On properties of solutions to nonlinear parabolic equations of the second order
The author studies sufficient conditions for finite time blow-up of positive solutions of the equation \(u_t=Lu+a_0(x,t)u^{\sigma_1}\) in \(\Omega\times(0,T)\) satisfying the nonlinear boundary condition \({\partial u\over\partial\nu}=b_0(x,t)u^{\sigma_2}\). Here \(\Omega\) is a bounded domain in \(\mathbb R^n\), \(L\) is an elliptic operator of the form \(Lu=\sum_{i,j}(a_{ij}(x,t)u_{x_j})_{x_i}+\sum_ia_i(x,t)u_{x_i}\) with bounded measurable coefficients, \(\sigma_1,\sigma_2\geq 1\). Let \(u_0\) be the first (positive) eigenfunction of the adjoint problem to the Neumann problem for the operator \(L\). If \(\sigma_1=\sigma_2>1\) and either \(a_0,b_0\geq 0\), \(\lim_{T\to\infty} \int_0^T(\int_\Omega a_0 dx+\int_{\partial\Omega} b_0 dS) dt = +\infty\) or \(a_{ij},a_i,b_0\) are independent of \(t\) and \(\int_\Omega a_0u_0 dx+\int_{\partial\Omega} b_0u_0 dS >0\) then all positive solutions blow up in finite time (moreover, the last condition is optimal up to the equality sign). The same conclusion is true if \(\sigma_1<2\sigma_2-1\) and \(b_0(x,t)\geq B\gg 1\). Blow-up of large positive solutions is studied for \(\sigma_1=1\), \(\sigma_2>1\) and \(b_0\) positive. A systematic analysis concerning blow-up and stationary solutions is contained in [\textit{M. Chipot}, \textit{M. Fila} and \textit{P. Quittner}, Acta Math. Univ. Comen. New Ser. 60, No. 1, 35-103 (1991; Zbl 0743.35038)] for the special case \(L=\Delta\), \(a_0(x,t)=-a\) and \(b_0(x,t)=1\). Some recent results and references concerning this particular case can be found in the paper by \textit{A. Rodríguez-Bernal} and \textit{A. Tajdine}: Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up (to appear in J. Differ. Equations).
scientific article; zbMATH DE number 1431892