Hi all, I
Fix $$(t,x) ∈ (0,\infty) \times R^n$$and consider auxillary function
$$w(s)=u(t+s,x+sb)$$
Then, $$\partial_s w(s)=(\partial_tu)(t+s,x+sb)\frac{d}{ds}(t+s)+<Du(t+s,x+sb)\frac{d}{ds}(x+sb)>$$
$$=(\partial_tu)(t+s,x+sb)+<b,Du(t+s,x+sb)>$$
$$=-cu(t+s,x+sb)$$...