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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 $${\partial u\over\partial \tau}={\partial^2 u\over \partial z^2}$$, $$u(0,z)= {1\over z}$$.

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