On linear and nonlinear heat equations in degenerating domains

Jenaliyev, Muvasharkhan; Ramazanov, M.I.; Yergaliyev, M.G.

doi:10.1063/1.5013968

Cited by 1 publication

(2 citation statements)

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“…It is known that by the Hopf-Cole transformation [4,13] the Burgers equation can be reduced to the heat equation [11,25,28]. Theorems on the existence, depending on the initial and boundary conditions, of a unique and non-unique solution to the inverse problem for the one-dimensional Burgers equation in a rectangular domain were proved in [11].…”

Section: Introductionmentioning

confidence: 99%

“…(2) but already in the angular domain were obtained in [25], and it was shown that boundary value problem (2) in the corresponding weight Lebesgue class, where the weight is determined by the nature of the degeneracy of the domain, along with the trivial solution, has a nontrivial solution. In [28] the results obtained in [25] were extended to the case of inhomogeneous boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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On an Inverse Problem With an Integral Overdetermination Condition for the Burgers Equation

Yergaliyev

Jenaliyev²,

Romankyzy³

et al. 2023

JMMCS

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In this paper we consider one inverse problem for the Burgers equation with an integral overdetermination and periodic boundary conditions in a domain that is trapezoid. Using an integral overdetermination, boundary and initial conditions, we reduce the inverse problem to the study of an already direct initial boundary value problem for the loaded Burgers equation. Next, we use a one-to-one transformation of independent variables to move from a trapezoid to a rectangular domain, where we study an auxiliary problem, for which the methods of Faedo-Galerkin, a priori estimates and functional analysis have been proved a theorem on its unique solvability in Sobolev classes. Note that the obtained a priori estimates are uniform with respect to the summation index of the approximate solution and do not depend on time. Further, on the basis of this theorem, due to the correspondence of spaces, theorems on the unique solvability of the original inverse problem are proved. Also, for the selected initial data, the paper presents graphs of the initial-boundary problem for the loaded Burgers equation and the desired function of the inverse problem, which together constitute the solution of the original inverse problem.

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