On asymptotically exact local estimates for compactly-supported solutions to a nonlinear parabolic equation with absorption
The aim of the article is to prove exact local estimates for a solution \(u\) to the Cauchy problem \[ u_t- (u^nu_x)_x + u^\beta=0, \quad n \geq 0,\;\beta>0, \quad u(x,0)= u_0(x), \] \(u_0(x)= 0\), \(|x|\geq l>0\); \(u_0(x) >0\), \(x\in (-l,l)\) and to describe the behaviour of \(\eta (t) = \sup \{x:u(x,t) > 0\}\) in the case of backward front movement (BFM). The main result is the following theorem: Let \(u(x,t)\) be a solution to the Cauchy problem with \(0< \beta <1\) and \(n\geq 1- \beta\). Suppose that there is an \(l_1<l\) such that \(u_0'(x) <0\) for \(x \in (l_1,l)\), \(u_0 \in C^2(l_1,l)\), and \((u^\gamma_0)_{|x=l} '=0\), \((u^\gamma_0)'' =O(1)\), \(x \to l -0\), for \(\gamma= 1- \beta\). Then we have BFM from the point \(x=l\) and for every sufficiently small \(\varepsilon> 0\) there exists \(\delta> 0\) such that \(\eta (t)\) satisfies the local estimates \[ u_0^{-1} \biggl(1/(1- \varepsilon) \bigl((1- \beta+ \varepsilon) t \bigr)^{1/(1-\beta)} \biggr) \leq\eta (t)\leq u_0^{-1} \biggl(1/(1 + \varepsilon) \bigl((1- \beta- \varepsilon) t \bigr)^{1/(1- \beta)} \biggr),\;t\in [0,\delta], \] where \(u_0^{-1}\) is the inverse function of \(u_0\) for \(x\in (l_1, l)\).
scientific article; zbMATH DE number 923554