Lipschitz stability for an inverse problem for the 2D-Sellers model on a manifold. (English) Zbl 1375.58021

Martinez, Patrick; Tort, Jacques; Vancostenoble, Judith

Riv. Mat. Univ. Parma (N.S.) 7, No. 2, 351-389 (2016).

The paper under review deals with an inverse problem that consists in recovering the so-called insolation function in the 2-D Sellers model on a Riemannian manifold \(\mathcal{M}\) that materializes the Earth’s surface. The nonlinear problem studied is \[ \begin{cases} u_t-\Delta_{\mathcal{M}}u=r(t)q(x)\beta(u)-\varepsilon(u)u|u|^3, & x\in \mathcal{M},\;t>0,\\ u(0,x)=u^0(x), & x\in \mathcal{M}, \end{cases} \] and under suitable assumptions on the data, the authors obtain Lipschitz stability results in the spirit of O. Y. Imanuvilov and M. Yamamoto [Inverse Probl. 14, No. 5, 1229–1245 (1998; Zbl 0992.35110)] in the case of the determination of the source term in the linear heat equation. The paper complements an analogous study by J. Tort and J. Vancostenoble [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 5, 683–713 (2012; Zbl 1270.35283)] in the case of the 1-D Sellers model.

Reviewer: Dian K. Palagachev (Bari)

MSC:

58J35	Heat and other parabolic equation methods for PDEs on manifolds
35K55	Nonlinear parabolic equations

Keywords:

PDEs on manifolds; 2-D Sellers model; nonlinear parabolic equations; climate models; inverse problems; Carleman estimates

Citations:

Zbl 0992.35110; Zbl 1270.35283