Time-space fractional Kirchhoff-type diffusion problems: existence and decay estimates of solutions

The present work is concerned with the following fractional diffusion problem in time and space. $${cases} ∂ ₜβu + M(\|u\|_{H₀α(Ω )}²)(-Δ )αu + (-Δ )α∂ ₜβu =&γ |u|ρu + g(x,t), \\ & x ∈ Ω , t > 0, \\ u(x,t) = 0 & x ∈ RN Ω , t > 0, \\ u(x,0) = u₀(x) & x ∈ Ω , {cases} $$ where $Ω ⊂ RN $ is a bounded domain with Lipschitz boundary, $(-Δ )α$ is the fractional Laplace operator $(0 < α < 1) $, we consider the Riemann-Liouville fractional time derivative $∂ ₜβ$, where the fractional order satisfies $0 < β < 1 $. By the Galerkin method, let $0 < ρ < 4α/N-2α $ when $γ < 0 $, and $0 < ρ < min \{4α/N, 2α/N-2α \} $ when $γ > 0 $, we establish the global existence and uniqueness of weak solutions for problems of Kirchhoff type. The decay behavior is analyzed through the application of differential inequality techniques.