Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as $t\to T<\mathrm{\infty }$ of all weak (energy) solutions of a class of equations with the following 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{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 ${\mathrm{\Delta }}_p(u)={\sum }_{i=1}^n{\left(\mathrm{\mid }{\mathrm{\nabla }}_{x}u{\mathrm{\mid }}^{p-1}{u}_{x_i}\right)}_{x_i}$, $p>0$, $\lambda >p$, $\mathrm{\Omega }$ is a bounded smooth domain, $b(t,x)\ge 0$. Particularly, in the case \begin{equation*} E(t)\leq 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 $t\to T$ in an arbitrary subdomain ${\mathrm{\Omega }}_0\subset \mathrm{\Omega }:{\bar{\mathrm{\Omega }}}_0\subset \mathrm{\Omega }$ and the sharp upper estimate of $u(t,x)$ when $t\to T$ has been obtained depending on $\mu >0$ and $s=dist(x,\mathrm{\partial }\mathrm{\Omega })$. In the case $b(t,x)>0$ $\mathrm{\forall }(t,x)\in (0,T)\times \mathrm{\Omega }$ sharp sufficient conditions on degeneration of $b(t,x)$ near $t=T$ that guarantee mentioned above boundedness for arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of solution when $t\to T$ has been obtained.