Let $u:\mathbb{R}\times(0,\infty)\to\mathbb{R}$ be a $C^2$ solution of the nonlinear heat equation $$ \frac{\partial{u}}{\partial{t}}(x,t)-\frac{\partial^2{u}}{\partial{x}^2}(x,t) +x^2u(x,t)+(u(x,t))^3 \quad\forall(x,t)\in\mathbb{R}\times(0,\infty). $$ Suppose that $$ \int_{\mathbb{R}}\left(|u(x,t)|^2+\left|\frac{\partial{u}}{\partial{x}}(x,t)\right|^2\right)dx $$ is finite and continuous for all $t>0$. I want to prove that $$ E(t):=\int_{\mathbb{R}} |u(x,t)|^2dx $$ converges to $0$ as $t\to\infty$.
I derived the energy equation $$ \int_{\mathbb{R}}u(x,T)^2dx +\int_{T_0}^T\int_{\mathbb{R}}2(u_x^2+x^2u^2+u^4)dx dt =\int_{\mathbb{R}}u(x,T_0)^2dx \quad (0<T_0<T), $$ so I know $E(t)$ is decreasing and converges to some value. I suspect $$ \int_{\mathbb{R}\setminus[-1,1]}u^2dx\le \int_{\mathbb{R}\setminus[-1,1]}x^2u^2dx $$ and $$ \left(\int_{-1}^1u^2dx\right)^2\le 2\int_{-1}^1u^4dx $$ are usefull.
