Propagation of singularities for large solutions of quasilinear parabolic equations

Yevgeniia A. Yevgenieva

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 1, pp. 131-144

Résumé

The quasilinear parabolic equation with an absorption potential is considered: \begin{equation*} \left(|u|^{q-1}u\right)_t-\Delta_p(u)=-b(t,x)|u|^{\lambda-1}u (t,x)\in(0,T)\times\Omega,\quad\lambda>p>q>0, \end{equation*} where $\Omega$ is a bounded smooth domain in ${R}^n$, $n\geqslant1$, $b$ is an absorption potential which is a continuous function such that $b(t,x)>0$ in $[0,T)\times\Omega$ and $b(t,x)\equiv0$ in $\{T\}\times\Omega$. In the paper, the conditions for $b(t,x)$ that guarantee the uniform boundedness of an arbitrary weak solution of the mentioned equation in an arbitrary subdomain $\Omega_0:\overline{\Omega}_0\subset\Omega$ are considered. Under the above conditions the sharp upper estimate for all weak solutions $u$ is obtained. The estimate holds for the solutions of the equation with arbitrary initial and boundary data, including blow-up data (provided that such a solution exists), namely, $u=\infty$ on $\{0\}\times\Omega$, $u=\infty$ on $(0,T)\times\partial\Omega$.

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