  Given a graph $G$ and assuming that some vertices of $G$ are infected, the
$r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected
if $v$ has at least $r$ infected neighbors. The $r$-percolation number,
$m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected
vertices in $G$ such that after continuously performing the $r$-neighbor
bootstrap percolation rule each vertex of $G$ eventually becomes infected. In
this paper, we consider percolation numbers of strong products of graphs. If
$G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected
graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$.
As a dichotomy, we present a family of strong products of $k$ connected graphs
with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these
results for strong products of graphs in which at least two factors have at
least three vertices. In addition, when all factors $G_i$ have at least three
vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$
for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families
of strong products of $k$ graphs such that their $2^{k}$-percolation numbers
are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at
least three vertices, we also characterize the strong prisms $G\boxtimes K_2$
for which this equality holds. Some of the results naturally extend to infinite
graphs, and we briefly consider percolation numbers of strong products of
two-way infinite paths.
