Localized peaking regimes for quasilinear parabolic equations

Andrey E. Shishkov, Evgeniya A. Evgenieva

Published 2018-02-11Version 1

This paper deals with asymptotic behavior as $t\rightarrow T<\infty$ of all weak (energy) solutions of a class of equations with model representative: \begin{equation*} (|u|^{p-1}u)_t-\Delta_p(u)+b(t,x)|u|^{\lambda-1}u=0 \quad (t,x)\in(0,T)\times\Omega,\,\Omega\in\mathbb{R}^n,\,n>1, \end{equation*} with prescribed global energy function \begin{equation*} E(t):=\int_{\Omega}|u(t,x)|^{p+1}dx+ \int_0^t\int_{\Omega}|\nabla_xu(\tau,x)|^{p+1}dxd\tau \rightarrow\infty\ \text{ as }t\rightarrow T. \end{equation*} Here $\Delta_p(u)=\sum_{i=1}^n\left(|\nabla_xu|^{p-1}u_{x_i}\right)_{x_i}$, $p>0$, $\lambda>p$, $\Omega$ --- bounded smooth domain, $b(t,x)\geqslant0$. Particularly, in the case \begin{equation*} E(t)\leqslant F_\mu(t)=\exp\left(\omega(T-t)^{-\frac1{p+\mu}}\right)\quad\forall\,t<T,\,\mu>0,\,\omega>0, \end{equation*} it is proved that solution $u$ remains uniformly bounded as $t\rightarrow T$ in arbitrary subdomain $\Omega_0\subset\Omega:\overline{\Omega}_0\subset\Omega$ and there is obtained some sharp upper estimate of $u(t,x)$ when $t\rightarrow T$ in dependence of $\mu>0$ and $s=dist(x,\partial\Omega)$. In the case $b(t,x)>0$ $\forall\,(t,x)\in(0,T)\times\Omega$ there are found sharp sufficient conditions on degeneration of $b(t,x)$ near $t=T$, guaranteeing for arbitrary (even large) solution mentioned above boundedness, and obtained sharp upper estimate of final profile of solution when $t\rightarrow T$.