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

We are interested in the Gevrey properties of formal power series solution time inhomogeneous semilinear heat equation with a power-law nonlinearity $1$-dimensional variable $t\in\mathbb{C}$ and $n$-dimensional spatial $x\in\mathbb{C}^n$ analytic initial condition coefficients at origin $x=0$. prove particular that inhomogeneity together $s$-Gevrey for any $s\geq1$. In opposite case $s<1$, we show is $1$-Gevrey most while $s$-Gevrey, give an explicit example which $s'$-Gevrey no $s'<1$.