Diffusion in nonhomogeneous environment with active diffusion interface conditions (Q2770833)

The authors study boundary value problems of the type NEWLINE\[NEWLINEp_i^{-1} (x)\bigl\{p_i(x)g_i (x,y_i)y_i'\bigr\}'= f_i(x,y_i);\;i-1\leq x\leq i,\;i=1,2,NEWLINE\]NEWLINE NEWLINE\[NEWLINEy_1'(0)= h_1\bigl(y_1(0)\bigr),\;y_2(2)=h_2\bigl(y_2'(2)\bigr),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{gathered} -\alpha p_1(1)g_1 \bigl(1,y_1(1) \bigr)y_1'(1)= H\bigl(y_1(1),y_2(1) \bigr)+h_3 \bigl(y_1(1) \bigr)-k_1,\\ -\beta p_2(1)g_2 \bigl(1,y_2(1) \bigr)y_2' (1)= H\bigl(y_1(1), y_2(1)\bigr)- h_4\bigl(y_2(1)\bigr)+ k_2,\end{gathered}NEWLINE\]NEWLINE where the given functions \(f_i,p_i,g_i\) and \(H\) are assumed to be continuous. They obtain a system of additional conditions on these functions, motivated by problems in population ecology, that guarantees the unique solution to the above problem.