Slawomir Michalik
Summability and fractional linear partial differential equations
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ABSTRACT. We consider the Cauchy problem for the Kowalevskaya type fractional linear partial differential equations
in two complex variables with constant coefficients. We show that solution is analytically continued in some directions with
exponential growth if and only if the similar properties satisfy the Cauchy data. Applying this result we study the summability
of formal power series solution of a Cauchy problem for general non-Kowalevskian linear partial differential equations in
normal form with constant coefficients. We obtain the necessary and sufficient conditions for the Borel summability in terms of
analytic continuation with an appropriate growth condition of the Cauchy data. Moreover, we show the similar characterisation of Borel summability in the case of non-Kowalevskian fractional equations.