Localized peaking regimes for quasilinear parabolic equations

Abstract

This paper deals with the asymptotic behavior as $t\to T<\infty$ of all weak (energy) solutions of a class of equations with the following model representative: \begin{equation*} (|u|^{p-1}u)_t-Δ_p(u)+b(t,x)|u|^{λ-1}u=0 \quad (t,x)\in(0,T)\timesΩ,\,Ω\in{R}^n,\,n>1, \end{equation*} with prescribed global energy function \begin{equation*} E(t):=\int_Ω|u(t,x)|^{p+1}dx+ \int_0^t\int_Ω|\nabla_xu(τ,x)|^{p+1}dxdτ\rightarrow\infty\ \text{ as }t\rightarrow T. \end{equation*} Here ${\Delta }_p(u)={\sum }_{i=1}^n{\left(|{\nabla }_{x}u{|}^{p-1}{u}_{x_i}\right)}_{x_i}$, $p>0$, $\lambda >p$, $\Omega$ is a bounded smooth domain, $b(t,x)\ge 0$. Particularly, in the case \begin{equation*} E(t)\leq F_μ(t)=\exp\left(ω(T-t)^{-\frac1{p+μ}}\right)\quad\forall\,t<T,\,μ>0,\,ω>0, \end{equation*} it is proved that solution $u$ remains uniformly bounded as $t\to T$ in an arbitrary subdomain ${\Omega }_0\subset \Omega :{\overline{\Omega }}_0\subset \Omega$ and the sharp upper estimate of $u(t,x)$ when $t\to T$ 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 $t\to T$ has been obtained.

27 pages