This article studies a distributed convex optimization problem with nonsmooth local objective functions subject to local inequality constraints and a coupled equality constraint. By combining the dual decomposition technique and subgradient flow method, a new distributed solution is developed in continuous time. Unlike the existing related continuous-time schemes either depending on specific initial conditions or on differentiability or strict (even strong) convexity of local cost functions, this study is free of initialization and takes into account general convex local objective functions which could be nonsmooth. Via nonsmooth analysis and set-valued LaSalle invariance principle, it is proved that a global optimal solution can be asymptotically obtained. Finally, the effectiveness of our algorithm is illustrated by numerical examples.