Applied Analysis and Differential Equations: Iasi, Romania, by Ovidiu Carja, Ioan I. Vrabie

By Ovidiu Carja, Ioan I. Vrabie

This quantity includes refereed examine articles written by means of specialists within the box of utilized research, differential equations and similar themes. recognized top mathematicians world wide and widespread younger scientists conceal a various diversity of themes, together with the main intriguing contemporary advancements. A large variety of themes of modern curiosity are handled: lifestyles, specialty, viability, asymptotic balance, viscosity options, controllability and numerical research for ODE, PDE and stochastic equations. The scope of the publication is vast, starting from natural arithmetic to numerous utilized fields reminiscent of classical mechanics, biomedicine, and inhabitants dynamics

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The fact that the derived cone is a proper generalization of the classical concepts in Differential Geometry and Convex Analysis is illustrated by the January 8, 2007 48 18:38 WSPC - Proceedings Trim Size: 9in x 6in icaade A. Cernea following results (Ref. 1): if X ⊂ Rn is a differentiable manifold and Tx X is the tangent space in the sense of Differential Geometry to X at x Tx X = {v ∈ Rn ; ∃ c : (−s, s) → X, of class C 1 , c(0) = x, c (0) = v} then Tx X is a derived cone; also, if X ⊂ Rn is a convex subset then the tangent cone in the sense of Convex Analysis defined by T Cx X = cl{t(y − x); t ≥ 0, y ∈ X} is also a derived cone.

G. Barles, Solutions de Viscosit´e des Equations de Hamilton-Jacobi, (SpringerVerlag, 1994). 7. -L. Lions, G. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, preprint. com We consider a reaction-diffusion system of the form   u (t) = Au(t) + F (u(t), v(t)), t ≥ 0 v (t) = Bv(t) + G(u(t), v(t)), t ≥ 0  u(0) = ξ, v(0) = η, where X and Y are real Banach spaces, K is a nonempty and locally closed subset in X × Y, A : D(A) ⊆ X → X, B : D(B) ⊆ Y → Y are the generators of two C0 -semigroups, {SA (t) : X → X; t ≥ 0} and {SB (t) : Y → Y ; t ≥ 0} respectively, F : K → X, G : K → Y, are continuous such that A + F and B + G are of compact type.

Let K ⊆ R×X ×Y be a locally closed set and let (F, G) : K → X ×Y be continuous. Let us assume that (A + F, B + G) is locally of β-compact type with respect the second argument. Then a necessary and sufficient condition in order that K be viable with respect to (A + F, B + G) is that, for each (τ, ξ, η) ∈ K lim inf h↓0 1 dist ((τ + h, SA (h)ξ + hF (τ, ξ, η), SB (h)η + hG(τ, ξ, η)); K) = 0. h 3. An Example for a Predator-pray System Let Ω ⊆ Rn , n = 1, 2, . . , be a bounded domain with C 2 boundary Γ, let δi ≥ 0, i = 1, 2, p > 0, q > 0, let f : R × R → R+ and g : R × R → R− be two continuous functions and let us consider the following general predatorpray system  ut (t, x) = δ1 ∆u − pu(t, x) + f (u(t, x), v(t, x)) (t, x) ∈ Qτ,∞    vt (t, x) = δ2 ∆v + qv(t, x) + g(u(t, x), v(t, x)) (t, x) ∈ Qτ,∞ (10)  u(t, x) = v(t, x) = 0 (t, x) ∈ Στ,∞ ,   u(τ, x) = ξ(x), v(τ, x) = η(x) x ∈ Ω, where 0 ≤ τ < T ≤ ∞, Qτ,T = (τ, T ) × Ω, Στ,T = (τ, T ) × Γ and ξ, η ∈ L2 (Ω).

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