Generalized quasilinearization and semilinear parabolic problems (Q1348202)

The authors extend the method of the generalized quasilinearization to initial boundary value problems for semilinear parabolic equations of the type \[ \frac{\partial u(x,t)}{\partial t} + A u(x,t) = f(x,t,u(x,t)) \quad \text{in} \;Q=\Omega \times (0,T), \] \[ u(x,t)=0 \quad \text{on} \;\partial \Omega \times (0,T),\qquad u(x,0)=0 \quad \text{in} \;\Omega, \] where \(A\) is a second-order strongly elliptic differential operator, \(f:Q \times \mathbb{R} \rightarrow \mathbb{R}\) is a Carathéodory function. The authors introduce the notions of weak upper and lower solutions and obtain monotone sequences which are convergent quadratically to the unique weak solution of the problem under consideration.