Consider the PDE $$ u_t = \Delta( (|x|^2 + 1) u) + |\nabla u| - 4nu, $$ where $x \in \mathbb{R}^n,$ $t > 0.$ Let $u_1$ and $u_2$ be smooth solutions such that $u_1 \leq u_2$ for all $x$ at $t = 0.$ Show that $u_1 \leq u_2$ for all $(x,t).$ Furthermore,
Let $u$ be a nonnegative, integrable smooth solution to the PDE above with the property that $|x|^2D^\alpha u \to 0$ as $|x| \to \infty,$ for any multi-index $\alpha.$ Show that $\int u(\cdot,t) \; dx$ exponentially decays to zero as $t \to \infty.$
I am posting this question here because I am completely stuck on the second part, the part in the highlighted box. Any suggestions, solutions, ideas are all appriciated. My attempted solution for the first part, the part that isn't in the highlighted box is as follows.
First Part Solution: Let $w = u_1 - u_2 - \epsilon e^{t},$ where this choice of $w$ has been inspired by previous problems I've done. If it can be shown that $w < 0$ for all $t,$ then we have that $u_1 \leq u_2$ as $\epsilon \to 0.$
Consider the PDE equality that $w_t$ must satisfy, \begin{align} w_t &= (u_1 - u_2)_t - \epsilon e^{t}\\ &= \Delta((|x|^2+1)(w + \epsilon e^{t})) + |\nabla u_1| - |\nabla u_2| - 4n(w + \epsilon e^{t}). \end{align}
We have that $w(x,0) < 0,$ so assume that at some positive time $t_0,$ $w(x,t_0) = 0.$ This point will be a local maximum, meaning $\Delta w \leq 0$ and the gradient will be $\nabla w = \nabla u_1 - \nabla u_2 = 0 \implies \nabla u_1 = \nabla u_2 \implies |\nabla u_1| = |\nabla u_2|.$ Additionally, we have that $w_t \geq 0$ since this is the first time $w$ hits a nonnegative value. We then have that $$ w_t - \Delta((|x|^2+1)(w + \epsilon e^{t})) + 4n\epsilon e^{t} = 0.$$ Simplifying further, $$ w_t - (1 + \epsilon e^t + |x|^2)\Delta w + 2n\epsilon e^t + 4n\epsilon e^{t} = 0,$$ but we see that the last line is actually a contradiction since all the terms are nonnegative and the last term is positive. Thus, $w$ never achieves the value $0$ for any $(x,t),$ and taking the limit as $\epsilon \to 0,$ we have that $u_1 \leq u_2$ for all time.
My idea for part 2: Define $$ e(t) := \int u \;dx,$$ and differentiate with respect to time. Integrating by parts is generally allowed due to the fact that $u$ decays sufficiently fast, \begin{align} {\dot e}(t) &= \int_{\mathbb{R}^n} \Delta( (|x|^2 + 1) u) + |\nabla u| - 4nu \; dx\\ &= \lim_{R \to \infty} \int_{\partial B_R(0)} \nabla ( (|x|^2 + 1) u) \cdot dS + \int_{\mathbb{R}^n} |\nabla u| - 4nu \;dx \end{align} and somehow show this is bounded by minus the original integral $-e(t).$ Then we would have that $e(t)$ decays exponentially.