Long time small solutions to nonlinear parabolic equations (Q810292)

The existence of a unique global classical solution to the Cauchy problem \[ u_ t=\Delta u+f(u,Du,D^ 2u,u_ t),\quad u(0)=u_ 0\text{ in } {\mathbb{R}}^ n\times {\mathbb{R}}_+ \] is obtained under the assumption that f is \(C^{1+r}\) for \(r>2/n\) and \(u_ 0\) is small in \(C^ 2\) and \(W^{1,2}\)-norms. The result follows from an a priori estimate for solutions to the linear heat equation by using the Banach fixed point theorem.