On the Cauchy problem of nonlinear degenerate parabolic equation (Q1904981)

We consider the Cauchy problem \[ u_t= \text{div}(a(|\nabla u|\nabla u))\text{ in }S_T= \mathbb{R}^N\times (0, T),\;u(x, 0)= u_0(x)\text{ on }\mathbb{R}^N.\tag{1} \] Here, we assume that \(a(s)\in C^{1+ \alpha}(0, \infty)\), \(a(s)> 0\) if \(s> 0\), \[ \int^s_0 \tau a(\tau)d\tau\leq Ca(s) s^2 \forall s> 0,\quad \lim_{s\to 0^+} sa(s)= 0, \] \[ \lim_{s\to \infty} {a(s)\over s^{p- 2}}> 0\quad\text{for some}\quad p> {2N\over N+ 1} \] and for \(x\in \mathbb{R}^N\), \(a(|x|) x_i\) is monotone. Existence and a priori estimates for solutions to (1) are proved.