Asymptotic behaviour of instantons on Cylinder Manifolds

IIn this article, we study the instanton equation on the cylinder over a closed manifold $X$ which admits non-zero smooth $3$-form $P$ and $4$-form $Q$. Our results are (1) if $X$ is a $\mathbf{\text{good}}$ manifold, i.e., $P,Q$ satisfying $d{\ast }_{X}P=d{\ast }_{X}Q=0$, then the instanton with integrable curvature decays exponentially at the ends, and, (2) if $X$ is a real Killing spinor manifold, i.e., $P,Q$ satisfying $dP=4Q$ and $d{\ast }_{X}Q=(n-3){\ast }_{X}P$, we prove that the solution of instanton equation is trivial under some mild conditions.