Parabolic Minkowski convolutions of solutions to parabolic boundary value problems

We introduce a new kind of convolution, which is a sort of parabolic version of the classical supremal convolution of convex analysis. This operation allows us to compare solutions of different parabolic problems in different domains. As examples of applications of our main result, we study the parabolic concavity of solutions to parabolic boundary value problems, analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term. We also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time, proving some optimality of the ball.