Modulation Theory for the Blowup of Vector-Valued Nonlinear Heat Equations

This paper is concerned with the blowup of solutions of the nonlinear vector-valued heat equation Ut − ΔU = Up − 1U, U(0) = U0, where U(x, t) = (u1(x, t), ..., um(x, t)) is a vector-valued function from Rn × (0, T) to Rm and 1 < p < (3n + 8)/(3n − 4). Working with the equation in similarity variables, and using modulation theory and ideas from center manifold theory, we obtain the asymptotic behavior of U in a backward space-time parabola near any blowup point.