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

Abstract

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