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Monotone solutions to quasilinear parabolic equations. (English. Russian original) Zbl 0814.35061

Sib. Math. J. 34, No. 4, 636-645 (1993); translation from Sib. Mat. Zh. 34, No. 4, 50-60 (1993).
General quasilinear parabolic equations on a bounded domain in \(\mathbb{R}^ N\) under linear boundary conditions are considered. Due to the maximum principle, the solution flows of such equations belong to the class of strongly monotone (order preserving) dynamical systems. The main result of the present contribution states that if the nonlinearities in the equation are real analytic and some growth and dissipative conditions are satisfied then most solutions (that is, solutions emanating from an open and dense set of initial conditions) become eventually monotone in time.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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