This article is cited in 3 papers

Procedings of the 3nd International Conference "Mathematical Physics and its Applications"
Equations of Mathematical Physics

Solutions of anisotropic parabolic equations with double non-linearity in unbounded domains

L. M. Kozhevnikova, A. A. Leont'ev

Sterlitamak Branch of Bashkir State University, Sterlitamak, 453103, Russia

Abstract: This work is devoted to some class of parabolic equations of high order with double nonlinearity which can be represented by a model equation
\begin{gather*} \frac{\partial}{\partial t}(|u|^{k-2}u)= \sum_{\alpha=1}^n(-1)^{m_\alpha-1}\frac{\partial^{m_\alpha}}{\partial x_\alpha^{m_\alpha}} \left[\left|\frac{\partial^{m_\alpha} u}{\partial x_\alpha^{m_\alpha}}\right|^{p_\alpha-2} \frac{\partial^{m_\alpha} u}{\partial x_\alpha^{m_\alpha}}\right],\\ m_1,\ldots, m_n\in \mathbb{N},\quad p_n\geq \ldots \geq p_1>k,\quad k>1. \end{gather*}
For the solution of the first mixed problem in a cylindrical domain $ D=(0,\infty)$ $\times\Omega, \;\Omega\subset \mathbb{R}_n,$ $n\geq 2,$ with homogeneous Dirichlet boundary condition and finite initial function the highest rate of decay established as $t \to \infty$. Earlier upper estimates were obtained by the authors for anisotropic equation of the second order and prove their accuracy.

Keywords: anisotropic equation, doubly nonlinear parabolic equations, existence of strong solution, decay rate of solution.

UDC: 517.957

MSC: 35K35, 35K61

Original article submitted 15/XI/2012
revision submitted – 10/III/2013

DOI: 10.14498/vsgtu1186

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