Limiting profile of solutions of quasilinear parabolic equations with flat peaking
| Autor: | Yevgeniia A. Yevgenieva |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Journal of Mathematical Sciences. 234:106-116 |
| ISSN: | 1573-8795 1072-3374 |
| DOI: | 10.1007/s10958-018-3985-8 |
| Popis: | 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 $$ {\displaystyle \begin{array}{cc}\varepsilon (t)\le F\upalpha (t){\upomega}_0{\left(T-t\right)}^{-\upalpha}& \forall t 0,\upalpha >\frac{1}{p+1}, $$ a sharp estimate of the profile of a solution has been obtained in a neighborhood of the blow-up time t = T. |
| Databáze: | OpenAIRE |
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