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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 Hs 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 Hr-topology is used then it is shown that the solution map becomes Hölder continuous in Hs. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces Bp,rs, p, r ∈ [1, ∞], s > max {3/2, 1 + 1/p} (which generalize the Sobolev spaces Hs) 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. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. 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. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we give the estimates of the blow-up rate, and obtain that the blow-up set is a single point x = 0 for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results. [ABSTRACT FROM AUTHOR]