Well-posedness and blow-up phenomena for the generalized Degasperis-Procesi equation

In this paper, we mainly study the Cauchy problem of the generalized Degasperis-Procesi equation. We establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which blow up in finite time.     

On solutions to the Degasperis-Procesi equation

In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u₀(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively.     

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.     

Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation

We establish the local well-posedness for the viscous Degasperis-Procesi equation. We show that the blow-up phenomena occurs in finite time. Moreover, applying the energy identity, we obtain a global existence result in the energy space.     

On the Cauchy problem for a generalized Degasperis-Procesi equation

ABSTRACT

We first establish the local well-posedness for a generalized Degasperis-Procesi equation in nonhomogeneous Besov spaces. Then we present a global existence result for the equation. Moreover, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.     

Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation

We consider the local and global existence of solutions for a generalized Boussinesq equation utt−uxx+uxxxx+(uk+1)_xx=0, k>4, with initial data in some homogenous Besov-type space.     

New solutions for the modified generalized Degasperis-Procesi equation

In this paper, using three distinct computational methods we obtain some new exact solutions for the generalized modified Degasperis-Procesi equation (mDP equation) u_t-u_xxt+(b+1)u²u_x=bu_xu_xx+uu_xxx. We show the graph of some of the new solutions obtained here with the aim to illustrate their physical relevance. Mathematica is used. Finally some conclusions are given.     

Traveling wave solutions of the Degasperis-Procesi equation

We classify all weak traveling wave solutions of the Degasperis-Procesi equation. In addition to smooth and peaked solutions, the equation is shown to admit more exotic traveling waves such as cuspons, stumpons, and composite waves.     

On the Cauchy problem for a model equation for shallow water waves of moderate amplitude

We prove the local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime.     

Non-uniform dependence on initial data for the periodic Degasperis-Procesi equation

In this paper, we show that the solution map of the periodic Degasperis-Procesi equation is not uniformly continuous in Sobolev spaces Hs(T) for s>3/2. This extends previous result for s?2 to the whole range of s for which the local well-posedness is known. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.     

Infinite propagation speed for the Degasperis-Procesi equation

We prove that any nontrivial classical solution of the Degasperis-Procesi equation will not have compact support if its initial data has this property.     

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.     

On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

Well-posedness for the Cauchy problem associated to the Hirota-Satsuma equation: Periodic case

We consider a system of Korteweg-de Vries (KdV) equations coupled through nonlinear terms, called the Hirota-Satsuma system. We study the initial value problem (IVP) associated to this system in the periodic case, for given data in Sobolev spaces Hs×Hs+1 with regularity below the one given by the conservation laws. Using the Fourier transform restriction norm method, we prove local well-posedness whenever s>−1/2. Also, with some restriction on the parameters of the system, we use the recent technique introduced by Colliander et al., called I-method and almost conserved quantities, to prove global well-posedness for s>−3/14.     

On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis–Procesi equation

We consider the problem of the existence of the global solutions and formation of singularities for a b-family of equations which includes the Camassa–Holm and Degasperis–Procesi equation. We also consider the problem of the uniformly continuity of Degasperis–Procesi equation.     

Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation

We consider the existence, both locally and globally in time, and the blow-up of solutions for the Cauchy problem of the generalized damped multidimensional Boussinesq equation.     

The Cauchy problem for the generalized hyperelastic rod wave equation

In this paper we consider the Cauchy problem for the generalized hyperelasticrod wave equation which includes the Camassa‐Holm equation and the hyperelastic rod wave equation. Firstly, by using the Kato's theory, we prove that the Cauchy problem for the generalized hyperelastic rod wave is locally well‐posed in Sobolev spaces with . Secondly, we give some conservation laws, some useful conclusions and the precise blow‐up scenario and show that the Cauchy problem for the generalized hyperelastic rod wave equation has smooth solutions which blows up in finite time. Thirdly, we give the blow‐up rate of the strong solutions to the Cauchy problem for the generalized hyperelastic rod wave equation. Finally, we give the lower bound of the maximal existence time of the solution and the lower semicontinuity of existence time of solutions to the generalized hyperelastic rod wave equation.     

Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation

For s < 3/2, it is shown that the Cauchy problem for the Degasperis-Procesi equation (DP) is ill-posed in Sobolev spaces H ^s . If 1/2 ≤ s < 3/2, then ill-posedness is due to norm inflation. This means that there exist DP solutions who are initially arbitrarily small and eventually arbitrarily large with respect to the H ^s norm, in an arbitrarily short time. Since DP solutions conserve a quantity equivalent to the L ²-norm, there is no norm inflation in H ⁰ for these solutions. In this case, ill-posedness is caused by failure of uniqueness. For all other s < 1/2, the situation is similar to H ⁰. Considering that DP is locally well-posed in H ^s for s > 3/2, this work establishes 3/2 as the critical index of well-posedness in Sobolev spaces.