Blow up behavior for a nonlinear parabolic equation of nondivergence form
The following Cauchy problem for a nonlinear parabolic equation is considered: \(u_t=u\left(u_{xx}+u^p\right),\,x\in\mathbb R,\,t>0;\; u(x,0)=u_0(x),\,x\in\mathbb R\), where \(p>1\) and \(u_0(x)>0\). To investigate the blow up behavior near the point \(t=T\), the variables \(y=x/(T-t)^\beta\), \(e^{-s}=T-t\), \(z(y,s) =(T-t)^\alpha u(x,t)\), where \(\alpha=1/p\), \(\beta=(p- 1)/2p\) are introduced. Then the above problem can be written in the form \[ \begin{gathered} z_s+\alpha z+\beta z_y=z\left(z_{yy}+z^p\right),\,y\in\mathbb R,\,s>s_0\equiv-\ln T,\\ z(y,s_0)=z_0(y)\equiv T^\alpha u_0(T^\beta y), \,y\in\mathbb R. \end{gathered}\tag{*} \] The set \[ \omega(z_0)=\left \{\varphi\in C^2(\mathbb R)\,| \,\exists s_j\to\infty, z(\cdot, s_j)\underset{K}\rightrightarrows \varphi(\cdot)\text{ as }{j}\infty\right\}, \] where \(K\) is a compact set in \(\mathbb R\) is called a \(\omega\)-limit set of the problem (*). In Theorem 1.1 under some assumptions on \(u_0(x)\) it is proved that the \(\omega\)-limit set of (*) is not empty and any \(\omega\)-limit is a solution of the equation \(\varphi_{yy}-\beta y\varphi^{-1} \varphi_y+ \varphi^p- \alpha=0\) such that \(\varphi_y \leq0\) and \(\varphi_{yy}+\varphi^p \geq0\) for \(y\in[0, \infty)\). A Lyapunov function is constructed. Theorem 1.2 states that, if moreover \(\lim_{x\to \infty}\inf x^{\alpha/\beta}u_0(x) =+\infty\), then the \(\omega\)-limit \(\varphi\) corresponding to \(u_0(x)\) is unique, and \(z(y,s)\) converges to \(\varphi(y)\) uniformly on any compact set as \(s\to\infty\).
scientific article; zbMATH DE number 2167834