Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

Abstract

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

Qualitative Properties of Solutions of Degenerate Parabolic Equations via Energy Approaches

“…To the best of our knowledge the first result in this direction for the heat equation with gradient damping was proven in [21]; see also [24]. We also quote on the subject of equations with nonlinear source terms depending on the gradient [3], [27], [22], [25], [26], [42], [23], [30], [40], [39], [59], [47], [29], [20], [48], [32].…”