Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions (Q1323104)

The nonlocal initial boundary value problem \[ \begin{aligned} Lu+ g(x,t,u) & = 0,\quad x\in \Omega,\quad t>0,\\ u(x,t) & = 0,\quad x\in \partial\Omega,\quad t>0,\\ u(x,0) & = \int^ \infty_ 0 h(x,t)u(x,t)dt+ f(x),\quad x\in \Omega,\end{aligned} \] is considered, where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(L\) is a uniformly parabolic operator with continuous and bounded coefficients. The solvability of the problem is shown using comparison arguments.