On approximate solutions of semilinear evolution equations

TL;DR

This paper introduces a general framework for analyzing approximate solutions of semilinear evolution equations in Banach spaces, providing a method to estimate the accuracy and existence interval of the true solution, with applications to Galerkin methods and nonlinear heat equations.

Contribution

It develops a theorem linking approximate and exact solutions through a control integral equation, enabling error estimation and existence interval determination.

Findings

The framework applies to various approximation methods, including Galerkin.

It provides bounds on the error and existence time for solutions.

The approach successfully analyzes the nonlinear heat equation, predicting blow-up times.

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.