The Pressure Equation in the Fast Diffusion Range
Emmanuel Chasseigne, Juan Luis Vázquez · Revista Matemática Iberoamericana · 2003
We consider the following degenerate parabolic equation v_{t}=v\Delta v-\gamma| abla v|^{2}\quad\text{in $\mathbb{R}^{N} \times(0,\infty)$,} whose behaviour depends strongly on the parameter \gamma . While the range \gamma 0 . Thus, the standard concepts of weak or viscosity solution do not produce uniqueness. Here we show that for \gamma>\max\{N/2,1\} the initial value problem is well posed in a precisely defined setting: the solutions are chosen in a class \mathcal{W}_s of local weak solutions with constant support; initial data can be any nonnegative measurable function v_{0} (infinite values also accepted); uniqueness is only obtained using a special concept of initial trace, the p -trace with p=-\gamma 0 , the support is constant in time, and not all of them can be obtained by the vanishing viscosity method. We also show that singular measures are not admissible as initial data, and study the asymptotic behaviour as t\to \infty .