The Properties of Solutions for a Generalized b-Family Equation with Peakons

This paper deals with the Cauchy problem for a shallow water equation with high-order nonlinearities, y _t +u ^m+1 y _x +bu ^m u _x y=0, where b is a constant, $m\in \mathbb{N}$ , and we have the notation $y:= (1-\partial_{x}^{2}) u$ , which includes the famous Camassa–Holm equation, the Degasperis–Procesi equation, and the Novikov equation as special cases. The local well-posedness of strong solutions for the equation in each of the Sobolev spaces $H^{s}(\mathbb{R})$ with $s>\frac{3}{2}$ is obtained, and persistence properties of the strong solutions are studied. Furthermore, although the $H^{1}(\mathbb{R})$ -norm of the solution to the nonlinear model does not remain constant, the existence of its weak solutions in each of the low order Sobolev spaces $H^{s}(\mathbb{R})$ with $1<s<\frac{3}{2}$ is established, under the assumption $u_{0}(x)\in H^{s}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$ . Finally, the global weak solution and peakon solution for the equation are also given.     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

Well-posedness and blowup phenomena for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

This paper deals with the Cauchy problem for a cross-coupled Camassa–Holm equation $$m_t=-(vm)_x-mv_x, n_t=-(un)_x-nu_x,$$ where \({n\doteq v-v_{xx}}\) , \({m\doteq u-u_{xx}+\omega}\) with a constant ω. The local well-posedness of solutions for the Cauchy problem of the cross-coupled Camassa–Holm equation in Sobolev space \({H^s(\mathbb{R})}\) with s > 5/2 is established. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and the blowup scenario of the solutions to the equation is also obtained.     

On the non-existence of global solutions to the Cauchy problem for semilinear parabolic equations and inequalities

We generalize and improve recent non-existence results for global solutions to the Cauchy problem for the inequality as well as for the equation u_t = Δu + |u|^q in the half-space . Received: 16 September 2005     

Non‐uniform dependence and persistence properties for coupled Camassa–Holm equations

This paper deals with the non‐uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well‐posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces H^s with s > 3/2. On the other hand, the persistence properties in weighted L^p spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd.     

On the periodic Cauchy problem for a coupled Camassa–Holm system with peakons

In this paper, we first prove that the solution map of the Cauchy problem for a coupled Camassa–Holm system is not uniformly continuous in \({H^{s}(\mathbb{T}) \times H^{s}(\mathbb{T}),s > \frac{3}{2}}\), the proof of which is based on well posedness estimates and the method of approximate solutions. Then we study the continuity properties of its solution map further and show that it is Hölder continuous in the \({H^\sigma(\mathbb{T}) \times H^\sigma(\mathbb{T})}\) topology with \({\frac{1}{2} < \sigma < s}\). Our results can also be carried out on the nonperiodic case.     

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.     

Restrictions on local inhomogeneous Strichartz estimates for the Schrödinger equation

We find new necessary conditions for the estimate ${||u||_{L^{q}_{t} (\mathbb{R}; L^{r}_{x} (\mathbb{R}^{n}))} \lesssim\,||F||_{L^{{\tilde{q}}^{\prime}}_{t}(\mathbb{R};L^{{\tilde{r}}^{\prime}}_{x}(\mathbb{R}^{n}))}}$ , where u =  u(t, x) is the solution to the Cauchy problem associated with the free inhomogeneous Schrödinger equation with identically zero initial data and inhomogeneity F =  F(t, x).     

Leading term at infinity of steady Navier‐Stokes flow around a rotating obstacle

Consider a viscous incompressible flow around a body in $\mathbb R^3$ rotating with constant angular velocity ω. Using a coordinate system attached to the body, the problem is reduced to a modified Navier‐Stokes system in a fixed exterior domain. This paper addresses the question of the asymptotic behavior of stationary solutions to the new system as |x| → ∞. Under a suitable smallness assumption on the velocity field, u, and the net force on the boundary, N, we prove that the leading term of u is the so‐called Landau solution U, a singular solution of the stationary Navier‐Stokes system in $\mathbb R^3$ with external force kωδ₀ and decaying as 1/|x|; here $k\in \mathbb R$ is a suitable constant determined by N and δ₀ is the Dirac measure supported in the origin.     

Global solutions for the generalized Camassa–Holm equation

In this paper, we study the Cauchy problem of the generalized Camassa–Holm equation. Firstly, we prove the existence of the global strong solutions provide the initial data satisfying a certain sign condition. Then, we obtain the existence and the uniqueness of the global weak solutions under the same sign condition of the initial data.     

