# Global existence and uniqueness of the solution to a nonlinear parabolic equation

## TL;DR

This paper proves the global existence and uniqueness of solutions for a nonlinear parabolic PDE with a power-type nonlinearity in three-dimensional space, ensuring solutions remain bounded over time.

## Contribution

It establishes the first rigorous proof of global well-posedness and boundedness for this class of nonlinear parabolic equations with smooth decaying initial data.

## Key findings

- Existence of a unique global solution for the PDE.
- Solutions are uniformly bounded independently of space and time.
- The solution's norm is controlled by a constant not depending on initial data specifics.

## Abstract

Consider the equation $$ u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1), $$ where $ u':=\frac {du}{dt}$, $ \rho=const >0, $ $x\in \mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\|u_0\|\:=\sup_{x\in \mathbb{R}^3, t\in \mathbb{R}_+} |u(x,t)|.$$ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate $$\|u(x,t)\|<c, $$ where $c>0$ does not depend on $x,t$.