| Abstract: |
| We deal with initial--boundary value problems to a class of quasilinear parabolic equations with the model representative:
\begin{equation}\label{eq1}
(|u|^{q-1}u)_t-\sum_{i=1}^n\left(|\nabla_xu|^{p-1}u_{x_i}\right)_{x_i}+ b(t,x)|u|^{\lambda-1}u=0,\quad n\geqslant1,
\end{equation}
where $\lambda>p\geqslant q>0$, absorption potential $b(t,x)\geqslant0$ degenerates on some manifold $\Gamma\in[0,\infty)\times\mathbb{R}^n$. We investigate qualitative properties of solutions of \eqref{eq1} with singularities belonging to $\Gamma$. As initial step of our analysis may be considered Cauchy problem for equation \eqref{eq1} with $p=q=1$, $b(t,x)=b(|x|):b(0)=0,\ b(s)>0\ \forall\,s>0$ (consequently, $\Gamma=\mathbb{R}^1_+\times\{0\}$) and initial condition
\begin{equation}\label{init1}
u(0,x)=k\delta(x),\quad k\in\mathbb{N},\ \delta(x)\text{ --- Dirac measure}.
\end{equation}
For solutions $u_k(t,x)$ of the mentioned problem in \cite{ShVer1}, \cite{MarSh2} there was obtained exact sufficient and necessary condition (criterium) on the flatness of function $b(s)$ near to $s=0$, which distinguish two possibilities:
\begin{enumerate}
\item[1)] $u_\infty(t,x):=\lim_{k\to\infty}u_k(t,x)$ is a very singular solution of the mentioned equation with strong singularity in the point $(0,0)$ and $u_\infty(0,x)=0$ $\forall\,x\neq0$;
\item[2)] $u_\infty(t,x)$ is a solution of the mentioned equation in $(0,\infty)\times\left(\mathbb{R}^n\setminus\{0\}\right)$ with blow-up set $\Gamma$ (razor blade solution): $u_\infty(t,0)=\infty$ $\forall\,t\geqslant0$.
\end{enumerate}
Elliptic version of these results is proved in \cite{ShVer3}, \cite{MarSh4}. We prove now some generalization of such results for general equation \eqref{eq1} and manifolds $\Gamma$.
\begin{thebibliography}{100}
\bibitem{ShVer1} Shishkov A., Veron L. {\it Singular solutions of some nonlinear
parabolic equations with spatially inhomogeneous absorption}, Calc.
Var. Part. Differ. Equat. 33 (2008), no.~3, 343--375.
\bibitem{MarSh2} Marcus~M., Shishkov~A.E. {\it Propagation of strong singularities in semilinear parabolic
equations with degenerate absorption}, Ann. Sc. Norm. Super. Pisa Cl. Sci. 16 (2016), no.~3, 1019--1047.
\bibitem{ShVer3} Shishkov A., Veron L. {\it Diffusion versus absorption in
semilinear elliptic equations}, J. Math. Anal. Appl. 352 (2009),
no.~1, 206--217.
\bibitem{MarSh4} Marcus M., Shishkov A. {\it Fading absorption in non-linear elliptic equations}, Ann. Inst. H. Poincare Anal Non Lineaire. 30 (2013), no.~2, 315--336.
\end{thebibliography} |
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