Localized peaking regimes for quasilinear parabolic equations

Abstract

This paper deals with the asymptotic behavior as $t\rightarrow T<\infty$ 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 $\Delta_p(u)=\sum_{i=1}^n\left(|\nabla_xu|^{p-1}u_{x_i}\right)_{x_i}$, $p>0$, $\lambda>p$, $\Omega$ is a bounded smooth domain, $b(t,x)\geq0$. 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 $t\rightarrow T$ in an arbitrary subdomain $\Omega_0\subset\Omega:\overline{\Omega}_0\subset\Omega$ and the sharp upper estimate of $u(t,x)$ when $t\rightarrow T$ has been obtained depending on $\mu>0$ and $s=dist(x,\partial\Omega)$. In the case $b(t,x)>0$ $\forall\,(t,x)\in(0,T)\times\Omega$ 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 $t\rightarrow T$ has been obtained.