The problem asks to find all $C^2$ solution for $$u_t-u_{xx}=t-x^2,\quad (t,x)\in \mathbb{R}^2$$ satisfying $$\lim_{|x|+|t|\to \infty}\frac{|u(x,t)|}{|x|^5+|t|^5}=0$$
The typical heat equation with $(t,x)\in [0,\infty)\times \mathbb{R}$ and initial condition $u|_{t=0}=0$admits simple heat kernel $H(x,t)={1 \over \sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$ and solves the equation by simple setting $u=(t-x^2)*H(x,t)$. But it is not obvious to me that a direct heat kernel analogues from the above case can be deduced to the $(t,x)\in \mathbb{R}^2$ case. It also seems not that realistic to apply Fourier transform to the equation since in this case we are going to dealing with nontrivial distributional equation, so I am not sure where to start. Any comment shall be greatly appreciated
