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On the Borel summability of divergent solutions of the heat equation. (English) Zbl 0958.35061
The authors consider the characteristic Cauchy problem for the complex heat equation \[ {\partial\over \partial\tau}u(\tau,z) ={\partial^2 u\over\partial z^2}(\tau,z),\;u(0,z)=\varphi(z)\tag{*} \] where \(\tau=t+i\nu\), \(z= x+iy\), and \(\varphi\) is a analytic in a domain \(D\) containing the origin. They study the Borel (respectively the fine Borel) summability of the formal solution of (*), namely \(\widehat u(\tau,z)=\sum \varphi^{(2n)}(z){z^n\over n!}\). It is well known that in general this solution is not convergent (already S. Kowalevskaya proved that a necessary condition, which is also sufficient, is that \(\varphi\) is an entire type of exponential order at most 2). In contrast with the case of an ordinary analytic differential equation where any formal solution is multisummable, for a partial differential equation the situation is somewhat different.
The authors begin with a summary of Gevrey asymptotic expansion and Borel summability for Gevrey formal power series. Let \(\theta \in\mathbb{R}\), \(\alpha>0\), and \(0<T \leq\infty\) and denote by \(S(\theta,\alpha,T)\) the sectorial domain \(\{\tau\in\mathbb{C} \setminus|\arg (\tau)-\theta|<{\alpha \over 2}\), \(0<|r|<T\}\). We will denote \(u(\tau,z)\sim\widehat u(\tau,z)\) in \(S(\theta, \alpha;T)\) for \(u(\tau,z)\) analytic on \(\cap_{\alpha'< \alpha}S(\theta,\alpha') \times B(r(\alpha'))\) where \(B(r(\alpha'))\) is the ball centered in 0 of radius \(r(\alpha')\) and \(\widehat u(\tau,z)=\sum^\infty_0 u_n(z) \tau^n\) with the \(u_n\)’s analytic functions on \(B(r(\alpha'))\) of \[ \max_{|z|< r_1}\left|u(\tau,z)-\sum^N_0u_n(z) \tau^n\right |\leq CK^NN!|\tau |^N\text{ for }\tau \in S', \tag{**} \] \(S'\) any subsector in \(S(\theta,\alpha,T)\), \(N=1,2,\dots\). The authors prove that if \(\widehat u(\tau,z)\) is the formal solution if (*) with \(\varphi\) just analytic at 0, then, for any \(\theta\in\mathbb{R}\) and any \(\alpha\), \(0<\alpha\leq\pi\) there exists an actual solution \(u(\tau,z)\) of (*) such that \(u(\tau,z) \sim_1\widehat u(\tau, z)\) in \(S(\theta,\alpha)\), and there are infinitely many such solutions. This result is the Gevrey version of S. Ouchi [Ann. Inst. Fourier 33, No. 1, 131-176 (1983; Zbl 0509.35013)].
The main result of the authors is the following theorem. Let \(v(s,z)=\sum^\infty_0 u_n(z){S^n\over n!}\) the formal Borel transform of the formal solution \(\widehat u(\tau,z)\). Then we have the following equivalence:
(1) \(v(s,z)\) is analytic in \(E_+(\theta, \omega) \times B(r)\), where \(E_+(\theta, \omega)= \{s\in\mathbb{C}, \text{dist} (s,e^{i \theta} \mathbb{R}_+) <\omega\}\) for some \(\omega,r\) positive constants, and satisfies the exponential growth condition as \(s\to\infty\) in \(E_+(\theta; \omega)\).
(2) The Cauchy data \(\varphi(z)\) is analytic in \(\Omega(\theta/2, \omega)=\{z \in\mathbb{C}, \text{dist}|z,e^{i\theta} \mathbb{R}\}\) and satisfies a growth condition of exponential \(\leq 2\) as \(z\to\infty\) in \(\Omega(\theta/2, \omega)\).
This allows to characterize Cauchy data with Borel summable formal solution in sectors of opening angles larger than \(\pi\). It is also proved that if the formal solution \(\widehat u(\tau,z)\) is finely Borel summable in a direction \(\theta\), with Cauchy data \(a_0(z)= \varphi(z)\) and \(u^\theta (\tau, z)\) its fine sum defined on \(O(\theta,\tau) \times\Omega (\theta/2,\omega)\) (here \(O(\theta,T)= \{\tau\in \mathbb{C}|+- Te^{i\theta} |<T\})\), we have Gevrey asymptotic estimates, and an initial representation of \(u^\theta (\tau,z)\) involving the heat kernel, a factorial series expansion for \(u^\theta (\tau,z)\), and finally new methods of deriving the heat kernel using the Borel sum of the following Cauchy problem \({\partial u\over\partial \tau}={\partial^2 u\over \partial z^2}\), \(u(0,z)= {1\over z}\).

35K05 Heat equation
40G10 Abel, Borel and power series methods
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