## A problem of asymptotic stabilization by the right-hand side.(English)Zbl 1152.65064

Summary: We consider the following problem for an operator $$S$$ acting in a Hilbert space $$H$$ and specifying for any $$u_0\in H$$ an evolutionary process $$u_{i+1} = S(u_i)$$, $$i = 0, 1,\dots$$: for given points $$z_0$$, $$a_0$$ construct corrections $$f_i\in{\mathcal F}$$ from a fixed subset $${\mathcal F}\subset H$$ so that positive semitrajectories $$\{a_{i+1}=S(a_i)+f_{i+1}\}^\infty_{i=0}$$ and $$\{z_{i+1}=S(z_i)\}^\infty_{i=0}$$ approach each other, i.e. $$\|a_i-z_i\|_2\to 0$$ for $$i\to\infty$$. Methods for construction of desired corrections $$f_i$$ are proposed under some restrictions, and results of numerical calculations are presented for the one-dimensional Chafee-Infante equation.

### MSC:

 65J15 Numerical solutions to equations with nonlinear operators 35K55 Nonlinear parabolic equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 47J25 Iterative procedures involving nonlinear operators 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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