By Toyohiko Aiki, Takanobu Okazaki (auth.), Isabel Narra Figueiredo, José Francisco Rodrigues, Lisa Santos (eds.)
This ebook collects refereed articles containing unique effects reporting the new contributions of the lectures and communications offered on the loose Boundary difficulties convention that happened on the collage of Coimbra, Portugal, from June 7 to twelve, 2005 (FBP2005). They care for the math of a large type of types and difficulties regarding nonlinear partial differential equations coming up in physics, engineering, biology and finance. one of the major subject matters, the talks thought of unfastened boundary difficulties in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in snapshot processing, in monetary arithmetic or in computations for inter-scale problems.
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Extra resources for Free Boundary Problems: Theory and Applications
Acerbi, G. A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian ﬂuids, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 21 (2004), pp. 25–60.  S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Analysis Serie A: Theory and Methods (to appear).  S. Antontsev and S. html. ) , A model porous medium equation with variables exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Analysis Serie A: Theory and Methods, 60 (2005), pp.
35K55; 35K65. Keywords. Nonlinear parabolic equation, nonstandard growth conditions, anisotropic nonlinearity, localization of solutions. 1. Statement of the problem Let Ω ⊂ Rn be a bounded simple-connected domain and 0 < T < ∞. 1) u(x, 0) = u0 (x) in Ω, where z = (x, t) ∈ Q ≡ Ω × (0, T ], Γ denotes the lateral boundary of the cylinder Q. 2) σ(z) ⊆ inf σ, sup σ ⊂ (σ − , σ + ) Q Q The ﬁrst author was partially supported by the research project DECONT, FCT/MCES (Portugal) at the “Centro de Matem´ atica”, Universidade da Beira Interior.
50 M. Aso, M. Fr´emond and N. 4) wµn ,λ → wλ weakly in L2 (0, T ; L2(Ω)), ⎪ 2 2 ⎪ ⎪ w → −∆ w weakly in L (0, T ; L (Ω)), A µ µ ,λ 0 λ n n ⎪ ⎪ ⎪ ⎪ in C([0, T ]; L2(Ω)), ⎪ βλ (wµn ,λ ) → βλ (wλ ) ⎪ ⎪ ⎪ in C([0, T ]; L2(Ω)), ⎪ f (θµn ,λ , wµn ,λ ) → f (θλ , wλ ) ⎩ p(θµn ,λ , Jµn wµn ,λ ) → p(θλ , wλ ) in C([0, T ]; L2(Ω)), as n → ∞. Also, put ξµ,λ := −wµ,λ − Aµ wµ,λ − βλ (wµ,λ ) + f (θµ,λ , wµ,λ ). e. 5) as n → ∞. 6) T (ξλ , wλ − p(θλ , wλ ))L2 (Ω) dt. e. on Q. e. e. e. 9) θλ (0) = θ0 , wλ (0) = w0 in L2 (Ω).