Localization of solutions to a doubly degenerate parabolic equation with a strongly nonlinear source. (English) Zbl 1252.35167
The authors study the localization of solutions to the Cauchy problem for a doubly degenerate parabolic equation with a strongly nonlinear source
\[
u_t=\text{div\,}(|\nabla u^m|^{p-2}\nabla u^l)+u^q,\quad (x,t)\in \mathbb R^N\times (0,T),
\]
where \(N \geq 1,\) \(p > 2\) and \(m, l, q > 1.\) In the case when \(q > l + m(p - 2),\) it is proved that the solution \(u(x, t)\) has strict localization if the initial data \(u_0(x)\) has a compact support, while \(u(x, t)\) has the property of effective localization if the initial data possesses radially symmetric decay. Moreover, when \(1 < q < l + m(p - 2),\) it turns out that the solution of the Cauchy problem blows up at any point of \(\mathbb R^N\) for arbitrary initial data with compact support.
Reviewer: Dian K. Palagachev (Bari)
MSC:
35K65 | Degenerate parabolic equations |
35B40 | Asymptotic behavior of solutions to PDEs |
35K59 | Quasilinear parabolic equations |
35B44 | Blow-up in context of PDEs |
35K15 | Initial value problems for second-order parabolic equations |
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