a multi dimensional quenching problem due to a concentrated nonlinear source in rn abstract let b be a n dimensional ball x r n x r centered at the origin with a radius r b be its closure and b be its boundary also let x denote the unit inward normal at x b and b x be the characteristic function which is 1 for x b and 0 for x r n b this article studies the following multi dimensional semilinear parabolic first initial boundary value problem with a concentrated nonlinear source on b u t u b x f u in r n 0 t u x 0 0 for x r n u x t 0 as x for 0 t t where and t are positive numbers f is a given function such that lim u c f u for some positive constant c and f u and its derivatives f u and f u are positive for 0 u c it is shown that the problem has a unique nonnegative continuous solution before quenching occurs and if u quenches in a finite time then it quenches everywhere on b only it is proved that u always quenches in a finite time for n 2 for n 3 it is shown that there exists a unique number such that u exists globally for and quenches in a finite time for thus quenching does not occur in infinite time a formula for computing is given a computational method for finding the quenching time is devised

a simple method and its applications to nonlinear partial differential equations abstract in this paper a simple method is proposed for constructing more general exact solutions of nonlinear partial differential equations we choose the camassa and holm degasperis and procesi equation and the generalized b family equations to illustrate the validity and advantages of the method as a result many new and more general exact solutions are obtained some previous results are extended

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

boundary control of viscosity degasperis procesi equation abstract this paper investigates the boundary control problem of the viscosity degasperis procesi equation in 0 1 results of the existence of a solution to the viscosity degasperis procesi equation in a short time interval under the boundary condition are given in the paper it proves the global exponential stability under the given boundary condition in l 2 h 1 h 2 h 3 space

on the integrable perturbations of the camassa holm equation we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure

on the behavior of the solution of the dissipative camassa holm equation with the arbitrary dispersion coefficient abstract in this paper we consider the dissipative camassa holm equation with arbitrary dispersion coefficient and compactly supported initial data we demonstrate the simple conditions on the initial data that lead to finite time blow up of the solution in finite time or guarantee that the solution exists globally also propagation speed for the equation under consideration is investigated

on the cauchy problem for the generalized shallow water wave equation abstract in this paper we mainly study the cauchy problem for the generalized shallow water wave equation in the sobolev space h s of lower order s using the crucial bilinear estimates in the fourier transform restriction spaces related to the shallow water wave equation we establish local well posedness in h s with any s 11 16

wave breaking for shallow water models with time decaying solutions abstract a family of camassa holm type equations with a linear term and cubic and quartic nonlinearities is considered local well posedness results are established via kato s approach conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is a time dependent monotonically decreasing function of time and bounded from above by the sobolev norm of the initial data under some conditions the existence of wave breaking phenomenon is investigated and necessary conditions for its existence are obtained in our framework the wave breaking is guaranteed among other conditions when the coefficient of the linear term is sufficiently small which allows us to interpret the equation as a linear perturbation of some recent camassa holm type equations considered in the literature

classification of travelling waves in the fornberg whitham equation abstract in this paper the fornberg whitham fw equation is investigated by using the improved qualitative method which combines characteristics of several methods to solve equations we classify all travelling wave solutions of this equation in specified regions of the parametric space besides already known solutions such as peaked and kink like wave solutions this equation also admits looped and cusped wave solutions especially some composite waves including fractal like waves and stumpons what is more interesting is that we give the limit behavior of all periodic solutions as the parameters trend to some special values

blow up of solutions for the dissipative dullin gottwald holm equation with arbitrary coefficients abstract we study the cauchy problem for the weakly dissipative dullin gottwald holm equation describing unidirectional propagation of surface waves in a shallow water regime u t 2 u x x t c 0 u x 3 u u x u x x x u 2 u x x 2 2 u x u x x u u x x x in the present paper we demonstrate the simple conditions on the initial data that lead to blow up of the solution in finite time

blow up phenomena for the weakly dissipative dullin gottwald holm equation in this paper we study the cauchy problem for the weakly dissipative dullin gottwald holm equation we establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time

towered waves and anti waves in the generalized degasperis procesi equation new traveling wave solutions of the generalized degasperis procesi equation are investigated the solutions are characterized by three parameters using an improved qualitative method abundant traveling wave solutions such as smooth waves peaked waves cusped waves compacted waves looped waves and fractal like waves are obtained especially some strange composite wave solutions such as towered waves and their anti waves are first given we also study the limiting behavior of all periodic solutions as the parameters trend to some special values

