Limiting profile of solutions of quasilinear parabolic equations with flat peaking

Abstract

The paper deals with energy (weak) solutions u (t; x) of the class of equations with the model representative

$$ \left(\left|u\right|{p}^{-1}u\right)t-\Delta p(u)=0,\kern0.5em \left(t,x\right)\in \left(0,T\right)\times \varOmega, \varOmega \in {\mathrm{\mathbb{R}}}^n,n\ge 1,p>0, $$

and with the following blow-up condition for the energy:

$$ \varepsilon (t):= {\int}_{\Omega}{\left|u\left(t,x\right)\right|}^{p+1} dx+{\int}_0^t{\int}_{\Omega}{\left|{\nabla}_xu\left(\tau, x\right)\right|}^{p+1} dx d\tau \to \infty \mathrm{as}\;t\to T, $$

where Ω is a smooth bounded domain. In the case of flat peaking, namely, under the condition

$$ {\displaystyle \begin{array}{cc}\varepsilon (t)\le F\upalpha (t){\upomega}_0{\left(T-t\right)}^{-\upalpha}& \forall t<T,\end{array}}{\upomega}_0>0,\upalpha >\frac{1}{p+1}, $$

a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T.