Analytic regularity of solutions to the Cauchy problem for degenerate parabolic equations (Q1848093)

The article is devoted to proving analytic regularity of solutions to the Cauchy for degenerate parabolic equations on \((0,T)\times \mathbb{R}^n\) \[ \begin{gathered} P(t,x,\partial_t, D_x)u(t,x) = f(t,x),\quad (t,x)\in (0,T)\times \mathbb{R}^n,\\ \partial_i^ju(0,x) = u_j(x),\quad x \in \mathbb{R}^n,\quad j = 0,\dots, m-1, \end{gathered} \] where \(D_x = i\partial_x\) and \[ P(t,x,\partial_t, D_x) = \partial_t^m + \sum_{j=1}^m\sum_{\alpha:\text{finite}}a_{j\alpha} (t,x)D_x^{\alpha}\partial_t^{m-j}. \] It is assumed that \(P\) is degenerate at \(t=0\), namely, the coefficients \(a_{j\alpha}(t,x)\) satisfy \(a_{j\alpha}(t,x) = t^{\sigma(j\alpha)}b_{j\alpha}(t,x)\), where \(\sigma(j\alpha)\) are nonnegative integers and \(b_{j\alpha}(t,x)\) belong to \(C^{\infty}([0,T]\); \(\gamma^{\langle s_0\rangle}(\mathbb{R}^n))\). Here \(\gamma^{\langle s_0\rangle}(\mathbb{R}^n)\) denotes the Gevrey class with exponent \(s > 0\). The main aim of the present article is to show that the solution of the above-stated problem is analytic with respect to the space variable \(x\) for \(t > 0\), if the initial values are in Gevrey classes.