Inverse Problems of Determination of the Right-Hand Side Term in the Degenerate Higher-Order Parabolic Equation on a Plane

Determination of the time-dependent thermal grooving coefficient

“…However, in anomalous cases where the surface diffsivity may vanish (e.g. at the initial time t = 0), the PDE in (2.3) becomes degenerate and the techniques of [19,20] may be applied. The solution (B(t), u(x, t)) of the inverse problem (2.3) and (2.4) is sought in the class M × (C 4,1 (Ω T ) ∩ C 3,0 (Ω T )).…”

“…x u(x, t) = g(x, t)p + r(x, t), (x, t) ∈ (0, L) × (0, T ), (1.1) were considered in [19] and [20] to determine the unknown right-hand side term p(t) or p(x), respectively from an integral observation. The well-posedness of these inverse problems were established even in the degenerate case when the coefficient a(x, t) is allowed to vanish on a zero-measure set.…”

Determination of the time-dependent effective ion collision frequency from an integral observation

“…The space-dependent diffusion coefficient σ(x) in (1.1) was identified in [27] from the same additional observation of [4], and the Lipschitz stability was obtained locally using the Bukhgeım-Klibanov method and Carleman estimates. Inverse source/load linear problems for recovering the free term f are not discussed herein, but we mention [13,14] for the Euler-Bernoulli equation and [21,22] for the more general 2m-order (m ∈ N * ) parabolic equation u t + (−1) m a(t)∂ 2m…”

Recovery of the time‐dependent zero‐order coefficient in a fourth‐order parabolic problem

“…For the following higher 2 m ‐order ($$$ m\in {\mathrm{\mathbb{N}}}^{\ast } $$$) parabolic problem: $$$$ \left(\begin{array}{cc}{\partial}_tu+{\left(-1\right)}^ma\left(x,t\right){\partial}_x^{2m}u=r(t)f\left(x,t\right),& \left(x,t\right)\in {Q}_T,\\ {}{\partial}_x^ju\left(0,t\right)={\partial}_x^ju\left(1,t\right)=0,\kern1em j=\overline{0,m-1},& t\in \left(0,T\right),\\ {}u\left(x,0\right)=g(x),& x\in \left(0,1\right),\end{array}\right. $$$$ the unknown time‐dependent source component term $$$ r(t) $$$ has been determined from some given integral observation in [10]. Under certain assumptions, the authors proved the well‐posedness of the solutions to such inverse problems even in the degenerate case, namely, $$$ a\left(x,t\right) $$$ may vanish on a zero‐measure set, by applying the contraction mapping theorem.…”

“…x ∈ (0, 1), the unknown time-dependent source component term r(t) has been determined from some given integral observation in [10]. Under certain assumptions, the authors proved the well-posedness of the solutions to such inverse problems even in the degenerate case, namely, a(x, t) may vanish on a zero-measure set, by applying the contraction mapping theorem.…”