Kinetics of multidimensional fragmentation

We present two classes of exact solutions to a geometric model which describes the kinetics of fragmentation of d-dimensional hypercuboid-shaped objects. The first class of exact solutions is described by a fragmentation rate a(x(l),...,x(d))=1 and daughter distribution function b(x(1),...,x(d)\x(1)',...,x(d)')=(alpha(1)+2)...(alpha(d)+2) x(1)(a1)...x(d)(alpha d)/x(1)'((alpha d+1)). The second class of exact solutions is described by a fragmentation rate a(x(1),...,x(d))=x(1)(beta 1+1)... x(d)(beta d+1)/2(d) and daughter distribution function b(x(1),...,x(d))\x(1)',...,x(d)')=2(d) delta(x(1)-x(1)'/2)...delta(x(d)-x(d)'/2). Each class of exact solutions is analyzed in detail for the presence of scalings solutions and the occurrence of shattering transitions; the results of these analyses are also presented.