## 4 Well-Posedness of the Inverse Problems
### The IP2
Let us sum up the results to obtain  
\[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]3 \[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]4 \[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]5  
We have, for each term,  
\[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]6\[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]7  
\[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]8  
\[\frac{\partial b}{\partial t}-v_{m}\frac{\partial^{2}b}{\partial x^{2}}-\beta (t)\frac{\partial U}{\partial x}=S(x,t), \tag{22}\]9  
\[R(x,t)=\frac{E^{\prime}(t)}{X}-u^{\prime}_{l}(t)-\frac{x}{X}g^{\prime}(t)- \frac{\
u}{X^{2}}g(t), \tag{23}\]0  
\[R(x,t)=\frac{E^{\prime}(t)}{X}-u^{\prime}_{l}(t)-\frac{x}{X}g^{\prime}(t)- \frac{\
u}{X^{2}}g(t), \tag{23}\]1  
\[R(x,t)=\frac{E^{\prime}(t)}{X}-u^{\prime}_{l}(t)-\frac{x}{X}g^{\prime}(t)- \frac{\
u}{X^{2}}g(t), \tag{23}\]2  
\[R(x,t)=\frac{E^{\prime}(t)}{X}-u^{\prime}_{l}(t)-\frac{x}{X}g^{\prime}(t)- \frac{\
u}{X^{2}}g(t), \tag{23}\]3  
The application of the trace inequalities implies  
\[R(x,t)=\frac{E^{\prime}(t)}{X}-u^{\prime}_{l}(t)-\frac{x}{X}g^{\prime}(t)- \frac{\
u}{X^{2}}g(t), \tag{23}\]4  
\[R(x,t)=\frac{E^{\prime}(t)}{X}-u^{\prime}_{l}(t)-\frac{x}{X}g^{\prime}(t)- \frac{\
u}{X^{2}}g(t), \tag{23}\]5  
Now, let us insert all the estimates in (26) to obtain  
\[ \begin{array}{l}\frac{d}{dt}\bigg{(}\bigg{\|}\frac{\partial U}{\partial x} \bigg{\|}^{2}+\bigg{\|}\frac{\partial b}{\partial x}\bigg{\|}^{2}\bigg{)}+C_ {1}\bigg{(}\bigg{\|}\frac{\partial^{2}U}{\partial x^{2}}\bigg{\|}^{2}+\bigg{\|} \frac{\partial^{2}b}{\partial x^{2}}\bigg{\|}\bigg{)}\\ \leq\bigg{(}X(\epsilon_{1}+\epsilon_{2})+\epsilon_{3}+\epsilon_{5} \bigg{(}\frac{1}{2}+\frac{1}{4\epsilon_{1}}\bigg{)}\bigg{)}\bigg{\|}\frac{ \partial^{2}U}{\partial x^{2}}\bigg{\|}\\ +(\epsilon_{4}+\epsilon_{6})\bigg{\|}\frac{\partial^{2}b}{\partial x^{2}} \bigg{\|}^{2}+\frac{1}{\epsilon_{5}}\bigg{(}\frac{1}{2}+\frac{1}{4\epsilon_{ 1}}\bigg{)}\bigg{\|}\frac{\partial U}{\partial x}\bigg{\|}^{2}+\frac{1}{4 \epsilon_{6}}\bigg{\|}\frac{\partial b}{\partial x}\bigg{\|}^{2}\\ \hskip 142.26378 pt+\frac{1}{4\epsilon_{3}}\|R\|^{2}+\frac{1}{4\epsilon_{4}}\|S \|^{2}.\end{array}  \tag{27}\]  
Next, we integrate the inequality (27) from \(0\) to \(t\leq T\) and apply to the result the [PERSON]'s inequality to find  
\[\bigg{\|}\frac{\partial U}{\partial x}\bigg{\|}_{0}^{2}+\bigg{\|}\frac{\partial b }{\partial x}\bigg{\|}_{0}^{2}\bigg{\|}\frac{\partial^{2}U}{\partial x^{2}} \bigg{\|}_{0}^{2}+\bigg{\|}\frac{\partial^{2}b}{\partial x^{2}}\bigg{\|}_{0}^{2}\]  
\[\leq C(t)\bigg{(}\|R\|_{2,Q_{T}}^{2}+\|S\|_{2,Q_{T}}^{2}+\Big{\|}U_{0}^{2}(x) \bigg{\|}_{2}^{2}+\|b_{0}(x)\|_{0}^{2}\bigg{)},\]  
where  
\[\|\cdot\|_{2,Q_{T}}^{2}=\int_{0}^{T}\|u_{0}\|^{2}d\tau,\ C(t)>0.\]  
From here follows the uniqueness and continuous dependence of the solution of problem IP2 from the input data. The local existence of a weak solution to the present NCIBVP follows from the general theory of parabolic equations; see, e.g., [5]. The existence of global weak solutions is a direct corollary of the a priori estimates (25). Next, using the smooth assumptions for the input data, it is easy to show (see [5]) that a weak solution is a classical one.