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

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 $x,t$.