Bulletin of the Iranian Mathematical Society https://doi.org/10.1007/s41980-018-0115-9 ORIGINAL PAPER Quasi Solution of a Nonlinear Inverse Parabolic Problem Amir Hossein Salehi Shayegan 1 · Ali Zakeri 1 · Touraj Nikazad 2 Received: 1 August 2017 / Accepted: 24 January 2018 © Iranian Mathematical Society 2018 Abstract In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to ℵ(u ) :≡ u t −∇· ( F (x , ∇u )) where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function F . At the first step of the methodology, we give a stability result corresponding to connectivity of F and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions. Keywords Quasi solution · Nonlinear inverse parabolic problem · Time-dependent problems Mathematics Subject Classification Primary 35K55; Secondary 35R30 · 49J20 1 Introduction We consider a nonlinear parabolic problem, which is an inverse problem of determining the unknown function F , as follows Communicated by Davod Khojasteh Salkuyeh. B Touraj Nikazad tnikazad@iust.ac.ir Amir Hossein Salehi Shayegan ah.salehi@mail.kntu.ac.ir Ali Zakeri azakeri@kntu.ac.ir 1 Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran 2 School of Mathematics, Iran University of Science and Technology, Tehran 16846-13114, Iran 123