Existence and uniqueness of solutions for a class of semilinear parabolic PDEs with non-Lipschitz coefficients (Q2581482)

The author studies the initial value problem for semilinear parabolic partial differential equation \(u_t=\Delta u+g(t,x,u)+ \text{div} (X(t,x,u))\) considered on a complete and smooth Riemannian manifold without boundary. Here, \(\Delta\) and div denote the Laplace-Beltrami operator and the divergence operator on \(M\), respectively. Under suitable (rather standard assumptions) on the nonlinearity \(g=g(t,x,u)\) and the vector field \(X=X(t,x,u)\), the unique global-in-time mild solution is constructed.