Blow-up solutions for a class of nonlinear parabolic equations with Dirichlet boundary conditions
There is studied the blow-up phenomenon of smooth local solutions to the following Cauchy-Dirichlet problem: \[ \begin{cases} u_t=\nabla(a(u)\nabla u)+f(x,u,q,t) &\text{ in } D\times(0,T),\;q\equiv|\nabla u|^2, \\ u=0 &\text{ on } \partial D\times (0,T),\;0<T<\infty, \\ u(x,0)=u_0(x)\geq 0 &\text{ in } \overline{D}, \end{cases} \] where \(D\subset{\mathbb R}^N\) is a smooth bounded domain, \(N\geq 2\). By constructing an auxiliary function and using Hopf's maximum principles on it, the sufficient condition for blow-up solutions are obtained and the upper bound of blow-up time is given under some suitable assumptions on \(a,\;f\) and initial date. But the proof of main result is not correct. Upper estimate for auxiliary function \(H\leq \beta\) does not follow from inequality \((2.18)\) and boundary conditions \((2.19),\;(2.20)\) if \(\beta>0\).
scientific article; zbMATH DE number 1880116