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Solutions to viscous Burgers equations with time dependent source term. (English) Zbl 1456.35077

Summary: 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 \(\infty\).

MSC:

35C15 Integral representations of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
35B09 Positive solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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