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

on the inverse scattering approach and action angle variables for the dullin gottwald holm equation abstract the poisson brackets for the scattering data of the dullin gottwald holm equation are computed then the action angle variables are expressed in terms of the scattering data we also discuss on the inverse scattering problem for this equation

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

low regularity solutions of the periodic fornberg whitham equation this paper studies low regularity solutions of the periodic fornberg whitham equation with initial value the existence and the uniqueness of solutions are proved the results are illustrated by considering the periodic peakons of the periodic fornberg whitham equation

local well posedness and stability of solitary waves for the two component dullin gottwald holm system abstract in this paper we study the cauchy problem for the two component dullin gottwald holm system with the initial data u 0 0 b p r s r b p r s 1 r 1 p r and s max 1 1 p 3 2 2 1 p which generalizes some previous local well posed results in sobolev spaces then we prove that all the smooth solitary wave solutions are orbitally stable under small disturbances

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

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

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

existence and uniqueness of low regularity solutions for the dullin gottwald holm equation we establish the local existence and uniqueness of solutions for the dullin gottwald holm equation with continuously differentiable periodic initial data the regularity conditions needed for the cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered

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

the limit behavior of the solutions to a class of nonlinear dispersive wave equations in this paper we study the limit behavior of the solutions to a class of nonlinear dispersive wave equations we also demonstrate that the solutions to eq 1 1 converge to the solution to the corresponding bbm equation as the parameter converges to zero and the convergence of solutions to eq 1 1 as 2 0 is studied in hs r s 32 given the discussion of the parameters in the nonlinear dispersive wave equation 1 1 one will obtain some conditions in which compacton and peakon occur

limit behavior of the global solutions to the degasperis procesi type equation abstract in this paper we study the degasperis procesi equation with a physically perturbation term a linear dispersion based on the global existence result we show that the solution of the degasperis procesi equation with linear dispersion tends to the solution of the corresponding degasperis procesi equation as the dispersive parameter goes to zero moreover we prove that smooth solutions of the equation have finite propagation speed they will have compact support if its initial data has this property

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

blow up phenomena for the weakly dissipative dullin gottwald holm equation revisited in this paper we investigate the cauchy problem for the weakly dissipative dullin gottwald holm equation we first establish the local well posedness result by kato s semigroup theory then we obtain the precise blow up scenario and present some blow up results for strong solutions to the equation finally we discuss the blow up rate of the wave breaking solutions this result complements the early one in the literature such as novruzov j math phys 54 092703 2013

on a dissipative form of camassa holm equation in this paper we consider several problems on the camassa holm equation with a weak dissipation term first we establish the local well posedness result and then we present several new blow up results and discuss the global existence of the solution for the periodic case finally we examine the persistence of decay properties for the weakly dissipative camassa holm equation and we prove that any nontrivial classical solution of the equation will not have compact support if its initial data have this property

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

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

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

global weak solutions and wave breaking phenomena to the periodic degasperis procesi equation with strong dispersion abstract considered herein is the initial value problem for the periodic degasperis procesi equation with a strong dispersive term that is an approximation to the incompressible euler equation for shallow water waves the existence and uniqueness of global weak solutions are established moreover the periodic peakon and shockpeakon solutions of the equation are constructed

l a priori bounds for gradients of solutions to quasilinear inhomogeneous fast growing parabolic systems abstract we prove boundedness of gradients of solutions to quasilinear parabolic systems the main part of which is a generalization to the p laplacian and its right hand side s growth depending on the gradient is not slower and generally strictly faster than p 1 this result may be seen as a generalization to the classical notion of a controllable growth of the right hand side introduced by campanato over gradients of p laplacian like systems energy estimates and a nonlinear iteration procedure of the moser type are cornerstones of the used method