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Extra resources for Equadiff 9: Conference on Differential Equations and their Applications : Proceedings
Ben ) over (0, T ) and pass to the limit for h → 0. ben ) Consequently, there is a constant C1 such that 1 αξ 2 T ∂t p 0 2 h dt + Eξ [pξ ](T ) ≤ Eξ [pξ ](0) + C1 . ben ) 31 Phase-Field equations These estimates allow to use the method proposed by  and used in . Define the following monotone function s |1 − (1 − 2r)2 |dr G(s) = . ben ) 0 We prove next lemma ) where Eξ [pξ ](0) ≤ M0 independently on Lemma 12. ben ξ. 5 . ben ) Ω Proof. We have shown that Eξ [pξ ](t) ≤ M0 + C1 , on 0, T . ben ) + C1 ).
Bog ) we shall get the positive solution to the problem ∗ ∂ ∂ρ ∂ν ∂ρ p ∗ N −1 + ρ ∂ν ∂ρ p + eλν+κ|ν | = 0, ν(1) = 0, ρ ∈ (0, 1) , ν (0) = 0, in the following form 1 τ ν(t, 0) = t 0 ρ τ N −1 e λν(ρ,0)−κν (ρ,0) 1 p dρ dτ. Supported by the Grant No. OTKA 019095 (Hungary). References  F. V. Atkinson, L. A. Peletier, Ground states of ∆u = f (u) and the Emden-Fowler equation, Archs. Ration. Mech. Analysis, 93 (1986), 103–107.  J. V. Baxley, Some singular nonlinear boundary value problems, SIAM J.
E. argument makes from the function v a continuous map from 0, T to ) and according to the assumption L1 (Ω). 5 G(1) − G(0) 2 . This concludes the proof. It remains to show the boundedness of the total variation of v (). The lower semicontinuity of the total variation in L1 -space together with the Lemma 12 yields |∇G∗ |dx ≤ M0 ess sup 0