Presentation of the Problems

Abstract

In this chapter we introduce the problems we are going to investigate in the next chapters of this part. They are mathematical models for diffusion-convectionreaction processes. We are particularly interested in coupled problems in which a small parameter is present. More precisely, let us consider in the rectangle Q_T = (a, c) × (0, T), −∞ < a < c < ∞, 0 < T < ∞, the following system of parabolic equations

$$ \left\{ {\begin{array}{*{20}c} {u_t+ \left( { - \varepsilon u_x+ \alpha _1 \left( x \right)u} \right)_x+ \beta _1 \left( x \right)u = f\left( {x,t} \right){\mathbf{ }}in{\mathbf{ }}Q_{1T} ,}\\ {v_t+ \left( { - \mu \left( x \right)v_x+ \alpha _2 \left( x \right)v} \right)_x+ \beta _2 \left( x \right)v = g\left( {x,t} \right){\mathbf{ }}in{\mathbf{ }}Q_T ,}\\ \end{array} } \right. $$

with which we associate initial conditions

$$ u\left( {x,0} \right) = u_0 \left( x \right),a \leqslant x \leqslant b;{\mathbf{}}v\left( {x,0} \right) = v_0 \left( x \right),b \leqslant x \leqslant c, $$

(2)

transmissions conditions at x = b:

$$ \left\{ {\begin{array}{*{20}c} {u\left( {b,t} \right) = v\left( {b,t} \right),}\\ { - \varepsilon u_x \left( {b,t} \right) + \alpha _1 \left( b \right)u\left( {b,t} \right) =- \mu \left( b \right)v_x \left( {b,t} \right) + \alpha _2 \left( b \right)v\left( {b,t} \right),{\mathbf{ }}0 \leqslant t \leqslant T,}\\ \end{array} } \right. $$

(3)

as well as one of the following types of boundary conditions:

$$ \left\{ {\begin{array}{*{20}c} {u\left( {b,t} \right) = v\left( {b,t} \right),}\\ { - \varepsilon u_x \left( {b,t} \right) + \alpha _1 \left( b \right)u\left( {b,t} \right) =- \mu \left( b \right)v_x \left( {b,t} \right) + \alpha _2 \left( b \right)v\left( {b,t} \right),{\mathbf{ }}0 \leqslant t \leqslant T,}\\ \end{array} } \right. $$

(3)

where $Q_{1T}= \left( {a,b} \right) \times \left( {0,T} \right),Q_{2T}= \left( {b,c} \right) \times \left( {0,T} \right),b \in \mathbb{R},a < b < c,\gamma$ , γ is a given nonlinear function and ε is a small parameter, 0 < ε ≪ 1.