Decay bounds for solutions of second order parabolic problems and their derivatives. III.
The initial-boundary value problem \[ \nabla (D(x)\nabla u)-u_t'=-f(u),\qquad x\in\Omega,\quad t\in(0,\tau), \] \[ u(x,0)=g(x)\geq0,\quad x\in\Omega, \qquad u(x,t)=0,\quad x\in\partial\Omega \] is considered. It is supposed that \(u(x,t)\) is a classical solution of this problem, where \(\Omega\) is a bounded convex domain in \(\mathbb R^n\), \(n\geq2\), and \(f\), \(g\), \(D\) are given nonnegative functions. Solution \(u(x,t)\) being nonnegative may (without appropriate data restriction) blow up at a finite time. Authors determine conditions on the data and on the geometry of \(\Omega\) sufficient to insure that the solution remains bounded for all time. Moreover, they establish exponential decay (in time) bounds for \(u(x,t)\) and \(| \nabla u| \) under these conditions. The particular case \(D(x)\equiv1\) has already been investigated in part I [Math. Models Meth. Appl. Sci. 5, 95--110 (1995; Zbl 0832.35019)]. For part II see Math. Inequal. Appl. 7, No. 4, 543--549 (2004; Zbl 1069.35506).
scientific article; zbMATH DE number 2155010