Cauchy Problem Solvability with the Data Specified on the Rectangle Boundary for a One-Dimensional Parabolic Equation
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inverse problem
Cauchy data
initial boundary problem
heat and mass transfer

How to Cite

Pyatkov S.G. Cauchy Problem Solvability with the Data Specified on the Rectangle Boundary for a One-Dimensional Parabolic Equation // Russian Journal of Cybernetics. 2022. Vol. 3, № 2. P. 40-46. DOI: 10.51790/2712-9942-2022-3-2-6.


We consider a one-dimensional parabolic equation in the (0,T) × (a,b) rectangle. The Cauchy data are specified on one of its lateral sides. The Cauchy condition is also specified at the initial moment. The solution to this problem is sought in the Sobolev space. A data class was constructed for which there is a unique solution to the Cauchy problem with the Cauchy data specified on the lateral side of the rectangle. The class is minimal, i. e., the smoothness conditions applied to the data cannot be weakened. They are necessary and sufficient for the existence of solutions in a given Sobolev class. The solution is regular which means that all derivatives in the equation belong to L2 space. The problem is ill-posed in the Hadamard sense. Mathematical models of this type arise when describing heat and mass transfer. There are many papers on such problems for both one-dimensional and multidimensional cases. The available sources mostly focus on numerical solutions, since the problem is found in many applications. Besides, there are uniqueness theorems and stability estimates, and existence theorems for solutions in Holmgren’s classes. We slightly refined the recent results and obtained an existence theorem for solutions in finite smoothness classes.
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