Upper bound for semilinear heat equation

Question

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $u \in C^{2,1}(\bar{\Omega} \times [0,\infty)) $ be a solution to $$ u_t=\Delta u-u^3 \; \; (x,t) \in \Omega \times [0,\infty) \\ u(x,t)=0 \; \; (x,t) \in \partial \Omega \times [0,\infty) $$ I wish to show that $\int_{\Omega}u^2(x,1)dx \leq |\Omega |$. My attempt: Consider $$ h(t)=\frac{1}{|\Omega |}\int_{\Omega}u^2(x,t)dx $$ Then we have $$ (\frac{1}{h(t)})'=\frac{-1}{h^2(t)}\frac{1}{|\Omega |}\int_{\Omega}\partial_tu^2(x,t)dx= \\ -\frac{1}{h^2(t)}\frac{2}{|\Omega |}\int_{\Omega}\partial_tu(x,t)u(x,t)dx= \\ -\frac{1}{h^2(t)}\frac{2}{|\Omega |}\int_{\Omega}(\Delta u-u^3)udx= \\ -\frac{1}{h^2(t)}\frac{2}{|\Omega |}\int_{\Omega}-|\nabla u|^2-u^4dx=\\ \frac{1}{h^2(t)}\frac{2}{|\Omega |}\int_{\Omega}|\nabla u|^2+u^4dx $$ But I can not continue here - how do I show that $$ h(1) \leq 1 $$ or $$ \frac{1}{h(1)} \geq 1 $$ Ive tried solving this ODE explicitly, but I got confused. Also I suspect a Gronwall-argument, but can not figure it out.