Data di Pubblicazione:
2004
Citazione:
On approximate solutions of semilinear evolution equations / C. Morosi, L. Pizzocchero. - In: REVIEWS IN MATHEMATICAL PHYSICS. - ISSN 0129-055X. - 16:3(2004), pp. 383-420.
Abstract:
A general framework is presented to discuss the approximate solutions
of an evolution equation in a Banach space, with a linear part generating a semigroup
and a sufficiently smooth nonlinear part. A theorem is presented, allowing
to infer from an approximate solution the existence of an exact solution. According to this theorem,
the interval of existence of the exact solution and the distance of the latter
from the approximate solution can be evaluated
solving a one-dimensional "control" integral equation, where the unknown
gives a bound on the previous distance as a function of time.
For example, the control equation can be
applied to the approximation methods based on the reduction of the evolution
equation to finite-dimensional manifolds: among them, the Galerkin method is
discussed in detail. To illustrate this framework, the nonlinear
heat equation is considered. In this case the control equation is used
to evaluate the error of the Galerkin approximation; depending on the
initial datum, this approach either grants global existence of the solution or
gives fairly accurate bounds on the blow up time.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Blow up; Differential equations; Nonlinear heat equation; Theoretical approximation
Elenco autori:
C. Morosi, L. Pizzocchero
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