Special Issue in honor of Alan C. Lazer.
Electron. J. Diff. Eqns., Special Issue 01 (2021), pp. 2389.
Spacetime analyticity of weak solutions to
semilinear parabolic systems with variable coefficients
Falko Baustian, Peter Takac
Abstract:
We study analytic smooth solutions of
a general, strongly parabolic semilinear Cauchy problem
of 2mth order in
with analytic coefficients (in space and time variables)
and analytic initial data (in space variables).
They are expressed in terms of holomorphic continuation of
global (weak) solutions to the system valued in
a suitable Besov interpolation space of
type
at every time moment
.
Given
,
it is proved that any
type solution
with analytic initial data possesses a bounded holomorphic continuation
into a complex domain in
defined by
,
and
,
where
are constants depending upon T'.
The proof uses the extension of a weak solution to a
type solution in a domain in
, such that this extension
satisfies the CauchyRiemann equations.
The holomorphic extension is obtained with a help from
holomorphic semigroups and maximal regularity theory for
parabolic problems in Besov interpolation spaces of
type imbedded (densely and continuously) into
an
type Lebesgue space.
Applications include risk models
for European options in Mathematical Finance.
Published October 6, 2021.
Math Subject Classifications: 35B65, 35K10, 32D05, 91G40.
Key Words: Spacetime analyticity; parabolic PDE; holomorphic semigroup;
Besov space; maximal regularity; Hardy space;
holomorphic continuation to a complex strip;
European option; bilateral counterparty risk.
Show me the PDF file (795 K),
TEX file for this article.
 Falko Baustian
Institut für Mathematik
Universität Rostock
Ulmenstraße 69, Haus 3
D18051 Rostock, Germany
email: falko.baustian@unirostock.de


Peter Takác
Institut für Mathematik
Universität Rostock
Ulmenstraße 69, Haus 3
D18051 Rostock, Germany
https://www.mathematik.unirostock.de/struktur/lehrstuehle/angewandteanalysis
email: peter.takac@unirostock.de

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