Abstract
We consider the Cauchy problem for a one-dimensional quasilinear degenerate heat equation with source. The property of monotone in time behavior of the weak solution at a fixed spatial point x?ϵ R is studied. It is shown that the conditions of a such a behavior depend on “a nonlinear interaction” between the nonlinear heat operator and the source of energy considered. There are two different cases: (i) the solution is monotone in time for any x0 which is far enough from the initial support and (ii) the solution is monotone in time if it becomes large enough. In a general situation the monotone in time behavior at a given point x — x0 is proved to depend on the shape of the initial function in some neighborhood of x = x0. Proofs are based on the method of intersection comparison of the solution and the continuous set of stationary solutions of the same equation.
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Galaktionov, V.A., Posashkov, S.A. (1993). On Some Monotonicity in Time Properties for a Quasilinear Parabolic Equation with Source. In: Ni, WM., Peletier, L.A., Vazquez, J.L. (eds) Degenerate Diffusions. The IMA Volumes in Mathematics and its Applications, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0885-3_5
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DOI: https://doi.org/10.1007/978-1-4612-0885-3_5
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