Large solutions of parabolic logistic equation with spatial and temporal degeneracies

Abstract

There is studied asymptotic behavior as \begin{document}$t\rightarrow T$\end{document} of arbitrary solution of equation \begin{document}$P_0(u):=u_t-\Delta u=a(t,x)u-b(t,x)|u|^{p-1}u \text{ in } [0,T)\times\Omega,$\end{document} where \begin{document}$\Omega$\end{document} is smooth bounded domain in \begin{document}$\mathbb{R}^N$\end{document} , \begin{document}$0 , \begin{document}$p>1$\end{document} , \begin{document}$a(\cdot)$\end{document} is continuous, \begin{document}$b(\cdot)$\end{document} is continuous nonnegative function, satisfying condition: \begin{document}$b(t, x)\geqslant a_1(t)g_1(d(x))$\end{document} , \begin{document}$d(x):=\textrm{dist}(x, \partial\Omega)$\end{document} . Here \begin{document}$g_1(s)$\end{document} is arbitrary nondecreasing positive for all \begin{document}$s>0$\end{document} function and \begin{document}$a_1(t)$\end{document} satisfies: \begin{document}$a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1}) \forall t 0a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1}) \forall t 0$\end{document} with some continuous nondecreasing function \begin{document}$\omega(\tau)\geqslant0$\end{document} \begin{document}$\forall\tau>0$\end{document} . Under additional condition: \begin{document}$\omega(\tau)\rightarrow\omega_0=\textrm{const}>0 \text{ as }\tau\rightarrow0$\end{document} it is proved that there exist constant \begin{document}$k:0 , such that all solutions of mentioned equation (particularly, solutions, satisfying initial-boundary condition \begin{document}$u|_\Gamma=\infty$\end{document} , where \begin{document}$\Gamma=(0, T)\times\partial\Omega\cup\{0\}\times\Omega$\end{document} ) stay uniformly bounded in \begin{document}$\Omega_0:=\{x\in\Omega:d(x)>k\omega_0^{\frac12}\}$\end{document} as \begin{document}$t\rightarrow T$\end{document} . Method of investigation is based on local energy estimates and is applicable for wide class of equations. So in the paper there are obtained similar sufficient conditions of localization of singularity set of solutions near to the boundary of domain for equation with main part \begin{document}$P_0(u)=(|u|^{\lambda-1}u)_t-\sum_{i=1}^N(|\nabla_xu|^{q-1}u_{x_i})_{x_i}$\end{document} if \begin{document}$0 .