A model containing both the Camassa-Holm and Degasperis-Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases, is investigated. Although the H¹-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space Hs with is established under the assumptions u₀∈Hs and ‖u0xL^∞_‖<∞. The local well-posedness of solutions for the equation in the Sobolev space Hs(R) with is also developed.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions

We study the global existence of solutions of the nonlinear degenerate wave equation (ρ?0)

     

The existence and uniqueness of the local solution for a Camassa-Holm type equation

A shallow water equation of Camassa-Holm type, containing nonlinear dissipative effect, is investigated. Using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself, we establish the existence and uniqueness of its local solution in Sobolev space H^s(R) with . Meanwhile, a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order Sobolev space H^s with are presented.     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Global solutions and blow-up phenomena to a shallow water equation

A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u₀∈Hs () and u₀∈L¹(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C¹([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.     

Three positive solutions of semilinear elliptic equations in the half space with a hole

In this paper, assume that h is nonnegative and ‖hL²_‖>0, we prove that if ‖hL²_‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in , where D is a C1,1 bounded domain in .     

The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations

In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L^p(Ω), and , respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L^p(Ω), and , the results in this part are new and appear to be optimal.     

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 . 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.     

Polynomial Pell's equation-II

The polynomial Pell's equation is X²−DY²=1, where D is a polynomial with integer coefficients and the solutions X,Y must be polynomials with integer coefficients. Let D=A²+2C be a polynomial in , where . Then for a prime, a necessary and sufficient condition for which the polynomial Pell's equation has a nontrivial solution is obtained. Furthermore, all solutions to the polynomial Pell's equation satisfying the above condition are determined.     

On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We present a systematic study of local solutions of the ODE of the form near t=0. Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ>0 and γ<0, where γ=∂f/∂x^′ at t=0. As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.     

A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

The -expansion method can be used for constructing exact travelling wave solutions of real nonlinear evolution equations. In this paper, we improve the -expansion method and explore new application of this method to (2+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation. New types of exact complex travelling wave solutions of (2+1)-dimensional BKP equation are found. Some exact solutions of (2+1)-dimensional BKP equation obtained before are special cases of our results in this paper.     

Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in R^N,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.