Abstract :
In this paper we study the global existence; asymptotic behavior, and blowing-up property of solutions of the initial-boundary value problem for the nonlinear integrodifferential reaction-diffusion equation ut + Lu = λu + μu ∫t0u(x, s)ds, with L a uniformly elliptic, self-adjoint operator, which arises in nuclear reactor dynamics. The estimates of asymptotic behavior and escape time given here are optimal in a sense. In particular, we find that if μ<0 then the solution exists globally and diminishes to the steady-state solution u = 0 faster than any exponential function e−αt (a > 0) does. If μ > 0, λ < 0, L = −(∂/∂xi) (aij(x) (∂/∂xj)) and the boundary condition is Neǔmann condition, then the solution may tend to zero as t→ + ∞ or blow up in finite time depending on the initial value ≤ λ2/2μ or > λ2/2μ.
