Strict positive property of the derivative of the solution of a partial differential equation (Q1087731)

Let \(Q\equiv \{(x,t):\) \(0<x<1\), \(0<t<T\}\). The author looks for a solution \(u\in C^{2,1}(\bar Q)\) of the equation \[ [a(t,u)]_ t=(k(x,t,u,u_ x)u_ x)_ x+d(x,t,u,u_ x)u_ x+f(x,t,u)\quad in\quad Q \] under initial and boundary conditions \[ u(x,0)=u_ 0(x),\quad u_ 0(x)\in H^{2+\alpha}[0,1],\quad u_ 0'(x)\geq 0, \] \[ Ku_{x|_{x=0}}=\psi_ 1(t)\geq 0,\quad ku_{x|_{x=1}}=\psi_ 2(t)\geq 0,\quad \psi_ 2(t)\in H^{1+\alpha /2}[0,T] \] and matching conditions of zero order. Conditions for the positiveness of the derivative of the solution of the problem under consideration are formulated.