Initial Boundary Value Problems of the Camassa–Holm Equation

In this paper we study initial boundary value problems of the Camassa–Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa–Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa–Holm equation on a compact interval possesses no nontrivial global classical solutions.     

A homotopy approach to solving the inverse mean curvature flow

In , n < 7, we treat the quasilinear, degenerate parabolic initial and boundary value problem which is the natural parabolic extension of Huisken and Ilmanen’s weak inverse mean curvature flow (IMCF). We prove long time existence and partial uniqueness of Lipschitz continuous weak solutions u(x,t) and show C ^1,α-regularity for the sets ∂{x| u(x,t) <  z }. Our approach offers a new approximation for weak solutions of the IMCF starting from a class of interesting and easily obtainable initial values; for these, the above sets are shown to converge against corresponding surfaces of the IMCF as t → ∞ globally in Hausdorff distance and locally uniformly with respect to the C ^1,α-norm.Research partially supported by the DFG, SFB 382 at Tübingen University     

Hopf-Lax type formula for non monotonic autonomous Hamilton Jacobi equations

We give an explicit formula for a solution of Hamilton-Jacobi equation u_t+H(u,Du)=0 in with initial condition u(x,0)=u₀(x), where p H(u,p) is convex, positively homogeneous of degree one and the Hamiltonian H need not satisfy the monotonicity condition in u.     

Schrödinger equations with concave and convex nonlinearities

We obtain the existence of infinitely many nodal solutions for the Schrödinger type equation on with Here, The nonlinearity f is symmetric in the sense of being odd in u, and may involve a combination of concave and convex terms.Received: November 11, 2003; revised: December 12, 2004Supported by NSFC:10441003     

A Super Camassa–Holm Equation with N‐Peakon Solutions

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero‐curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1‐peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.     

Non-uniform dependence on initial data for the periodic modified Camassa–Holm equation

In this paper, we study the periodic Cauchy problem for the modified Camassa–Holm equation $$m_t+um_x+2u_xm=0,\quad m=(1-\partial_x^2)^2u$$ , and show that the solution map is not uniformly continuous in Sobolev spaces ${H^s(\mathbb T)}$ for s > 7/2. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      18. Time periodic solutions to a nonlinear wave equation with <Emphasis Type="Italic">x</Emphasis>-dependent coefficients      Shuguan Ji 《Calculus of Variations and Partial Differential Equations》2008,32(2):137-153 In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients on under the boundary conditions a 1 y(0, t)+b 1 y x (0, t) = 0, ( for i = 1, 2) and the periodic conditions y(x, t + T) = y(x, t), y t (x, t + T) = y t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients. This work was supported by the 985 Project of Jilin University, the Specialized Research Fund for the Doctoral Program of Higher Education, and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University.      19. Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems      Mario Marino  Antonino Maugeri 《Journal of Global Optimization》2008,40(1-3):185-196 In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to (see Theorem 3).      20. Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and $$L^\infty ({\mathbb {R}}^n)$$L∞(Rn): Real Method      Ciqiang Zhuo  Dachun Yang  Wen Yuan 《Journal of Geometric Analysis》2018,28(3):2288-2311 Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space \(H^{p(\cdot )}({\mathbb {R}}^n)\) and the space \(L^{\infty }({\mathbb {R}}^n)\): \((H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}\) \(WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)\) denotes the variable weak Hardy space. As an application, the variable weak Hardy space \(WH^{p(\cdot )}({\mathbb {R}}^n)\) with \(p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )\) is proved to coincide with the variable Lebesgue space \(WL^{p(\cdot )}({\mathbb {R}}^n)\).

Time periodic solutions to a nonlinear wave equation with <Emphasis Type="Italic">x</Emphasis>-dependent coefficients

In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients on under the boundary conditions a ₁ y(0, t)+b ₁ y _x (0, t) = 0, ( for i = 1, 2) and the periodic conditions y(x, t + T) = y(x, t), y _t (x, t + T) = y _t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients. This work was supported by the 985 Project of Jilin University, the Specialized Research Fund for the Doctoral Program of Higher Education, and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University.     

Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to (see Theorem 3).     

Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and $$L^\infty ({\mathbb {R}}^n)$$L∞(Rn): Real Method

Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space \(H^{p(\cdot )}({\mathbb {R}}^n)\) and the space \(L^{\infty }({\mathbb {R}}^n)\): \((H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}\) \(WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)\) denotes the variable weak Hardy space. As an application, the variable weak Hardy space \(WH^{p(\cdot )}({\mathbb {R}}^n)\) with \(p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )\) is proved to coincide with the variable Lebesgue space \(WL^{p(\cdot )}({\mathbb {R}}^n)\).