arXiv subjects
Publications and source records attributed to Andrey E. Shishkov.
Regimes with a singular peaking for a wide class of quasilinear second order parabolic equations are studied. On the basis of energy methods, precise estimates of a final profile of a weak solution in a neighborhood of the peaking time are established depending on the rate of increase of a global energy of this solution. Key words: quasilinear parabolic equations, peaking regimes, energy solutions.
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-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 $Δ_p(u)=\sum_{i=1}^n\left(|\nabla_xu|^{p-1}u_{x_i}\right)_{x_i}$, $p>0$, $λ>p$, $Ω$ is a bounded smooth domain, $b(t,x)\geq0$. Particularly, in the case \begin{equation*} E(t)\leq F_μ(t)=\exp\left(ω(T-t)^{-\frac1{p+μ}}\right)\quad\forall\,t 0,\,ω>0, \end{equation*} it is proved that solution $u$ remains uniformly bounded as $t\rightarrow T$ in an arbitrary subdomain $Ω_0\subsetΩ:\overlineΩ_0\subsetΩ$ and the sharp upper estimate of $u(t,x)$ when $t\rightarrow T$ has been obtained depending on $μ>0$ and $s=dist(x,\partialΩ)$. In the case $b(t,x)>0$ $\forall\,(t,x)\in(0,T)\timesΩ$ 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.