$L^2$ estimate and a nonlinear logarithmic transformation

Question

Let $\phi$ satisfy $$ \begin{align*} \begin{cases} \phi_t = \phi_{xx}, & t>0, \ x \in (0,1) \\ \phi(0,\cdot) = \phi_0, & x \in (0,1)\\ \phi(t,0) = a(t), \ \phi(t,1) = b(t), & t >0. \end{cases} \end{align*} $$ and $\psi$ satisfy $$ \begin{align*} \begin{cases} \psi_{xx} = 0, & x \in (0,1) \\ \psi(0) = \alpha, \ \psi(1) = \beta, & t >0. \end{cases} \end{align*} $$ Suppose that we know for some $t \in (0,T)$ that $$ \|\phi(t,\cdot)-\psi\|_{L^2} + \|a-\alpha\|_{L^2} + \|b-\beta\|_{L^2} \le C(e^{-t}-e^{-(T-t)}) \quad\quad\quad (*) $$ Can we prove that the same inequality holds for $$\tilde \phi = -2 \,\partial_x(\log(\phi))$$ and $$\tilde \psi = -2 \,\partial_x(\log(\psi))$$ ? If not, how can we estimate $$\|\tilde \phi(t,\cdot)-\tilde\psi\|_{L^2}$$ assuming (*)?