In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial s...In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.展开更多
We study for a class of symmetric Levy processes with state space R^n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t〉o and （δt）t〉o. The first family of metrics describes...We study for a class of symmetric Levy processes with state space R^n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t〉o and （δt）t〉o. The first family of metrics describes the diagonal term pt （0）; it is induced by the characteristic exponent ψ of the Levy process by dr（x, y） = √tψ（x - y）. The second and new family of metrics 6t relates to √tψ through the formula exp（-δ^2t（x,y））=F[e^-tψ/pt（0）]（x-y）,where Y denotes the Fourier transform. Thus we obtain the following ＂Gaussian＂ representation of the tran- sition density： pt（x） = pt（O）e^-δ^2t（x,0） where pt（O） corresponds to a volume term related to √tψ and where an ＂exponential＂ decay is governed by 5t2. This gives a complete and new geometric, intrinsic interpretation of pt（x）.展开更多
基金partially supported by the National Natural Science Foundation of China(12071439)the Zhejiang Provincial Natural Science Foundation of China(LY19A010016)the Natural Science Foundation of Jiangxi Province(20212BAB201016)。
文摘In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.
文摘We study for a class of symmetric Levy processes with state space R^n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t〉o and （δt）t〉o. The first family of metrics describes the diagonal term pt （0）; it is induced by the characteristic exponent ψ of the Levy process by dr（x, y） = √tψ（x - y）. The second and new family of metrics 6t relates to √tψ through the formula exp（-δ^2t（x,y））=F[e^-tψ/pt（0）]（x-y）,where Y denotes the Fourier transform. Thus we obtain the following ＂Gaussian＂ representation of the tran- sition density： pt（x） = pt（O）e^-δ^2t（x,0） where pt（O） corresponds to a volume term related to √tψ and where an ＂exponential＂ decay is governed by 5t2. This gives a complete and new geometric, intrinsic interpretation of pt（x）.
