Formal solutions of Burgers type equations

Abstract

We study formal power series solutions to the initial value problem for the Burgers type equation $\partial_t u-\Delta u = X\big(f(u)\big)$ with polynomial nonlinearity $f$ and a vector field $X$, and prove that they belong to the formal Gevrey class $G^2$. Next we give counterexamples showing that the solution, in general, is not analytic in time at $t=0$. We also prove the existence of non-constant globally analytic solutions.