On self-similar blow-up in evolution equations of Monge-Ampère type

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge-Ampère (M-A) equation with power source, with the following basic 2D model where in two-dimensions and p > 1 is a fixed exponent. For a class of 'dominated concave' and compactly supported radial initial data, the Cauchy problem is shown to be locally well posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented.The parabolic analogy of the parabolic M-A equation in 3D for which is a cubic operator is and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similar blow-up in a cubic radial model related to the fourth-order M-A equation where the cubic operator is the catalecticant 3 × 3 determinant is also briefly discussed.