The role of forward self-similar solutions in the Cauchy problem for semi-linear heat equations with exponential nonlinearity

TL;DR

This paper investigates the borderline solutions of semi-linear heat equations with exponential nonlinearity, using approximation techniques based on power-type equations to establish existence results.

Contribution

It introduces a novel approach to prove existence of solutions at the critical threshold for exponential nonlinearities by leveraging solutions of power-type equations.

Findings

Existence of solutions at the critical threshold established.

Approximation method using power-type equations demonstrated.

Framework applicable to similar nonlinear PDEs.

Abstract

We consider the Cauchy problem for semi-linear heat equations with exponential nonlinearity. The main purpose of this paper is to prove the existence of solutions lying on the borderline between global existence and blow-up infinite time. The existence has been shown for semi-linear heat equations with power type nonlinearity. We explain the main strategy to prove the existence. By using the definition of exponential function, we approximate the solution to exponential type equation by that of power type equation. Then we can use directly the knowledge for power type equation.