Electron. J. Differential Equations, Vol. 2021 (2021), No. 02, pp. 1-16.

Solutions to viscous Burgers equations with time dependent source term

Satyanarayana Engu, Manas R. Sahoo, Venkatramana P. Berke

We study the existence and uniqueness of weak solutions for a Cauchy problem of a viscous Burgers equation with a time dependent reaction term involving Dirac measure. After applying a Hopf like transformation, we investigate the associated two initial boundary value problems by assuming a common boundary. The existence of the boundary data is shown with the help of Abel's integral equation. We then derive explicit representation of the boundary function. Also, we prove that the solutions of associated initial boundary value problems converge uniformly to a nonzero constant on compact sets as t approaches infinity.

Submitted January 19, 2019. Published January 7, 2021.
Math Subject Classifications: 35C15, 35K05, 35K20, 35B09, 35B40.
Key Words: Abel integral equation; Hopf transformation; heat equation; large time asymptotic; weak solutions.
DOI: https://doi.org/10.58997/ejde.2021.02

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Satyanarayana Engu
Department of Mathematics
National Institute of Technology, Warangal
Telangana-506004, India
email: satya@nitw.ac.in
Manas R. Sahoo
School of Mathematical Sciences
National Institute of Science Education and Research
HBNI, Jatni, Khurda, Bhubaneswar 752050, India
email: manas@niser.ac.in
Venkatramana P. Berke
Department of Mathematical and Computational Sciences
National Institute of Technology Karnataka, Surathkal
Shrinivas Nagar, Mangalore-575025, India
email: venkat.nitk19@gmail.com

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