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Any large solution of a non-linear heat conduction equation becomes monotonic in time

Published online by Cambridge University Press:  14 November 2011

Victor A. Galaktionov  and
Sergey A. Posashkov
Victor A. Galaktionov
Affiliation:
Keldysh Institute of Applied Mathematics, Miusskaya pl. 4, 125047 Moscow, U.S.S.R
Sergey A. Posashkov
Affiliation:
Keldysh Institute of Applied Mathematics, Miusskaya pl. 4, 125047 Moscow, U.S.S.R
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Synopsis

In this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) >0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 1-2 , 1991 , pp. 13 - 20
Copyright
Copyright © Royal Society of Edinburgh 1991

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