Electron. J. Diff. Equ., Vol. 2016 (2016), No. 157, pp. 1-13.

Existence of solutions to Burgers equations in domains that can be transformed into rectangles

Yassine Benia, Boubaker-Khaled Sadallah

Abstract:
This work is concerned with Burgers equation $\partial _{t}u+u\partial_x u-\partial _x^2u=f$ (with Dirichlet boundary conditions) in the non rectangular domain $\Omega =\{(t,x)\in R^2;\ 0<t<T,\; \varphi_1(t)<x<\varphi _2(t)\}$ (where $\varphi _1(t)<\varphi _2(t)$ for all $t\in [ 0;T]$). This domain will be transformed into a rectangle by a regular change of variables. The right-hand side lies in the Lebesgue space $L^2(\Omega )$, and the initial condition is in the usual Sobolev space $H_0^{1}$. Our goal is to establish the existence, uniqueness and the optimal regularity of the solution in the anisotropic Sobolev space.

Submitted April 15, 2016. Published June 21, 2016.
Math Subject Classifications: 35K58, 35Q35.
Key Words: Semilinear parabolic problem; Burgers equation; existence; uniqueness; anisotropic Sobolev space.

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Yassine Benia
Department of Mathematics
University of Tiaret, B.P. 78
14000, Tiaret, Algeria
email: benia.yacine@yahoo.fr
Boubaker-Khaled Sadallah
Lab. PDE and Hist Maths
Dept of Mathematics, E.N.S.
16050, Kouba, Algiers, Algeria
email: sadallah@ens-kouba.dz

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