Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement

Abstract A semilinear parabolic problem of second order with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe’s method, derive a priori estimates and prove convergence of iterates towards the exact solution.

[1]  Davide Guidetti,et al.  Convergence to a stationary state for solutions to parabolic inverse problems of reconstruction of convolution kernels , 2007, Differential and Integral Equations.

[2]  C. Simader,et al.  Direct methods in the theory of elliptic equations , 2012 .

[3]  Michael V. Klibanov,et al.  Global convexity in a three-dimensional inverse acoustic problem , 1997 .

[4]  L. A. Li︠u︡sternik,et al.  Elements of Functional Analysis , 1962 .

[5]  Jozef Kačur,et al.  Method of rothe in evolution equations , 1986 .

[6]  Michael Renardy,et al.  Mathematical problems in viscoelasticity , 1987 .

[7]  Y. Y. Belov,et al.  Inverse Problems for Partial Differential Equations , 2002 .

[8]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[9]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[10]  Ronald E. Miller.  Optimization: Foundations and Applications , 1999 .

[11]  Pablo Pedregal.  Introduction to Optimization , 2003 .

[12]  Davide Guidetti,et al.  A GLOBAL IN TIME EXISTENCE AND UNIQUENESS RESULT FOR A SEMILINEAR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEM IN SOBOLEV SPACES , 2007 .

[13]  Karel Rektorys,et al.  The method of discretization in time and partial differential equations , 1982 .

[14]  W. Rudin.  Real and complex analysis , 1968 .

[15]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[16]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[17]  Vincenzo Vespri,et al.  Some global in time results for integro-differential parabolic inverse problems , 2006 .

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  F. Colombo,et al.  Some Results on the Identification of Memory Kernels , 2011 .

[20]  Fabrizio Colombo,et al.  On applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type , 2019, Evolution Equations.

[21]  D. Bainov,et al.  Integral Inequalities and Applications , 1992 .

[22]  Michael V. Klibanov,et al.  Globally strongly convex cost functional for a coefficient inverse problem , 2013, 1312.2886.

[23]  J. Delleur.  The handbook of groundwater engineering , 1998 .

[24]  M. I. Ismailov,et al.  Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions , 2011, Appl. Math. Comput..

[25]  A. I. Prilepko,et al.  Methods for solving inverse problems in mathematical physics , 2000 .