On the stability of stationary solutions of nonlinear parabolic systems

In this paper the linearization method is developed for the problem of stability of stationary solutions of nonlinear autonomous parabolic systems of general form. Estimates, uniform with respect to time, are established for Hölder norms of the solutions of the linearized systems. These estimates are then used in the study of the nonlinear problems. Theorems are proved concerning the stability of the first approximation and the conditional stability, generalizing the well-known Lyapunov theorems from the theory of ordinary differential equations.
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