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Register a new userSingular limit of the generalized Burgers equation with absorption
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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 $underset{ptoinfty}limunderset{mtoinfty}lim u_{m,p} eunderset{mtoinfty}limunderset{ptoinfty}lim u_{m,p}$.
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Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation begin{eqnarray*} u_t+u_{xxx}+epsilon |partial_x|^{2alpha}u+(u^2)_x=0, u(0)=phi, end{eqnarray*} where $0<epsilon,alphaleq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s (s>s_alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $epsilon in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $epsilon$ tends to 0.
We study the problem of global exponential stabilization of original Burgers equations and the Burgers equation with nonlocal nonlinearities by controllers depending on finitely many parameters. It is shown that solutions of the controlled equations are steering a concrete solution of the non-controlled system as $trightarrow infty$ with an exponential rate.
Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for these equations with smooth initial conditions.
In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $ u_t - Delta u + |u|^alpha u =0$, where $u=u(t,x)in {mathbb R}, $ $(t,x)in (0,infty)times{mathbb R}^N$ and $alpha>0$. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables $x_1,; x_2,; cdots,; x_m$ for some $min {1,2, cdots, N}$, such as $u_0 = (-1)^mpartial_1partial_2 cdots partial_m|cdot|^{-gamma} in {{mathcal S}({mathbb R}^N)}$, $0 < gamma < N$. In fact, we show global well-posedness for initial data bounded in an appropriate sense by $u_0$, for any $alpha>0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $Omega_m = {x in {{mathbb R}^N} : x_1, cdots, x_m > 0}$, and then to extend the results by reflection to solutions on ${{mathbb R}^N}$ which are antisymmetric. We show that the large time behavior depends on the relationship between $alpha$ and $2/(gamma+m)$, and we consider all three cases, $alpha$ equal to, greater than, and less than $2/(gamma+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.
The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d $le$ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.
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