Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

Preprint English OPEN
Nakamura, Gen ; Vashisth, Manmohan (2017)
  • Subject: Mathematics - Analysis of PDEs

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\geq 3$. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\sqrt{\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotropic wave equation perturbed by a divergence with respect to $x$ of a vector whose components are quadratics with respect to $\nabla_x u(t,x)$ by ignoring the terms with smallness $O(|\nabla_x u(t,x)|^3)$. We will show that we can uniquely determine $\gamma(x)$ and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval. More precisely the boundary measurements are given as the so-called the hyperbolic Dirichlet to Neumann map.
  • References (23)
    23 references, page 1 of 3

    [1] Belishev, Mikhail I. Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Problems 13 (1997), no. 5, R1R45.

    [2] Choulli, Mourad; Ouhabaz, El Maati; Yamamoto, Masahiro. Stable determination of a semilinear term in a parabolic equation. Commun. Pure Appl. Anal. 5 (2006), no. 3, 447462.

    [3] Denisov, Alexander M. An inverse problem for a quasilinear wave equation. (Russian) Di er. Uravn. 43 (2007), no. 8, 1097{1105, 1151{1152; translation in Di er. Equ. 43 (2007), no. 8, 11231131.

    [4] Grasselli, Maurizio. Local existence and uniqueness for a quasilinear hyperbolic inverse problem. Appl. Anal. 32 (1989), no. 1, 1530.

    [5] Hervas, David; Sun, Ziqi. An inverse boundary value problem for quasilinear elliptic equations. (English summary) Comm. Partial Di erential Equations 27 (2002), no. 11-12, 24492490.

    [6] Isakov, Victor. Uniqueness of recovery of some systems of quasilinear elliptic and parabolic partial di erential equations. Nonlinear problems in mathematical physics and related topics, II, 201212, Int. Math. Ser. (N. Y.), 2, Kluwer/Plenum, New York, 2002.

    [7] Isakov, Victor. Inverse problems for partial di erential equations. Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006. xiv+344 pp.

    [8] Isakov, Victor; Nachman, Adrian I. Global uniqueness for a twodimensional semilinear elliptic inverse problem. Trans. Amer. Math. Soc. 347 (1995), no. 9, 33753390.

    [9] Kang, Kyeonbae; Nakamura, Gen. Identi cation of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map. Inverse Problems 18 (2002), no. 4, 10791088.

    [10] Katchalov, Alexander; Kurylev, Yaroslav; Lassas, Matti. Inverse boundary spectral problems. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123. Chapman & Hall/CRC, Boca Raton, FL, 2001.

  • Metrics
    views in OpenAIRE
    views in local repository
    downloads in local repository
Share - Bookmark