On the role of the data on the regularity of the solutions to some evolution equations

In this paper we study the influence of the initial data and the forcing terms on the regularity of the solutions to a class of evolution equations including as model cases linear and semilinear parabolic equations together with the nonlinear p-Laplacian equation. We focus our study to the regularity (in terms of belonging to appropriate Lebesgue spaces) of the gradient of the solutions. We will prove that there are cases where the regularity of the solutions as soon as $t>0$ is not influenced at all by the initial data. We also derive estimates for the gradient of these solutions which are independent of the initial data and reveal, once again, that for this class of evolution problems the real "actor of the regularity" are the forcing terms.