Vol. 84, No. 1, 1979

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Asymptotic stability and nonexistence of global solution for a semilinear parabolic equation

Chia-Ven Pao

Vol. 84 (1979), No. 1, 191–197

The aim of this paper is to investigate the existence and nonexistence of a nonnegative global solution for a semilinear parabolic system in a bounded domain. It is shown that for a certain class of initial functions the corresponding solution of the initial boundary value problem has a finite escape time, while for another class of initial functions a unique solution exists for all time and diminishes to zero. This result leads to an explicit estimate for the stability and the instability regions of the trivial steady-state solution. In the case of nonexistence of global solutions, an estimate for the finite escape time is also given.

Mathematical Subject Classification 2000
Primary: 35K60
Secondary: 35B40
Received: 4 March 1977
Published: 1 September 1979
Chia-Ven Pao