On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition
The paper deals with the blow-up behavior of positive solutions to the following initial-boundary value problem \[ \begin{cases} u_t=\Delta u^m, &x \in {\mathbb R}^N_+, t>0,\\ -{{\partial u^m}\over {\partial x_1}}=u^p, &x_1=0, t>0,\\ u(x,0)=u_0(x), &x\in {\mathbb R}^N_+, \end{cases} \tag{*} \] where \({\mathbb R}^N_+=\{(x_1,x')\colon\;x'\in {\mathbb R}^{N-1},\;x_1>0\},\) \(m>1,\) \(p>0\) and \(u_0(x)\) is a nonnegative and bounded function locally supported near some point of \({\mathbb R}^N_+\) and satisfying the compatibility conditions \[ -{{\partial u_0^m(x)}\over {\partial x_1}}=u_0^p(x)\quad \text{ as} x_1=0. \] A number \(p_0\) is called the critical global existence exponent when \(\forall\;p>p_0\) there exist nonglobal (in time) solutions to \((*)\), while, if \(p\in (0,p_0)\) then every solution to \((*)\) is global. Further, \(p_c\) is called the critical Fujita exponent if for \(p\in (p_0,p_c)\) any nontrivial solution to \((*)\) blows up in a finite time while, for \(p>p_c\), there exist globally small-initial-data solutions but the large-initial-data ones blow-up. The one-dimensional case \((N=1)\) of \((*)\) was studied by \textit{V. A.~Galaktionov} and \textit{H. A.~Levine} [Isr. J. Math. 94, 125-146 (1996; Zbl 0851.35067)] who proved \(p_0=(m+1)/2,\) \(p_c=m+1.\) Adopting Galaktionov-Levine's super- sub-solutions approach, the authors show \(p_0=(m+1)/2,\) \(p_c=m+1/N.\)
scientific article; zbMATH DE number 2006332
scientific article