Existence and blow-up of global solutions for a class of fractional Lane–Emden heat flow system

Abstract

In this paper, we consider a class of Lane–Emden heat flow system with the fractional Laplacian { u t + ( − Δ ) α 2 u = N 1 ( v ) + f 1 ( x ) ,   ( x , t ) ∈ Q , v t + ( − Δ ) α 2 v = N 2 ( u ) + f 2 ( x ) ,   ( x , t ) ∈ Q , u ( x , 0 ) = a ( x ) , v ( x , 0 ) = b ( x ) ,   x ∈ R N , where 0 < α ≤ 2 , N ≥ 3 , Q := R N × ( 0 , + ∞ ) , f i ( x ) ∈ L l o c 1 ( R N ) ( i = 1 , 2 ) are nonnegative functions. We study the relationship between the existence, blow-up of the global solutions for the above system and the indexes p , q in the nonlinear terms N 1 ( v ) , N 2 ( u ) . Here, we first establish the existence and uniqueness of the global solutions in the supercritical case by using Duhamel's integral equivalent system and the contraction mapping principle, and we further obtain some relevant properties of the global solutions. Next, in the critical case, we prove the blow-up of nonnegative solutions for the system by utilizing some heat kernel estimates and combining with proof by contradiction. Finally, by means of the test function method, we investigate the blow-up of negative solutions for the Cauchy problem of a more general higher-order nonlinear evolution system with the fractional Laplacian in the subcritical case.