A Bernstein type theorem for parabolic kk-Hessian equations

We are concerned with the characterization of entire solutions to the parabolic kk-Hessian equation of the form −utFk(D2u)=1−utFk(D2u)=1 in Rn×(−∞,0]Rn×(−∞,0]. We prove that for 1≤k≤n1≤k≤n, any strictly convex–monotone solution u=u(x,t)∈C4,2(Rn×(−∞,0])u=u(x,t)∈C4,2(Rn×(−∞,0]) to −utFk(D2u)=1−utFk(D2u)=1 in Rn×(−∞,0]Rn×(−∞,0] must be a linear function of tt plus a quadratic polynomial of xx, under some growth assumptions on uu.