Regularity of viscosity solutions of a degenerate parabolic equation (Q2781245)

In the present article, it is studied the following Cauchy problem: NEWLINE\[NEWLINE u_t = u\Delta u - \gamma|\nabla u|^2,\quad \text{in } \mathbb{R}^N \times\mathbb{R}^+ NEWLINE\]NEWLINE with initial data NEWLINE\[NEWLINE u(x,0) = u_0(x)\quad\text{in } \mathbb{R}^N. NEWLINE\]NEWLINE Here \(\gamma\) is a nonnegative constant and the initial function \(u_0\) belongs to \(C(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)\) and \(u_0 \geq 0\) in \(\mathbb{R}^N\). The authors investigate the regularity of the viscosity solution of the Cauchy problem for \(\gamma \geq \sqrt{2N} - 1\), and as a result, obtain certain conditions under which \(u\) is Lipschitz continuous function.