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Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model.

This paper is devoted to a new integrable two-component Novikov equation with Lax pairs and bi-Hamiltonian structures. Ons the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. On the other hand, we prove that the strong solutions maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

In present paper, we study the Cauchy problem for a generalized Camassa-Holm equation, which was discovered by Novikov. Our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted spaces.

This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood–Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

In this paper, we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force. We first establish the local well-posedness in a range of the Besov spaces, as well as the local well-posedness in the critical space. Then, we study the Gevrey regularity of the shallow-water model by using a generalized Cauchy-Kovalevsky theorem, which implies that the shallow-water model admits analytical solutions locally in time and globally in space. Moreover, we obtain a precise lower bound of the lifespan and the continuity of the solution. Finally, working with moderate weight functions that are commonly used in time-frequency analysis, some persistence results to the shallow-water model are illustrated.

In this paper, we are concerned with the Cauchy problem of the multi-component Novikov equation. We establish the local well-posedness in a range of the Besov spaces by using Littlewood–Paley decomposition and transport equation theory. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well posed in Sobolev...

In this paper, we are concerned with \begin{document}$N$\end{document} -Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation, which was proposed by by J. Hunter and R. Saxton [SIAM J. Appl. Math. 51(1991) 1498-1521]. Using the approximate solution method, the local well-posedness of the model equation is obtained in Sobolev spaces $H^{s}$ with $s > 3/2$, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. However, if a weaker $H^{r}$-topology is used then it is shown that the solution map becomes Holder continuous in $H^{s}$.

This paper deals with the Cauchy problem for the generalized Camassa–Holm equation. The existence of weak solutions for the generalized Camassa–Holm equation in lower order Sobolev space with is obtained.

In this paper, we are concerned with the Cauchy problem of the new integrable four-component system with cubic nonlinearity. We establish the local well-posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.

This paper deals with a fully parabolic chemotaxis-growth system with generalized volume-filling effect and sublinear secretion $$\begin{aligned} \left\{ \begin{array}{ll} u_t=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (\psi (u)\nabla v)+ru-\mu u^{2}, &{}\quad (x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta v-v+g(u), &{}\quad (x,t)\in \Omega \times (0,\infty ), \end{array} \right. \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^{2}\), where \(\varphi (u)\) is a nonlinear diffusion function, \(\psi (u)\) is a chemotactic sensitivity and g(u) is a production rate of the chemoattractant. Under some suitable assumptions on the nonlinearities \(\varphi (u)\), \(\psi (u)\) and g(u), we study the global boundedness and decay of solutions for the system.

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation. The local well-posedness of the model equation is obtained in Besov spaces (which generalize the Sobolev spaces Hs) by using Littlewood-Paley decomposition and transport equation theory. Moreover, the local well-posedness in critical case (with ) is considered.

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev (Comm Part Diff Equ 30:761–781, 2005). Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s},s>3/2\) for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Finally, it is shown that the solution map for the generalized Camassa–Holm equation is Holder continuous in \(H^{r}\)-topology.

We are concerned with the Cauchy problem of the new integrable three-component system with cubic nonlinearity. We establish the local well-posedness in a range of the Besov spaces. Then the precise blow-up scenario for strong solutions to the system is derived.

We study the Cauchy problem associated with a new integrable two-component system with cubic nonlinearity, which was recently proposed by Song, Qu and Qiao. We establish the local well-posedness in a range of the Besov spaces. Moreover, for analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend a result by Danchin, and Himonas et al. to more complex equations.

This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces B p , r s , p , r ∈ [ 1 , ∞ ] , s > max { 3 2 , 1 + 1 p } (which generalize the Sobolev spaces H s ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with s = 3 2 , p = 2 , r = 1 ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated.

In this paper, we study the Cauchy problem of the periodic two-component hyperelastic-rod wave equation. We first establish the local well-posedness and precise blow-up scenario. Then we obtain several blow-up results and the blowup rate of strong solutions. Moreover, we present two global existence results for strong solutions. Finally, we consider analyticity of solutions and the initial–boundary value problem.

In this paper, we are concerned with the Cauchy problem for the generalized Camassa–Holm equation, which was proposed by Hakkaev and Kirchev [S. Hakkaev, K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation, Comm. Partial Differential Equations 30 (2005) 761–781]. We establish the local well-posedness for a range of Besov spaces. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extends some results of Danchin [R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (2001) 953–988] and Himonas and Misiolek [A. Himonas, G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575–584] to more general equations.

In this article we give a complete description of the set of blow-up points of solutions of the problem We obtain that the blow-up set B(u) satisfies . The proof is based on the analysis of the asymptotic behaviour of self-similar representation and on the comparison methods.

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

In this paper, we consider the positive solution of the Cauchy problem for the following doubly degenerate parabolic equation $$u_t-{\rm div}(|\nabla u|^{p} \nabla u^m)=u^q$$ with p > 0, q > 1,m > 1, and initial value decaying at infinity and give a new secondary critical exponent for the existence of global and nonglobal solutions. Furthermore, the large time behavior and the life spans of solutions are also studied.

In this article, we investigate the blow-up properties of the positive solutions to a degenerate parabolic system with nonlocal boundary condition. We give the criteria for finite time blow-up or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate for small weighted nonlocal boundary.

We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case , and Wang et al. (2007), which studied the single equation.

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin's cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.

This paper is concerned with the global existence and nonexistence of solution to the quasi-linear parabolic equation with nonlinear boundary condition. We obtain the critical global existence curve by constructing self-similar supersolutions and subsolutions and using the comparison principle.