Decay of mass for the equation \(u_t=\Delta u-a(x)u^p|\nabla u|^q\)
The paper deals with the quasilinear Cauchy problem \[ u_t =\Delta u-a(x)u^p |\nabla u|^q, x\in {\mathbb R}^d,\;t>0, \quad u(x,0)=\phi(x)\geq 0,\;x\in {\mathbb R}^d, \] with \(a>0,\) \(0\not\equiv \phi\in L^1({\mathbb R}^d)\cap C_b^{3,\alpha}({\mathbb R}^d)\) and \(p\geq 0\) and \(q\geq 1\) or \(p> 1\) and \(q=0\). It is proved a unique positive classical solvability of the above problem. Moreover, the author shows that \[ \lim_{t\to\infty} \int_{{\mathbb R}^d} u(x,t) dx=0 \] if \(a(x)\) is of the order \(|x|^m\) as \(|x|\to\infty\) and \(dp+(d+1)q\leq d+2+m.\)
scientific article; zbMATH DE number 1538227