Asymptotics for Nonlinear Integral Equations with Generalized Heat Kernel and Time Dependent Coefficients Using Renormalization Group Technique

TL;DR

This paper uses the Renormalization Group method to analyze the long-time behavior of nonlinear integral equations with generalized heat kernels and time-dependent coefficients, classifying nonlinearities and their impact on asymptotics.

Contribution

It applies the RG technique to nonlinear integral equations with generalized heat kernels, providing insights into how different nonlinearities affect long-term asymptotics.

Findings

Irrelevant nonlinearities do not alter the linear asymptotic behavior.

The method classifies nonlinearities based on their influence on asymptotics.

Future work will address marginal nonlinearities with logarithmic decay.

Abstract

In this paper we employ the Renormalization Group (RG) method to study the long-time asymptotics of a class of nonlinear integral equations with a generalized heat kernel and with time-dependent coefficients. The nonlinearities are classified and studied according to its role in the asymptotic behavior. Here we prove that adding nonlinear perturbations classified as irrelevant, the behavior of the solution in the limit $t \to\infty$ remains unchanged from the linear case. In a companion paper, we will include a type of nonlinearities called marginal and we will show that, in this case, the large time limit gains an extra logarithmic decay factor.