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Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. (English) Zbl 1219.35367

Summary: We consider initial value/boundary value problems for fractional diffusion-wave equation: \(\partial_t^{\alpha}u(x,t)=Lu(x,t)\), where \(0<\alpha \leqslant 2\), where \(L\) is a symmetric uniformly elliptic operator with \(t\)-independent smooth coefficients. First, we establish the unique existence of the weak solution and the asymptotic behavior as the time \(t\) goes to \(\infty \) and the proofs are based on the eigenfunction expansions. Second, for \(\alpha \in (0,1)\), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time; (ii) the uniqueness in determining an initial value; and (iii) the uniqueness of solution by the decay rate as \(t\rightarrow \infty \); (iv) stability in an inverse source problem of determining \(t\)-dependent factor in the source by observation at one point over (\(0,T\)).

MSC:

35R30 Inverse problems for PDEs
35R11 Fractional partial differential equations
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[1] Adams, E. E.; Gelhar, L. W., Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis, Water Resources Res., 28, 3293-3307 (1992)
[2] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[3] Agarwal, O. P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam., 29, 145-155 (2002) · Zbl 1009.65085
[4] Brezis, H., Analyse Fonctionnelle (1983), Masson: Masson Paris · Zbl 0511.46001
[5] Cannon, J. R.; Esteva, S. P., An inverse problem for the heat equation, Inverse Problems, 2, 395-403 (1986) · Zbl 0624.35078
[6] Cheng, J.; Nakagawa, J.; Yamamoto, M.; Yamazaki, T., Uniqueness in an inverse problem for one-dimensional fractional diffusion equation, Inverse Problems, 25, 115002 (2009) · Zbl 1181.35322
[7] Clément, P.; Londen, S.-O.; Simonett, G., Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Equations, 196, 418-447 (2004) · Zbl 1058.35136
[8] Courant, R.; Hilbert, D., Methods of Mathematical Physics, vol. 1 (1953), Interscience: Interscience New York
[9] Eidelman, S. D.; Kochubei, A. N., Cauchy problem for fractional diffusion equations, J. Differential Equations, 199, 211-255 (2004) · Zbl 1129.35427
[10] Fujita, Y., Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27, 309-321 (1990), 797-804 · Zbl 0790.45009
[11] Gejji, V. D.; Jafari, H., Boundary value problems for fractional diffusion-wave equation, Aust. J. Math. Anal. Appl., 3, 1-8 (2006) · Zbl 1093.35041
[12] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001
[13] Ginoa, M.; Cerbelli, S.; Roman, H. E., Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A, 191, 449-453 (1992)
[14] Gorenflo, R.; Mainardi, F., Fractional calculus: Integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag New York), 223-276 · Zbl 1438.26010
[15] Hatano, Y.; Hatano, N., Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resources Res., 34, 1027-1033 (1998)
[16] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag: Springer-Verlag Berlin · Zbl 0456.35001
[17] Isakov, V., Inverse Problems for Partial Differential Equations (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1092.35001
[18] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[19] Kochubei, A. N., A Cauchy problem for evolution equations of fractional order, J. Differential Equations, 25, 967-974 (1989) · Zbl 0696.34047
[20] Kochubei, A. N., Fractional order diffusion, J. Differential Equations, 26, 485-492 (1990) · Zbl 0729.35064
[21] Lions, J. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications, vols. I, II (1972), Springer-Verlag: Springer-Verlag Berlin · Zbl 0227.35001
[22] Luchko, Y., Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351, 218-223 (2009) · Zbl 1172.35341
[23] Y. Luchko, Initial-boundary value problems for the generalized time-fractional diffusion equation, in: Proceedings of 3rd IFAC Workshop on Fractional Differentiation and Its Applications (FDA08), Ankara, Turkey, 05-07 November 2008.; Y. Luchko, Initial-boundary value problems for the generalized time-fractional diffusion equation, in: Proceedings of 3rd IFAC Workshop on Fractional Differentiation and Its Applications (FDA08), Ankara, Turkey, 05-07 November 2008.
[24] Luchko, Y., Some uniqueness and existence results for the initial-boundary value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59, 1766-1772 (2010) · Zbl 1189.35360
[25] Mainardi, F., On the initial value problem for the fractional diffusion-wave equation, (Rionero, S.; Ruggeri, T., Waves and Stability in Continuous Media (1994), World Scientific: World Scientific Singapore), 246-251
[26] Mainardi, F., The time fractional diffusion-wave equation, Radiophys. and Quantum Electronics, 38, 13-24 (1995)
[27] Mainardi, F., Fractional diffusive waves in viscoelastic solids, (Wegner, J. L.; Norwood, F. R., Nonlinear Waves in Solids (1995), ASME/AMR: ASME/AMR Fairfield), 93-97
[28] Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9, 23-28 (1996) · Zbl 0879.35036
[29] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag New York), 291-348 · Zbl 0917.73004
[30] Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Phys. A, 278, 107-125 (2000)
[31] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[32] Miller, K. S.; Samko, S. G., Completely monotonic functions, Integral Transforms Spec. Funct., 12, 389-402 (2001) · Zbl 1035.26012
[33] Nakagiri, S., Identifiability of linear systems in Hilbert spaces, SIAM J. Control Optim., 21, 501-530 (1983)
[34] Nigmatullin, R. R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133, 425-430 (1986)
[35] Oldham, K. B.; Spanier, J., The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (1974), Academic Press: Academic Press New York · Zbl 0292.26011
[36] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0516.47023
[37] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[38] Pollard, H., The completely monotonic character of the Mittag-Leffler function \(E_\alpha(- x)\), Bull. Amer. Math. Soc., 54, 115-116 (1948)
[39] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., Integrals and Series, vol. I (1990), Gordon & Breach Science Publishers: Gordon & Breach Science Publishers New York · Zbl 0967.00503
[40] Prüss, J., Evolutionary Integral Equations and Applications (1993), Birkhäuser: Birkhäuser Basel · Zbl 0793.45014
[41] Roman, H. E.; Alemany, P. A., Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27, 3407-3410 (1994) · Zbl 0827.60057
[42] Saitoh, S.; Tuan, V. K.; Yamamoto, M., Reverse convolution inequalities and applications to inverse heat source problems, JIPAM. J. Inequal. Pure Appl. Math., 3, 5 (2002), Article 80 · Zbl 1029.44002
[43] Saitoh, S.; Tuan, V. K.; Yamamoto, M., Convolution inequalities and applications, JIPAM. J. Inequal. Pure Appl. Math., 4, 3 (2003), Article 50 · Zbl 1058.44004
[44] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Philadelphia · Zbl 0818.26003
[45] Schmidt, E. J.P. Georg; Weck, N., On the boundary behavior of solutions to elliptic and parabolic equations - with applications to boundary control for parabolic equations, SIAM J. Control Optim., 16, 593-598 (1978) · Zbl 0388.93027
[46] Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004
[47] Schneider, W. R., Completely monotone generalized Mittag-Leffler functions, Expo. Math., 14, 3-16 (1996) · Zbl 0843.60024
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