Asymptotics for a parabolic problem of Kirchhoff type with singular critical exponential nonlinearity

Abstract

The main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as t$t$ goes to infinity for the following class of parabolic Kirchhoff equations: ut+∥u∥(θ−1)Ns(−Δ)N/ssu=λ|u|q−2uexpα0|u|NN−s|x|γinΩ,t>0,u=0inRN∖Ω,t>0,u(x,0)=u0(x)inΩ,$$\begin{eqnarray*} \hspace*{13pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{llc}u_{t}+\Vert u\Vert ^{\frac{(\theta -1)N}{s}}(-\Delta)^{s}_{N/s}u=\frac{\lambda |u|^{q-2}u\exp {\left(\alpha _{0}|u|^{\frac{N}{N-s}}\right)}}{|x|^{\gamma }} &\text{in}\ &\Omega,\;t>0, \\ u =0 &\text{in} & \mathbb {R}^{N}\backslash \Omega,\;t > 0, \\ u(x,0)=u_{0}(x)& \text{in} &\Omega, \end{array} \right.} \end{eqnarray*}$$where ∥u∥Ns=∫R2N|u(x,t)−u(y,t)|N/s|x−y|2Ndxdy,$$\begin{equation*} \hspace*{7pc}\Vert u\Vert ^{\frac{N}{s}}=\int _{\mathbb {R}^{2N}}\frac{|u(x,t)-u(y,t)|^{N/s}}{|x-y|^{2N}}\,dxdy, \end{equation*}$$Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N \, (N\ge 2)$ is a bounded domain with a Lipschitz boundary, 0∈Ω$0\in \Omega$, α0,λ>0$\alpha _{0},\lambda >0$, θ≥1$\theta \ge 1$, γ∈[0,N)$\gamma \in [0, N)$, q>Nθ/s$q>N\theta /s$, and (−Δ)N/ss$(-\Delta)_{N/s}^{s}$ is the fractional N/s$N/s$‐Laplacian operator, s∈(0,1)$s\in (0,1)$.