A proof that strong solutions of the Dullin-Gottwald-Holm equation vanishing on an open set of the (1+1) space-time are identically zero is presented. In order to do it, we use a geometrical approach based on the conserved quantities of the equation to prove a unique continuation result for its solutions. We show that this idea can be applied to a large class of equations of the Camassa-Holm type.

Conserved quantities, continuation and compactly supported solutions of some shallow water models

Wave breaking for shallow water models with time decaying solutions

In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000): $$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu+\alpha \partial _x^7u+\beta \partial _x^5u+\gamma \partial _x^3u+\mu \partial _xu+\lambda u\partial _xu=0,&{}\quad x \in {\mathbb {R}},\quad t\in {\mathbb {R}}, \\ u(x,0)=u_0(x),&{}\quad x\in {\mathbb {R}} \end{array}\right. \end{aligned}$$
 with the initial data in the Sobolev space $$H^s({\mathbb {R}}).$$
 Benefited from the ideas of Huo and Jia (Z Angew Math Phys 59:634–646, 2008), Zhang et al. (Acta Math Sci 37B(2):385–394, 2017) and Zhang and Huang (Math Methods Appl Sci 39(10):2488–2513, 2016) that is, using Fourier restriction norm method, Tao’s [k, Z]-multiplier method and the contraction mapping principle, we prove that IVP is locally well-posed for the initial data $$u_0\in H^s({\mathbb {R}})$$
 with $$s\ge -\frac{5}{8}$$
 . Moreover, based on the local well-posedness and conservation law, we establish the global well-posedness for the initial data $$u_0\in H^s({\mathbb {R}})$$
 with $$s=0$$
 .

Low Regularity for the Higher Order Nonlinear Dispersive Equation in Sobolev Spaces of Negative Index

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation...

Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space

In this paper, we investigate the initial value problem (IVP henceforth) associated with the perturbed nonlinear Schrodinger equations with the Kerr law nonlinearity. First, by using Fourier restriction norm method and Tao’s [k, Z]-multiplier method, we establish a trilinear estimate on the Bourgain space $$X_{s,b}.$$
 Then, combining the trilinear estimate with the contraction mapping principle, we prove that IVP is locally well-posed for the initial data $$(u_0(x),v_0(x))\in H^s({\mathbb {R}})\times H^s({\mathbb {R}})$$
 with $$s\ge \frac{1}{4}$$
 .

A trilinear estimate with application to the perturbed nonlinear Schrödinger equations with the Kerr law nonlinearity

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)].

Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited

Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

/pdf/well-posedness-and-decay-property-for-the-generalized-damped-inufe8h4ld.pdf

Mingbao Sun

Papers

Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited

Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

Well-posedness and decay property for the generalized damped Boussinesq equation with double rotational inertia