The Evolutionary Case

Abstract

In the rectangle Q_T = (a, c) × (0, T) we consider the following partial differential system

$$ \left\{ {\begin{array}{*{20}c} {u_t- \left( {\varepsilon u_x- \alpha _1 \left( x \right)u} \right)_x+ \beta _1 \left( x \right)u = f\left( {x,t} \right){\mathbf{ }}in{\mathbf{ }}Q_{1T,} }\\ {v_t- \left( {\mu \left( x \right)v_x- \alpha _2 \left( x \right)v} \right)_x+ \beta _2 \left( x \right)v = g\left( {x,t} \right){\mathbf{ }}in{\mathbf{ }}Q_{2T,} }\\ \end{array} } \right. $$

(1)

with which we associate initial conditions

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

(2)

the following natural transmissions conditions at b:

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

(3)

as well as one of the following boundary conditions:

$$ \begin{gathered} u\left( {a,t} \right) = v\left( {c,t} \right) = 0,{\mathbf{}}0 \leqslant t \leqslant T; \hfill \\ u_x \left( {a,t} \right) = v\left( {c,t} \right) = 0,{\mathbf{}}0 \leqslant t \leqslant T; \hfill \\ u\left( {a,t} \right) = 0,{\mathbf{}} - v_x \left( {c,t} \right) = \gamma \left( {v\left( {c,t} \right)} \right),{\mathbf{ }}0 \leqslant t \leqslant T; \hfill \\ \end{gathered} $$

(4)

where Q_1T = (a, b) × (0, T), Q_2T = (b, c) × (0, T), a, b, c ∈ ℝ, a < b < c, T > 0, and ε is a small parameter, 0 < ε ≪ 1.