Global Existence of Classical Solutions for a Class of Reaction-Diffusion Systems
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Global Existence of Classical Solutions for a Class of Reaction-Diffusion Systems El-Haj Laamri
Received: 29 July 2010 / Accepted: 25 March 2011 / Published online: 7 April 2011 © Springer Science+Business Media B.V. 2011
Abstract In this paper, we use duality arguments “ à la Michel Pierre” to establish global existence of classic solutions for a class of parabolic reaction-diffusion systems modeling, for instance, the evolution of reversible chemical reactions. Keywords Reaction-diffusion systems · Global existence · Classical solutions · Weak solutions · Blow-up
1 Introduction This paper is motivated by the general question of global existence in time of solutions to the following reaction-diffusion system
(S )
⎧ γ α β ⎪ ⎪ ut − d1 u = w − u v ⎪ ⎪ ⎪ ⎪ vt − d2 v = wγ − uα v β ⎪ ⎪ ⎪ ⎪ γ α β ⎪ ⎪ ⎨ wt − d3 w = −w + u v ∂u ∂v ∂w (t, x) = ∂n (t, x) = ∂n (t, x) = 0 ∂n ⎪ ⎪ ⎪ ⎪ u(0, x) = u0 (x) ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ v(0, x) = v0 (x) ≥ 0 ⎪ ⎪ ⎪ ⎩ w(0, x) = w0 (x) ≥ 0
(0, +∞) × ,
(E1 )
(0, +∞) × ,
(E2 )
(0, +∞) × ,
(E3 )
(0, +∞) × ∂, x ∈ , x ∈ , x ∈ ,
where is a bounded regular open subset of RN , (d1 , d2 , d3 , α, β, γ ) ∈ (0, +∞)3 × [1, +∞)3 . E.-H. Laamri () Institut Elie Cartan, Université Henri Poincaré, Nancy 1, B.P. 239, 54 506 Vandoeuvre-lès-Nancy, France e-mail: [email protected]
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E.-H. Laamri
Note that the system (S ) satisfies two main properties, namely: (P ) the nonnegativity of solutions of (S ) is preserved for all time; (M) the total mass of the components u, v, w is a priori bounded on all finite intervals (0, t). If α, β and γ are positive integers, system (S ) is intended to describe for example the evolution of a reversible chemical reaction of type αU + βV γ W where u, v, w stand for the density of U , V and W respectively. This chemical reaction is typical of general reversible reactions and contains the major difficulties encountered in a large class of similar problems as regards global existence of solutions. Let us make precise what we mean by solution. By classical solution to (S ) on QT = (0, T ) × , we mean that, at least (i) (u, v, w) ∈ C ([0, T ); L1 ()3 ) ∩ L∞ ([0, τ ] × )3 , ∀τ ∈ (0, T ); (ii) ∀k, = 1, . . . , N , ∀p ∈ (1, +∞) ∂t u, ∂t v, ∂t w, ∂xk u, ∂xk v, ∂xk w, ∂xk x u, ∂xk x v, ∂xk x w, u, v, w ∈ Lp ((0, T ) × ) ; (iii) equations in (S ) are satisfied a.e. (almost everywhere). By weak solution to (S ) on QT = (0, T )×, we essentially mean solution in the sense of distributions or, equivalently here, solution in the sens of the variation of constants formula with the corresponding semigroups. More precisely u(t) = Sd1 (t)u0 + Sd1 (t − s)(wγ (s) − uα (s)v β (s)) ds, 0
v(t) = Sd2 (t)v0 +
Sd2 (t − s)(wγ (s) − uα (s)v β (s)) ds, 0
w(t) = Sd3 (t)u0 +
0
Sd3 (t − s)(−wγ (s) + uα (s)v β (s)) ds,
where Sdi (.) is the semigroup generated in L1 () by −di with homogeneous Neumann boundary condition, 1 ≤ i ≤ 3. By just integrating the sum (E and time, and taking into 1 ) + (E2 ) + 2(E3 ) in space account the boundary conditions (d1 u + d2 v
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