Sharp regularity for the degenerate doubly nonlinear parabolic equation (Q2003935)

In this paper the author studies sharp regularity estimates for bounded weak solutions to the inhomogeneous degenerate doubly nonlinear equations of the type \[ u_t-\operatorname{div}(m|u|^{m-1}|Du|^{p-2}Du) = f . \] Fo such kind of equations, Harnack estimates and \(C^{0, \alpha}\)-estimates are well known but, till now, no quantitative information is known for the exponent \(\alpha\). In this paper the author provides sharp Hölder estimates for weak solutions to the inhomogeneous equation. More precisely, the author proves that \(u\in C^{0, \alpha}\) for \(\alpha\) depending explicitly only on the optimal Hölder regularity for the homogeneous case, the integrability of \(f\) in space and time, and the nonlinear diffusion parameters \(p\) and \(m\). The result is proved by using, in a real very smart way, some intrinsic scaling properties of the solutions.