On a fully nonlinear parabolic equation and the asymptotic behaviour of its solutions

Abstract

Abstract : This paper deals with the first boundary problem associated with the fully nonlinear equation u sub t = Min(thi, delta u) on the set omega x (0, infinity), where omega is a domain of R to the N power and thi(x) is a given obstacle such that thi > or = 0 on omega. Formulating the problem (occurring in heat control) as an Evolution Variational Inequality, H. Brezis obtained the existence and uniqueness of weak solutions in the space H sub 0 to the 1st (omega) as well as weak convergence to an unknown equilibrium point of the equation (when t goes to infinity). The strong convergence of the solution to the zero equilibrium point is shown here, provided the obstacle is positive and subharmonic. If in addition thi(x) > or = gamma > 0 then the asymptotic behaviour is completely described in the sense that the solution satisfies the linear heat equation u sub t = delta u on (T sub O, infinity) x omega, T sub 0 being a finite time. To do this the results are first presented for strong solutions (that is, those which satisfy the equation a.e.). The fact that under more regularity on the initial datum the weak solution is also a strong one and certain useful comparison principles are proved by using the theory of accretive operators in Banach spaces. (Author)