On a Second Order Linear Parabolic Equation with Variable Coefficients in a Non-Regular Domain of R³

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Математика и физика. Mathematics & Physics. 2018 11 (4)
Boulkouane, Ferroudj; Kheloufi, Arezki
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Boulkouane, Ferroudj: Facult´e des science de la nature et de la vie Universit´e de Bejaia, 6000 B´ejaia Alg´erie; ; Kheloufim, Arezki: Faculty of Technology, Lab. of Applied Mathematics Bejaia University, 6000 B´ejaia Alg´eria
parabolic equations; non-regular domains; variable coefficients; anisotropic Sobolev spaces

This paper is devoted to the study of the following variable-coefficient parabolic equation in non-divergence form @tu ����� Σ2 i=1ai(t;x 1; x2)@iiu +Σ2i=1bi(t; x1; x2)@iu + c(t; 1; x2)u = f(t; x1; x2); subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-regular domain of the form Q ={(t; x1) 2 R2 : 0 < t < T; φ1 (t) < x1 < φ2 (t)} ]0; b[ where φk; k = 1; 2 are "smooth" functions. One of the main issues of this work is that the domain can possibly be non-regular, for instance, the singular case where φ1 coincides with φ2 for t = 0 is allowed. The analysis is performed in the framework of anisotropic Sobolev spaces by using the domain decomposition method. This work is an extension of the constant-coefficients case studied in [15].

Paper at repository of SibFU