On the quasireversibility of an inverse boundary value problem for nonequilibrium sorption dynamics (Q1878731)

Let \(Q_T=\{(x,t)| x\in (0,l), t\in [0,T]\}\), \(f\in C^2\), \(f(0)=0\), \(0<m\leq f'(x)\leq N\leq\infty\), and let \(l,T, v, \varepsilon\), \(D, \lambda\), \(\gamma\) be given constant. The author studies the following problem: It is required to find a function \(\mu\in C^2[0,T]\) with \(\mu(0)=\mu'(0)=\mu''(0)=0\) entering the mixed boundary-value problem \[ vc_{xx}+\varepsilon c_t+a_t=Dc_xx,\, c(x,0)=0,\, x\in [0,l],\tag{1} \] \[ c(0,t)=\mu(t),\, C(l,t) + \lambda c_x(l,t)=0, \, t\in [0,T], \tag{2} \] \[ a_t=\gamma[f(c)-a],\, (x,t)\in Q_t,\, a(x,0)=0,\, x\in [0,l] \tag{3} \] by using an additional information given by \(\phi(t)=c(l,t), 0\leq t\leq T\), where \(\phi\) is a function of class \(C^2\) such that \(\phi(0)=\phi'(0)=\phi''(0)=0\). The author suggests an approximate method for solution of this problem based on passing from the diffusion equation (1) to a hyperbolic equation with a small parameter standing by the higher derivative. The convergence of the method is proved. Also, the uniqueness of solutions of the original problem is proved.