On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping

Abstract

In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending dampingwhere (d = 2, 3), the frictional coefficient is with and , is a constant, , , , and is sufficiently small. One can totally divide the range of and into the following four cases:Case 1: , for d = 2, 3;Case 2: , for d = 2, 3;Case 3: , for d = 2;Case 4: , for d = 2, 3.We show that there exists a global smooth solution in Case 1, and Case 2 with , while in Case 3 and Case 4, for some classes of , the solution will blow up in finite time. Therefore, and appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution in d−dimensional compressible Euler equations with time-depending damping.