Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides

S. N. Sidorov

Ufa mathematical journal, Tome 11 (2019) no. 1, pp. 75-89

Résumé

We consider direct and inverse problems on determining time-dependent factors in the right hand sides for a mixed parabolic-hyperbolic equation with a degenerate hyperbolic part in a rectangular area. As a preliminary, we study a direct initial boundary problem for this equation. By the method of spectral analysis we establish the uniqueness criterion for the solution and the solution is constructed as a sum over the system of the eigenfunctions of the corresponding one-dimensional Sturm-Liouville spectral problem. In justifying the convergence of the series, the problem of small denominators arises. Because of this, we prove the estimates for the distance from the zero to the small denominators with a corresponding asymptotics. These estimates allow us to justify the convergence of the constructed series in the class of regular solutions of this equation. On the base of the solution to the direct problem, we formulate and study three inverse problems on finding time-dependent factors in the right hand side only by the parabolic or hyperbolic part of the equation, and also as the factors in the both sides of the equation are unknown. Using the formula of solution to the direct initial boundary problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. By means of the theory of integral equations, the corresponding theorems of uniqueness and the existence of solutions of the stated inverse problems are proved. At that, the solutions of inverse problems are constructed explicitly, as sums of orthogonal series.

Détail

Keywords: equation of mixed parabolic-hyperbolic type, initial boundary value problem, inverse problems, uniqueness, series, small denominators, integral equations.
Mots-clés : existence

Bibliographie

[1] I.M. Gel'fand, “Some questions of analysis and differential equations”, Uspekhi Matem. Nauk, 14:3 (1959), 3–19 (in Russian) | Zbl

[2] Ya. S. Uflyand, “On oscillation propagation in compound electirc lines”, Inzhener.-Phis. Zhurn., 7:1 (1964), 89–92 (in Russian)

[3] I.T. Lozanovskaya, Ya.S. Uflyand, “A class of problems in mathematical physics with a mixed eigenvalue spectrum”, Sov. Phys., Dokl., 10 (1965), 918–919 | MR | Zbl

[4] T.D. Dzhuraev, A. Sopuev, M. Mamazhanov, Boundary value problems for equations of parabolic-hyperbolic type, Izdat. “Fan” Uzbek. SSR, Tashkent, 1986 (in Russian) | MR

[5] O.A. Ladyzhenskaya, L. Stupyalis, “On mixed type equations”, Vestnik LGU. Ser. Matem. Mekh. Astr., 19:4 (1965), 38–46 (in Russian) | Zbl

[6] L. Stupyalis, “Initial-boundary value problems for equations of mixed type”, Proc. Steklov Inst. Math., 127 (1975), 135–170 ; 24:8 (1988), 1379–1386 | MR | MR | Zbl

[7] N.Yu. Kapustin, “Tricomi problem for a parabolic-hyperbolic equation with degeneracy in the hyperbolic part. I”, Diff. Equat., 23:1 (1987), 51–55 ; N.Yu. Kapustin, “Tricomi problem for a parabolic-hyperbolic equation with degeneracy in the hyperbolic part. II”, Diff. Equat., 24:8 (1988), 898–903 | MR | MR | Zbl | Zbl

[8] N.Yu. Kapustin, E.I. Moiseev, “On the spectral problem from the theory of the parabolic-hyperbolic heat equation”, Dokl. Math., 55:1 (1997), 80–83 | MR | Zbl

[9] K.B. Sabitov, Direct and inverse problems for mixed parabolic-hyperbolic equations, Nauka, M., 2016 (in Russian)

[10] K.B. Sabitov, “Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation”, Math. Notes, 102:3 (2017), 378–395 | DOI | DOI | MR | Zbl

[11] K.B. Sabitov, L.Kh. Rakhmanova, “Initial-boundary value problem for an equation of mixed parabolic-hyperbolic type in a rectangular domain”, Diff. Equat., 44:9 (2008), 1218–1224 | DOI | MR | Zbl

[12] K.B. Sabitov, “Tricomi problem for a mixed parabolic-hyperbolic equation in a rectangular domain”, Math. Notes, 86:2 (2009), 249–254 | DOI | DOI | MR | Zbl

[13] K.B. Sabitov, “Initial-boundary value problem for a parabolic-hyperbolic equation with power-law degeneration on the type change line”, Diff. Equat., 47:10 (2011), 1490–1497 | DOI | MR | MR | Zbl

