Gevrey index theorem for the inhomogeneous $n$-dimensional heat equation with a power-law nonlinearity and variable coefficients

Abstract

We are interested in the Gevrey properties of the formal power series solution in time of the inhomogeneous semilinear heat equation with a power-law nonlinearity in $1$-dimensional time variable $t\\in\\mathbb{C}$ and $n$-dimensional spatial variable $x\\in\\mathbb{C}^n$ and with analytic initial condition and analytic coefficients at the origin $x=0$. We prove in particular that the inhomogeneity of the equation and the formal solution are together $s$-Gevrey for any $s\\geq1$. In the opposite case $s<1$, we show that the solution is $1$-Gevrey at most while the inhomogeneity is $s$-Gevrey, and we give an explicit example in which the solution is $s $-Gevrey for no $s <1$.