ACTA issues

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

Pascal Remy

Acta Sci. Math. (Szeged) 87:1-2(2021), 121-139

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 $s\geq 1$. In the opposite case $s<1$, we show that the solution is generically $1$-Gevrey while the inhomogeneity is $s$-Gevrey, and we give an explicit example in which the solution is $s'$-Gevrey for no $s'<1$.

DOI: 10.14232/actasm-020-571-9

AMS Subject Classification (1991): 35C10, 35K05, 35K55, 40A30, 40B05

Keyword(s): Gevrey order, heat equation, inhomogeneous partial differential equation, nonlinear partial differential equation, formal power series, divergent power series

received 21.3.2020, revised 30.9.2020, accepted 29.10.2020. (Registered under 321/2020.)