 h 1 r u 0 l 1 r and 1 x 2 u 0 does not change sign the existence and uniqueness of global weak solution for the equation are established

explicit solution of a two dimensional deterministic inventory problem we determine the optimal ordering policy for impulse control of a deterministic two product inventory system subject to constant demand rates linear storage and shortage costs and economies of joint ordering the optimal cost is explicitly obtained as the smoothest solution of a two dimensional quasi variational inequality exact construction of the boundary of the continuation set leads to the optimal ordering policy extensions to independent products average cost and n product case n 2 are also briefly considered

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

global existence and blow up solutions for a nonlinear shallow water equation considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by dullin gottward and holm blow up can occur only in the form of wave breaking a wave breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow up rate is established the blow up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined

new exact periodic wave solutions for the dullin gottwald holm equation abstract in this paper the dullin gottwald holm equation is studied using semi inverse method and integral bifurcation method new periodic waves such as peakon like periodic wave compacton like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion the exact parametric representations of these waves are also presented

precise asymptotics in the deviation probability series of self normalized sums abstract let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and w n 2 x 1 2 x n 2 in the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self normalized sums s n w n for positive functions g x x x and x we obtain the precise asymptotics for the following deviation probabilities of self normalized sums n 1 g n n e s n w n 2 i s n w n n n the results given can be considered as the generalization of that in the complete moment convergence law of iterated logarithm and large deviation for self normalized sums

on permanent and breaking waves in hyperelastic rods and rings we prove that the only global strong solution of the periodic rod equation vanishing in at least one point t0 x0 r s1 is the identically zero solution such conclusion holds provided the physical parameter of the model related to the finger deformation tensor is outside some neighborhood of the origin and applies in particular for the camassa holm equation corresponding to 1 we also establish the analogue of this unique continuation result in the case of non periodic solutions defined on the whole real line with vanishing boundary conditions at infinity our analysis relies on the application of new local in space blowup criteria and involves the computation of several best constants in convolution estimates and weighted poincare inequalities

on the integrability of degasperis procesi equation control of the sobolev norms and birkhoff resonances abstract we consider the dispersive degasperis procesi equation u t u x x t c u x x x 4 c u x u u x x x 3 u x u x x 4 u u x 0 with c r 0 in 15 the authors proved that this equation possesses infinitely many conserved quantities we prove that there are infinitely many of such constants of motion which control the sobolev norms and which are analytic in a neighborhood of the origin of the sobolev space h s with s 2 both on r and t by the analysis of these conserved quantities we deduce a result of global well posedness for solutions with small initial data and we show that on the circle the formal birkhoff normal form of the degasperis procesi at any order is action preserving

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

optimal control of the viscous degasperis procesi equation this paper studies the problem for optimal control of the viscous degasperis procesi equation the existence and uniqueness of weak solution to the equation are proved in a short interval the optimal control of the viscous degasperis procesi equation under boundary condition is given and the existence of optimal solution to the equation is proved

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

orbital stability of peakons for the degasperis procesi equation with strong dispersion abstract in this paper we study the orbital stability of the peakons for the degasperis procesi equation with a strong dispersive term on the line using the method in z lin y liu stability of peakons for the degasperis procesi equation comm pure appl math 62 2009 125 146 we prove that the shapes of these peakons are stable under small perturbations some previous results are extended

wave breaking phenomenon for a modified two component dullin gottwald holm equation this paper is concerned with the wave breaking phenomenon for a modified two component dullin gottwald holm shallow water system we give sufficient conditions on the initial data to guarantee blow up of solutions in finite time

classification of bounded travelling wave solutions for the dullin gottwald holm equation abstract in this paper we classify all bounded travelling wave solutions for the integrable dullin gottwald holm equation it is shown that it decomposes in two known cases the camassa holm and the korteweg de vries equation for the former the classification is similar to the one presented in 2005 12 while for the latter it is only possible to obtain smooth solutions