A stability result for a nonlinear hyperbolic inverse problem (Q1814208)

The inverse problem corresponding to \[ u_{tt}(x,t)-q\left(\int^ L_ 0| u_ x(y,t)|^ 2dy\right)u_{xx}(x,t)=0\quad (x,t)\in(0,L)\times(0,T) \] \[ u(x,0)=u_ 0(x), u_ t(x,0)=u_ 1(x),\quad x\in[0,L],\quad u(0,t)=u(L,t)=0,\quad t\in[0,T] \] consists here of determining the coefficient function \(q\) knowing \(u(x_ 0,t)=a(t)\), \(t\in[0,T]\), at some point in the interior of the space interval, \(x_ 0\in (0,L)\). Regularity assumptions are \[ u_ 0\in W^ 3_ \infty((0,L), u_ 1\in W^ 2_ \infty((0,L)), a\in W^ 3_ \infty((0,T)), q\in W^ 1_ \infty \] strictly positive on its domain of definition. The author derives the following stability result for two solutions \((u_ 1,q_ 1)\), \((u_ 2,q_ 2)\) corresponding to two sets of data \[ \begin{split}\| u_ 2-u_ 1\|_{W_ \infty^{2+\lambda}((0,L)\times (0,T_ 0))}+\| q_ 2-q_ 1\|_{W^ \lambda_ \infty(G_{T_ 0})} \\ \leq C(\| u_{02}- u_{01}\|^ \lambda _{L_ \infty((0,L))}+\| u_{12}- u_{11}\|^ \delta_{L_ \infty}+\| a_ 2-a_ 1\|^ \gamma_{L_ \infty((0,T_ 0))})\end{split} \] for all \(\lambda\in[0,1)\), \(\gamma:=(4(1-\lambda))^{-1}\), \(\delta:=(2(1-\lambda))^{-1}\). The proof is based on the method of characteristics. Assuming certain constant bounds from below and above for occuring functions and their norms, the resulting integral equations are transformed into inequalities allowing for the application of Gronwall's lemma. As the data is known at \(x_ 0\in(0,L)\) the PDE can be applied at this space point.