Global in-time rough large data solution to complex-valued semilinear damped evolution equations

2025/07/11 by Chen, Wenhui, Reissig, Michael
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.08272

Abstract

We study the semilinear Cauchy problem for complex-valued damped evolution equations ∂_t²u+(-Δ)^σu+(-Δ)^δ∂_tu=u^p, u(0,x)=u₀(x), ∂_tu(0,x)=u₁(x), with δ∈[0,σ], σ∈ℝ₊ and p∈ℕ₊\backslash\1\, where the initial data belong to the rough space E^α_s endowed with the norm ‖f‖_E^αs=‖⟨ξ⟩^s 2^α|ξ|\widehatf(ξ)‖_L² with αlt;0, s∈ℝ. Concerning (u₀,u₁)∈ E^α_s+κ× E^α_s when s\geqslant(n)/(2)-(2κ+κ-2δ)/(p-1)-κ with κ=min\2δ,σ\ and κ=max\2δ,σ\ whose Fourier transforms are supported in a suitable subset of first octant, we prove a global in-time existence result without requiring the smallness of rough initial data.

Citations

• Global wellposedness of general nonlinear evolution equations for distributions on the Fourier half space
• Remarks on the Complex Euler Equations
• Global Cauchy problems for the nonlocal (derivative) NLS in E^s_σ
• Complex valued semi-linear heat equations in super-critical spaces E^s_σ
• A test function method for evolution equations with fractional powers of the Laplace operator
• Navier-Stokes Equation in Super-Critical Spaces E^s_p,q
• L¹ estimates for oscillating integrals and their applications to semi-linear models with σ-evolution like structural damping
• An application of L¹ estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models
• Gevrey class regularity for the solutions of the Navier-Stokes equations

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