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We study the compact support property for solutions of the followingstochastic partial differential equations: $$\partial_t u =a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad(t,x)\in (0,\infty)\times{\bf{R}}^d,$$ where $\dot{F}$ is a spatiallyhomogeneous Gaussian noise that is white in time and colored in space, and$h(t, x, u)$ satisfies $K^{-1}|u|^{\lambda}\leq h(t, x, u)\leq K(1+|u|)$ for$\lambda\in(0,1)$ and $K\geq 1$. We show that if the initial data $u_0\geq 0$has a compact support, then, under the reinforced Dalang's condition on$\dot{F}$ (which guarantees the existence and the H\"older continuity of a weaksolution), all nonnegative weak solutions $u(t, \cdot)$ have the compactsupport for all $t>0$ with probability 1. Our results extend the works byMueller-Perkins [Probab. Theory Relat. Fields, 93(3):325--358, 1992] and Krylov[Probab. Theory Relat. Fields, 108(4):543--557, 1997], in which they show thecompact support property only for the one-dimensional SPDEs driven byspace-time white noise on $(0, \infty)\times \bf{R}$.

The compact support property for solutions to the stochastic partial differential equations with colored noise