2022/04/24 by Jie Chen, Chen, Jie, Baoxiang Wang +3 · 1 citation
Engineering · Mathematics · #35K58 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2204.11277
openalex publication_date 2022/04/24 · openalex created_date 2022/11/19 · openalex updated_date 2026/07/28
We consider the Cauchy problem for the complex valued semi-linear heat equation ∂t u - Δu - um =0, u (0,x) = u0(x), where m≥ 2 is an integer and the initial data belong to super-critical spaces Esσ for which the norms are defined by ‖f‖Esσ = ‖⟨ ξ⟩σ2s|ξ|f(ξ)‖L2, σ∈ ℝ, slt;0. If s<0, then any Sobolev space Hr is a subspace of Esσ, i.e., ∪r ∈ ℝ Hr ⊂ Esσ. We obtain the global existence and uniqueness of the solutions if the initial data belong to Esσ (s<0, σ≥ d/2-2/(m-1)) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in Esσ are not required for the global solutions. Moreover, we show that the error between the solution u and the iteration solution u(j) is Cj/(j !)2. Similar results also hold if the nonlinearity um is replaced by an exponential function eu-1.