A new nonlinear dispersive partial differential equation with cubic nonlinearity, which includes the famous Novikov equation as special case, is investigated. We first establish the local wellposedness in a range of the Besov spaces B s p,r , p, r ∈ [1, ∞], s > max{ 3 2 , 1 + 1 p } but s = 2 + 1 p (which generalize the Sobolev spaces H s ), well-posedness in H s with s > 3 2 , is also established by applying Kato's semigroup theory. Then we give the precise blow-up scenario. 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 peakon solutions to the equation are global weak 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.
Abstract. The paper deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical Fujita curve is conjectured with the aid of some new results.
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