[14] K.B. Sabitov, S.N. Sidorov, “On a nonlocal problem for a degenerating parabolic-hyperbolic equation”, Diff. Equat., 50:3 (2014), 352–361 | DOI | DOI | MR | Zbl

[15] K.B. Sabitov, S.Sidorov, “Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type”, J. Math. Sci., 236:6 (2019), 603–640 | DOI

[16] K.B. Sabitov, E.M. Safin, “The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain”, Russian Math. Iz. VUZ, 54:4 (2010), 48–54 | DOI | MR | Zbl

[17] K.B. Sabitov, E.M. Safin, “The Inverse problem for an equation of mixed parabolic-hyperbolic type”, Math. Notes, 87:6 (2010), 880–889 | DOI | DOI | MR | Zbl

[18] K.B. Sabitov, S.N. Sidorov, “Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition”, Russian Math. (Iz. VUZ), 59:1 (2015), 39–50 | DOI | MR | Zbl

[19] M.M. Lavrent'ev, K.G. Reznitskaya, V.G. Yakhno, One-dimensional inverse problems of mathematical physics, Nauka, Novosibirsk, 1982 (in Russian) | MR

[20] V.G. Romanov, Inverse problems of mathematical physics, Nauka, M., 1984 (in Russian)

[21] V.G. Romanov, S.I. Kabanikhin, Inverse problems of geoelectricity, Nauka, M., 1991 (in Russian) | MR

[22] A.M. Denisov, Introduction in the theory of inverse problems, Moscow State Univ., M., 1994 (in Russian) | MR

[23] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker Inc., New York–Basel, 1999, 709 pp. ; Monograph and Textbooks in Pure and Applied Mathematics, 222, Marcel Dekker Inc., New York, 2000 | MR | MR

[24] Isakov V., Inverse problems for partial differential equations, Springer, New York, 2006, 358 pp. | MR | Zbl

[25] S.I. Kabanikhin, Inverse and ill-posed problems. Theory and applications, de Gruyter, Berlin, 2012 ; 23, No 11 | MR | Zbl | MR | Zbl

[26] A.I. Prilepko, V.V. Solov'ev, “Solvability theorems and Rothe's method for inverse problems for a parabolic equation. I”, Diff. Equat., 23:10 (1987), 1230–1237 ; A.I. Prilepko, V.V. Solov'ev, “Solvability theorems and Rothe's method for inverse problems for a parabolic equation. II”, Diff. Equat., 23:11 (1987), 1341–1349 | MR | MR | MR | Zbl | Zbl

[27] A.I. Prilepko, I.A. Vasin, “Uniqueness of solutions in some nonlinear inverse problems of hydrodynamics”, Diff. Equat., 26:1 (1990), 97–105 | MR | Zbl

[28] A.I. Prilepko, A.B. Kostin, “Estimation of the spectral radius of an operator and the solvability of inverse problems for evolution equations”, Math. Notes, 53:1 (1993), 63–66 | DOI | MR | MR | Zbl

[29] V.V. Solov'ev, “Finding the source and the coefficients of a parabolic equation in the multi-dimensional case”, Differ. Equat., 31:6 (1995), 992–1001 | MR | Zbl

[30] V.V. Solov'ev, “Global existence of a solution to the inverse problem of determining the source term in a quasilinear equation of parabolic type”, Diff. Equat., 32:4 (1996), 538–547 | MR | Zbl

[31] A.B. Kostin, “The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation”, Sb. Math., 204:10 (2013), 1391–1434 | DOI | DOI | MR | Zbl

[32] Kozhanov A. I., “A nonlinear loaded parabolic equation and a related inverse problem”, Math. Notes, 76:6 (2004), 784–795 | DOI | MR | Zbl

[33] Kozhanov A. I., Safiullova R. R., “Linear inverse problems for parabolic and hyperbolic equations”, J. Inverse Ill-Posed Probl., 18:1 (2010), 1–18 | DOI | MR | Zbl

[34] V.I. Arnol'd, “Small denominators. I. Mapping the circle onto itself”, Izv. RAN. Ser. Matem., 25:1 (1961), 21–86 (in Russian)

[35] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher transcendental functions, v. 2, Robert E. Krieger Publ. Co., Malabar, Florida, 1981 | MR

[36] K.B. Sabitov, Functional, differential and integral equations, Vysshaya Schkola, M., 2005 (in Russian)