×

Single-point blow-up for a doubly degenerate parabolic equation with nonlinear source. (English) Zbl 1223.35089

The authors consider positive solutions of the Cauchy problem for the doubly degenerate equation
\[ u_t-\text{div}(|\nabla u^m|^\sigma\nabla u^m)=u^\beta, \quad \text{for } x\in\mathbb R^N,\;t>0, \]
where \(\sigma>0\), \(m>1\), \(\beta>m(1+\sigma)\), \(N\geq1\). The authors study the set of blow-up points and the behavior of \(u\) at the blow-up point. They prove single-point blow-up for a large class of radial decreasing solutions. The upper and lower estimates of the blow-up solution near the single blow-up point are also obtained.

MSC:

35B44 Blow-up in context of PDEs
35K65 Degenerate parabolic equations
35B09 Positive solutions to PDEs
35K59 Quasilinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI