Inverse problem for semilinear parabolic equation

M. I. Ivanchov

In this paper we consider the problem of finding the pair $ (a(t), u(x,t)) $ from the equation $u_t=a(t)u_{xx}+b(x,t,u,u_x), $ an initial condition, Dirichlet's boundary condition and given heat flux as overdetermination condition. Existence and uniqueness of a smooth solution are proved.

1. N.V. Muzylyov, Theorems of uniqueness for some inverse problems of heat conduction, J. Differential Equations, 20, No 2 (1980), pp.388-400 (in Russian).

2. N.V. Muzylyov, On uniqueness of simultaneous determination of coefficients of conductivity and volume heat capacity, U.S.S.R. Comput. Math.and Math. Phys., 23, No 1 (1983), pp.102-108 (in Russian).

3. N.V. Muzylyov, On uniqueness of solution of an inverse problem for nonlinear heat conduction, U.S.S.R. Comput. Math. and Math. Phys., 25, No 9 (1985), pp.1346-1352 (in Russian).

4. N.V. Muzylyov, On uniqueness of simultaneous determination of coefficient in a quasilinear parabolic equation and coefficient in a boundary condition, J. Differential Equations, 27, No 12 (1991), pp.2124-2128 (in Russian).

5. A.Lorenzi, Determination of a time-dependent coefficient in a quasilinear parabolic equation, Ricerche Mat., 32, No 2 (1983), pp. 263-284.

6. A.Lorenzi, Identification of the thermal conductivity in the nonlinear heat equation, Inverse Problems, 3 (1987), pp. 437-451.

7. J.R.Cannon, Y.Lin, Determination of a parameter p(t) in some quasilinear parabolic differential equations, Inverse Problems, 4 (1988), pp. 35-45.

8. J.R.Cannon, Y.Lin, Determination of a parameter p(t) in a Hölder class for some semilinear parabolic equations, Inverse Problems, 4 (1988), pp. 596-605.

9. J.R.Cannon, Y.Lin, An inverse problem of finding a parameter in a semilinear heat equation, J. Math. Anal. Appl., 145 (1990), pp. 470- 484.

10. Y.Lin, An inverse problem for a class of quasilinear parabolic equations, SIAM J. Math. Anal., 22, No 1 (1991), pp. 146-156.

11. O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural'ceva, Linear and quasilinear equations of parabolic type, Moscow, Nauka (1967).

12. M.Ivanchov, Inverse problems for equations of parabolic type, VNTL Publishers (2003).

13. N.I.Ivanchov, N.V.Pabyrivska On the determination of two time-dependent coefficients in a parabolic equation, Siberian Math. J., 43, No 2 (2002), pp. 406-413.

14. N.I.Ivanchov, On the determination of time-dependent major coefficient in a parabolic equation, Siberian Math. J., 39, No 3 (1998), pp. 539-550.

	Inverse problem for semilinear parabolic equation
Author	M., I., Ivanchov ivanchov@franko.lviv.ua Department of Mechanics and Mathematics, Ivan Franko,National University of Lviv,1 Universytetska st., 79602 Lviv, Ukraine
Abstract	In this paper we consider the problem of finding the pair $ (a(t), u(x,t)) $ from the equation $u_t=a(t)u_{xx}+b(x,t,u,u_x), $ an initial condition, Dirichlet's boundary condition and given heat flux as overdetermination condition. Existence and uniqueness of a smooth solution are proved.
Keywords	inverse problem, semilinear parabolic equation, smooth solution
DOI	doi:10.30970/ms.29.2.181-191
Reference	1. N.V. Muzylyov, Theorems of uniqueness for some inverse problems of heat conduction, J. Differential Equations, 20, No 2 (1980), pp.388-400 (in Russian). 2. N.V. Muzylyov, On uniqueness of simultaneous determination of coefficients of conductivity and volume heat capacity, U.S.S.R. Comput. Math.and Math. Phys., 23, No 1 (1983), pp.102-108 (in Russian). 3. N.V. Muzylyov, On uniqueness of solution of an inverse problem for nonlinear heat conduction, U.S.S.R. Comput. Math. and Math. Phys., 25, No 9 (1985), pp.1346-1352 (in Russian). 4. N.V. Muzylyov, On uniqueness of simultaneous determination of coefficient in a quasilinear parabolic equation and coefficient in a boundary condition, J. Differential Equations, 27, No 12 (1991), pp.2124-2128 (in Russian). 5. A.Lorenzi, Determination of a time-dependent coefficient in a quasilinear parabolic equation, Ricerche Mat., 32, No 2 (1983), pp. 263-284. 6. A.Lorenzi, Identification of the thermal conductivity in the nonlinear heat equation, Inverse Problems, 3 (1987), pp. 437-451. 7. J.R.Cannon, Y.Lin, Determination of a parameter p(t) in some quasilinear parabolic differential equations, Inverse Problems, 4 (1988), pp. 35-45. 8. J.R.Cannon, Y.Lin, Determination of a parameter p(t) in a Hölder class for some semilinear parabolic equations, Inverse Problems, 4 (1988), pp. 596-605. 9. J.R.Cannon, Y.Lin, An inverse problem of finding a parameter in a semilinear heat equation, J. Math. Anal. Appl., 145 (1990), pp. 470- 484. 10. Y.Lin, An inverse problem for a class of quasilinear parabolic equations, SIAM J. Math. Anal., 22, No 1 (1991), pp. 146-156. 11. O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural'ceva, Linear and quasilinear equations of parabolic type, Moscow, Nauka (1967). 12. M.Ivanchov, Inverse problems for equations of parabolic type, VNTL Publishers (2003). 13. N.I.Ivanchov, N.V.Pabyrivska On the determination of two time-dependent coefficients in a parabolic equation, Siberian Math. J., 43, No 2 (2002), pp. 406-413. 14. N.I.Ivanchov, On the determination of time-dependent major coefficient in a parabolic equation, Siberian Math. J., 39, No 3 (1998), pp. 539-550.
Pages	181-191
Volume	29
Issue	2
Year	2008
Journal	Matematychni Studii
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