Consider a stochastic heat equation (SHE) driven by multiplicative space-time white noise. It is known that if the initial function had compact support, then the solution is bounded almost surely for all time t. On the other hand, if the initial function is constant, then the solution is not bounded for all time t>0. A natural question is that "is there some decay rate of the initial function at infinity which tells us that the solution is bounded or unbounded for some or even all t”? A short answer is “Yes” and, in this talk, we describe precisely the decay rate of the initial function at infinity which tells us boundedness and unboundedness for some or all time t. This is based on ongoing work with Le Chen and Davar Khoshnevisan.