Global solution and blow-up of critical heat equation with nonlocal interaction

Abstract We consider the Cauchy problem of the critical nonlocal heat equation u t - Δ ⁢ u = ( | x | - μ ∗ | u | 2 μ ∗ ) ⁢ | u | 2 μ ∗ - 2 ⁢ u in ⁢ ℝ N × ( 0 , T max ) , u_{t}-\Delta u=(|x|^{-\mu}\ast|u|^{2^{\ast}_{\mu}})|u|^{2^{\ast}_{\mu}-2}u% \quad\text{in }\mathbb{R}^{N}\times(0,T_{\max}), where N ≥ 3 {N\geq 3} , 0 < μ < min ⁡ { 4 , N } {0<\mu<\min\{4,N\}} and 2 μ ∗ = 2 ⁢ N - μ N - 2 {2^{\ast}_{\mu}=\frac{2N-\mu}{N-2}} is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. The aim of this paper is to study the behaviour of the solutions in time. More precisely, we prove the existence and decay of global solutions and blow-up in finite time. Furthermore, the global uniform bound of the global solutions in H ˙ 1 ⁢ ( ℝ N ) {\dot{H}^{1}(\mathbb{R}^{N})} and the asymptotic behaviour of them are given.