Singular Limit of the Generalized Burgers Equation with Absorption

  • Kin Ming Hui
  • Sunghoon KimEmail author


We prove the convergence of the solutions \(u_{m,p}\) of the equation \(u_t+(u^m)_x=-\,u^p\) in \({{\mathbb {R}}}\times (0,\infty )\), \(u(x,0)=u_0(x)\ge 0\) in \({{\mathbb {R}}}\), as \(m\rightarrow \infty \) for any \(p>1\) and \(u_0\in L^1({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})\) or as \(p\rightarrow \infty \) for any \(m>1\) and \(u_0\in L^{\infty }({{\mathbb {R}}})\). We also show that in general \(\underset{p\rightarrow \infty }{\lim }\underset{m\rightarrow \infty }{\lim }u_{m,p}\ne \underset{m\rightarrow \infty }{\lim }\underset{p\rightarrow \infty }{\lim }u_{m,p}\).


Singular limit Generalized Burgers equation Generalized Burgers equation with absorption Kruzkov sense solution Non-commutativity of singular limit 

Mathematics Subject Classification

35B40 35F20 35L02 



Sunghoon Kim was supported by the Research Fund, 2020 of The Catholic University of Korea.


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Authors and Affiliations

  1. 1.Institute of MathematicsAcademia SinicaTaipeiTaiwan, R.O.C.
  2. 2.Department of Mathematics, School of Natural SciencesThe Catholic University of KoreaBucheon-siRepublic of Korea

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