Global solutions to some reaction diffusion systems with a balance law

M. KIRANE Universite de Picardie Jules Verne 60 000, Beauvais France

            Mokhtar.kirane@aub.u-picardie.fr

We obtained results on existence and large time behaviour of global classical nonnegative solutions of

u_{t} − aΔu − bΔv = −f(u)g(v) in R^{+} × Ω  (0.1)
v_{t} − cΔu − dΔv = +f(u)g(v) in R^{+} × Ω  (0.2)

with boundary and initial conditions

\frac{∂u}{∂ν} = \frac{∂v}{∂ν} = 0 on R^{+} × S  (0.3)
0 ≤ u_{0}(x) ≤ M := max_{x∈Ω} u_{0}(x),  0 ≤ v_{0}(x) ≤ N := max_{x∈Ω} v_{0}(x).  (0.4)

The nonlinearities f(u) and g(v) are nonnegative differentiable functions, f(0) = 0.

Example: The Frank-Kamenetskii approximation to an m^{th}-order exothermic chemical reaction of Arrhenius type where the reaction term is

f(u)g(v) = u^{m}e^{v} with m ≥ 1.

Results

1. Case of lower-triangular matrices : b = 0.

Theorem 1. There exists a unique nonnegative global bounded solution (u(t, x), v(t, x)) of problem (1) − (4) for any nonlinearities f(u) and g(v) whenever

a > d,  c ≥ 0  and  v_{0}(x) ≥ (c/(a − d))u_{0}(x).

Moreover (u(t, x), v(t, x)) ⟶ (k_{1}, k_{2}) as t → +∞ uniformly in Ω, where

k_{1} ≥ 0, k_{2} ≥ 0, f(k_{1})g(k_{2}) = 0, k_{1} + k_{2} = \frac{1}{|Ω|} ∫_{Ω}(u_{0}(x) + v_{0}(x)) dx.

Theorem 2. Assume that

1. f(v) is bounded in a neighbourhood of v = −∞, and

2. c ≥ d − a > 0

hold true. Then there exists a unique global solution (u(t, x), v(t, x)) of problem

(1) − (4).

2. Case of a full matrix.

Notation. N1. γ_{1} := (d − a − \sqrt{(d − a)^{2} + 4bc}) /2b.

N2. γ_{2} := (d − a + \sqrt{(d − a)^{2} + 4bc}) /2b.

Theorem 3. Assume that

H1. w_{10}(x) := w_{1}(0, x) = v_{0}(x) − γ_{1}u_{0}(x) ≥ 0; H2. w_{20}(x) := w_{2}(0, x) = v_{0}(x) − γ_{2}u_{0}(x) ≥ 0; H3. c > d − a > 0, 0 < b << 1; H4. f(v) = v^{n}, n ≥ 1 is an odd integer; and H5. |d + b − a − c| < (γ_{1} + 1)/(γ_{1}C_{p}).

hold true. Then there exists a unique global solution to problem (1) − (4).