Construction of blow-up solution for 5 dimentional critical fujita type equation with different blow-up speed

TL;DR

This paper constructs blow-up solutions for a 5-dimensional critical heat equation, demonstrating the existence of type 2 solutions with two different blow-up rates at two points using the inner-outer gluing method.

Contribution

It introduces a novel construction of type 2 blow-up solutions with multiple blow-up rates for a 5D critical heat equation.

Findings

Existence of blow-up solutions at two points with different rates

Application of inner-outer gluing method to construct solutions

Demonstration of complex blow-up behavior in critical heat equations

Abstract

We are concerned with blow-up solutions of the 5-dimensional energy critical heat equation $u_t=\Delta u + | u |^{\frac{4}{3}}u$. Our main result is to show that the existence of type 2 solutions blows up at 2 points with 2 different blow-up rates. The inner-outer gluing method has been employed.