The Generalized Riemann Zeta heat flow

We consider the PDE flow associated to Riemann zeta and general Dirichlet $L$-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet $L$-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet $L$-function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet $L$-functions and data initially on the right of a possible pole. Additional global well-posedness and convergence results are proved in the case of the defocusing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the ``focusing'' model, and prove blow-up of solutions near the pole $s=1$.