On determination of a time-dependent leading coefficient in a parabolic equation
type: Item
instance of: scholarly article
title: On determination of a time-dependent leading coefficient in a parabolic equation
zbMATH Open document ID: 0903.35086
DOI: 10.1007/BF02673901
author: N. I. Ivanchov
published in: Siberian Mathematical Journal
publication date: 1998-11-22T00:00:00Z
review text: The author considers the equation \[ u_t=a(t)u_{xx} + b(x,t)u_x +c(x,t)u + f(x,t),\quad (x,t)\in Q_{T}=(0,h)\times (0,T). \] Two inverse problems are studied. The first is to find a function \(a(t)\) and a solution to this equation that satisfies the conditions of the first initial-boundary value problem, \[ u(x,0)=\varphi(x), \;x\in [0,h],\quad u(0,t)=\mu_1(t),\;t\in [0,T], \] \[ u(h,t)=\mu_2(t), \;t\in [0,T]\tag{1} \] and the additional condition \[ a(t)u_x(0,t)=\mu_3(t), \;t\in [0,T]. \tag{2} \] The second problem differs from the first by the boundary conditions (1) and (2) which are replaced by the conditions \[ u_x(h,t)=\mu_2(t),\;\;u_x(0,t)=\mu_3(t),\;\;t\in [0,T]. \] Under appropriate conditions on the coefficients and the data of the problems, it is established that there exists a unique local (in time) solution \((a(t),u(x,t))\) to these problems which belongs to the Hölder class \(H^{1+\alpha/2}[0,t_0]\times H^{2+\alpha,1+\alpha/2}(\overline{Q_{t_0}})\). As a whole, the author's results are similar to the known results for the case \(b\equiv c\equiv 0\).
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1225829
zbMATH Keywords: one-dimensional parabolic equation
zbMATH Keywords: first initial-boundary value problem
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
cites work: Inverse problems of the determination of the coefficient in parabolic equations. I
cites work: On inverse problems of determining a coefficient in a parabolic equation. II
cites work: Q5719594
cites work: Determination of a time-dependent coefficient in a quasi-linear parabolic equation
cites work: On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity
cites work: Q5548525
cites work: Q5560475
cites work: Q5332414
description: scientific article; zbMATH DE number 1225829