Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation
The qualitative properties of nonegative solutions of the Cauchy problem for the viscous Hamilton-Jacobi equation in \((x,t) \in \mathbb R^n \times (0,\infty)\) \[ \begin{aligned} u_t-\Delta u + | \nabla u| ^q & =0, \tag{1}\\ u(x,0) & =u_0(x) \geq 0, \tag{2}\end{aligned} \] under the critical assumption \[ \frac{n+2}{n+1}<q<2 \] are studied. The existence of general selfsimilar solutions of the equation (1), the asymptotic properties of (radially symmetric) selfsimilar solutions of (1) is proved. The behavior of solutions with nonintegrable initial data satisfying, rougly speaking, \(u_0(x) \sim | x| ^{-b}\) as \(| x| \to \infty\) with \(a=\frac{2-q}{q-1}<b<n,\) is established. For the integrable nonegative initial data \(0 \leq u_0 \in L^1(\mathbb R^n)\) the large time asymptotics of solutions is determined by the heat kernel. Namely \(u(t,.) \sim M_{\infty}G(.,t),\) where \[ G(x,t)=(4\pi t)^{-n/2} \exp \biggl(-\frac{| x| ^2}{4t}\biggr), \] and \( M_{\infty}=\lim_{t \to \infty} \int_{\mathbb R^n}u(x,t)dx \) is finite and strictly positive provided \(q>\frac{n+2}{n+1}.\)
scientific article; zbMATH DE number 2111543
scientific article