On Dissipative Nonlinear Evolutional Pseudo-Differential Equations

2017/08/31 by Mingjuan Chen, Chen, Mingjuan, Baoxiang Wang +5
Mathematics · #35K55 #35S30 #42B35 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.FA #msc:35K55 #msc:35S30 #msc:42B35 #msc:42B37

paper · pdf · doi:10.48550/arxiv.1708.09519

39 Pages

arxiv created 2017/08/31 · arxiv updated 2017/09/01

Abstract

First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation ∂_t u + A(x,D) u = F((∂^α_x u)_{|α|≤ κ}), u(0,x)= u₀(x), where A(x,D) is a dissipative pseudo-differential operator and F(z) is a multi-polynomial. We will develop the uniform decomposition techniques in both physical and frequency spaces to study its local well posedness in modulation spaces M^s_p,q and in Sobolev spaces H^s. Moreover, the local solution can be extended to a global one in L² and in H^s (s>κ+d/2) for certain nonlinearities.

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