Abstract: Using linear stability theory (LST) and direct numerical simulation (DNS), the linear evolution and oblique breakdown of Mack's second mode in Mach 4.5 boundary layer flows over convex walls are investigated through comparative analyses with flat plate boundary layer flows, revealing the influence mechanism of convex curvature on different stages of the second mode transition. Analysis of the linear growth stage indicates that convex curvature significantly suppresses linear disturbance growth, with the suppression intensity increasing as the radius of curvature decreases; convex curvature shifts the unstable frequency band of the second mode upward, but reduces the maximum growth rate; LST can qualitatively reflect the change of stability through the influence of curvature on basic flow. For the analysis of nonlinear saturation and breakdown, convex curvature has almost no suppression effect on disturbance amplitudes at the nonlinear saturation stage, but prolongs this stage, thereby delaying breakdown; meanwhile, convex curvature promotes the generation of higher-order harmonics at the nonlinear saturation stage. Furthermore, statistical analysis indicates that from the nonlinear saturation stage to the turbulent stage, convex curvature significantly weakens the intensity of Reynolds shear stress and turbulent kinetic energy generation, and simultaneously reduces skin friction.