By Messoud Efendiev

ISBN-10: 3034806140

ISBN-13: 9783034806145

ISBN-10: 3034806159

ISBN-13: 9783034806152

This publication offers with the modeling, research and simulation of difficulties bobbing up within the lifestyles sciences, and particularly in organic strategies. The types and findings awarded consequence from extensive discussions with microbiologists, medical professionals and scientific employees, physicists, chemists and commercial engineers and are in keeping with experimental info. They bring about a brand new category of degenerate density-dependent nonlinear reaction-diffusion convective equations that at the same time contain different types of degeneracy: porous-medium and fast-diffusion sort degeneracy. to this point, this classification continues to be no longer in actual fact understood within the mathematical literature and hence specially attention-grabbing. the writer either derives sensible lifestyles technological know-how types and their above-mentioned governing equations of the degenerate kinds and systematically stories those periods of equations. In every one concrete case well-posedness, the dependence of suggestions on boundary stipulations reflecting a few homes of our environment, and the large-time habit of suggestions are investigated and in a few circumstances additionally studied numerically.

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Example text

Sv i + (1 − s)ui , . . , v k · ∂xl v i , i and integrating for all 1 ≤ l ≤ n and 1 ≤ i ≤ k. Multiplying the i-th equation by w− over Ω yields i −∂t w− 2 L2 (Ω) =− Ω i i aii (u) · w− · w− dx − Ω n i w− + Ω l=1 i + w− + Ω 2 2 · a˜ ii v 1 , . . , sv i + (1 − s)ui , . . , v k · v i dx i i · w− γiil (u) · ∂xl w− · γ˜iil v 1 , . . , sv i + (1 − s)ui , . . , v k · ∂xl v i dx i fi (u) − fi (v) · w− dx. Taking into account the homogeneous Dirichlet boundary conditions we derive for the first term of the right-hand side − Ω i i aii (u) · w− · w− dx = Ω i i · ∇w− ∇ aii (u) · w− dx 2 = Ω i aii (u) · ∇w− dx k + j =1 Ω i i ∂j aii (u) · ∇uj · w− · ∇w− dx.

U , . . , vk , i for 1 ≤ l ≤ n. Moreover, for the interaction term follow the monotonicity conditions fi v 1 , . . , u , . . , v k ≤ fi u1 , . . , u , . . , uk , i i whenever i = j , for all 1 ≤ i, j ≤ k. Sufficiency. 26) takes the form ⎧ n ⎪ ⎪ ⎪ ∂t w i = aii (u) · ui − aii (v) · v i − γiil (u) · ∂xl ui − γiil (v) · ∂xl v i ⎪ ⎪ ⎨ l=1 + fi (u) − fi (v), ⎪ ⎪ ⎪ ⎪ w| = w0 , ⎪ ⎩ t=0 w|∂Ω = 0, for 1 ≤ i ≤ k with initial data w0 ∈ K + . e. in Ω corresponding solution w = w(·, t; w0 ) that (w0i )− = 0 implies w− for t > 0 and all 1 ≤ i ≤ k.

1) is order preserving with respect to if and only if the matrices a and γ are diagonal and the reaction term f satisfies fi y 1 , . . , y , . . , y k ≤ fi z 1 , . . , y , . . , z k , i whenever y j ≤ zj , j = i, i for all 1 ≤ i, j ≤ k. 1 and assume u0 v0 . We prove that the order is preserved by the corresponding solutions u = u(·, t; u0 ) and v = v(·, t; v0 ), that is u v for t > 0, if and only if the matrices a and γ are diagonal and the reaction term fulfils the stated monotonicity conditions.

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Evolution Equations Arising in the Modelling of Life Sciences by Messoud Efendiev

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