On the convergence rates for solutions of some chemical interfacial reaction problems (Q1912839)

We treat some diffusion equations with a nonlinear system of boundary conditions, which appear in chemical engineering. Our concern is to investigate the asymptotic behavior (as \(z\to\infty\)) of solutions to the following initial boundary value problem: \[ a(x){\partial u\over\partial z}= {\partial^2u\over\partial x^2},\quad b(x){\partial v\over\partial z}= {\partial^2v\over\partial x^2}\quad\text{for }(x,z)\in I\times(0,\infty); \] \[ {\partial u\over\partial x} (0,z)= R_1(u(0,z),v(0,z)),\quad {\partial v\over\partial x} (0,z)= R_2(u(0,z),v(0,z)), \] \[ {\partial u\over\partial x} (1,z)=0,\;{\partial v\over\partial x} (1,z)=0\text{ for }z\in(0,\infty);\;u(x,0)=\phi_1(x),\;v(x,0)=\phi_2(x)\text{ for }x\in I. \] Here \(I\) and \(\overline I\) denote \((0,1)\) and \([0,1]\), respectively; \(a(x)\) and \(b(x)\) are given functions satisfying \(a\in C^\infty(\overline I)\), \(b\in C^\infty(\overline I)\), \(a(x)>0\), \(b(x)>0\) for \(x\in[0,1)\), \(a(1)=b(1)=0\); \(\phi_i(x)\) \((i=1,2)\) are nonnegative initial data; \(R_i(u,v)= k_iR_0(u,v)\) \((i=1,2)\), where \(k_i\) \((i=1,2)\) are positive constants and \(R_0(u,v)= u^mv^n\) with positive integers \(m\) and \(n\).