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On the solution of nonlinear problems for parabolic equations by the method of successive approximations. (English. Russian original) Zbl 0596.65088
Sov. Math., Dokl. 31, 181-184 (1985); translation from Dokl. Akad. Nauk SSSR 280, 1040-1043 (1985).
The parabolic equation is $$\partial u/\partial t+\tau (t)u+\Phi (u)u=f$$, where $$\tau$$ (t) is a linear second order space operator and $$\Phi$$ (u) a mildly nonlinear space operator. The iteration process consists of equations of the type $$\partial u_ k/\partial_{t+1}(t)u_ k=f- r(t)u_{k-1}-\Phi (u_{k-1})u_{k-1}.$$ Some sufficient conditions for the existence of a unique solution u of the first initial-boundary value problem and the convergence of the iteration process to u in the energy norm are given. Two other theorems state similar behavior under somewhat more restricted preconditions and for the third initial-boundary value problem.
Reviewer: P.Laasonen
##### MSC:
 65Z05 Applications to the sciences 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
##### Keywords:
method of successive approximations; iteration; convergence