Singular limit of the generalized Burgers equation with absorption

We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $Rtimes (0,infty)$, $u(x,0)=u_0(x)ge 0$ in $R$, as $mtoinfty$ for any $p>1$ and $u_0in L^1(R)cap L^{infty}(R)$ or as $ptoinfty$ for any $m>1$ and $u_0in L^{infty}(R)$ . We also show that in general $underset{ptoinfty}limunderset{mtoinfty}lim u_{m,p} eunderset{mtoinfty}limunderset{ptoinfty}lim u_{m,p}$.