We prove the global existence of non-negative variational solutions to the “drift diffusion” evolution equation ¶_t u+ div (uD(2 \fracDÖuÖu-f))=0{{\partial_t} u+ div \left(u{\mathrm{D}}\left(2 \frac{\Delta \sqrt u}{\sqrt u}-{f}\right)\right)=0} under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, non-negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the (perturbed) Fisher information functional \fancyscript F^f(u):=\frac 12ò|D logu|² u dx+òfu dx{\fancyscript F^f(u):=\frac 12\int \left|{\mathrm{D}} \log u\right|^2 {u} dx+\int fu dx} with respect to the Kantorovich–Rubinstein–Wasserstein distance between probability measures. We also study long-time behavior of the solutions, proving their exponential decay to the equilibrium state g = e^−V characterized by -DV+\frac12 |D V|²=f,    òe^-V dx=òu₀dx,{-\Delta V+\frac12 \left|{\mathrm{D}} V\right|^2=f,\quad \int {\rm e}^{-V} dx=\int u_{0}dx,} when the potential V is uniformly convex: in this case the functional \fancyscript F^f{\fancyscript F^f} coincides with the relative Fisher information \fancyscript F^f(u)=\frac12\fancyscript I(u|g) = ò|Dlog(u/g)|²u dx{\fancyscript F^f(u)=\frac12\fancyscript I(u|g)= \int \left|{\mathrm{D}}\log(u/g)\right|^2u dx}.