Article

Ufa Mathematical Journal
Volume 5, Number 1, pp. 63-82

Decay of solution of anisotropic doubly nonlinear parabolic equation in unbounded domains

Kozhevnikova L.M., Leontiev A.A.

DOI:10.13108/2013-5-1-63

This work is devoted to a class of parabolic equations with a double nonlinearity whose representative is a model equation $$(|u|^{k-2}u)_t=\sum_{\alpha=1}^n(|u_{x_{\alpha}} |^{p_{\alpha}-2}u_{x_{\alpha}})_{x_\alpha},\quad p_n\geq \ldots \geq p_1>k,\quad k\in(1,2).$$ For the solution of Dirichlet initial boundary value problem in a cylindrical domain $D=(0,\infty)$ $\times\Omega$, ${\Omega\subset \mathbb{R}_n}$, $n\geq 2$, with homogeneous Dirichlet boundary condition and compactly supported initial function, precise estimates the decay rate as $t\rightarrow\infty$ are established. Earlier these results were obtained by the authors for $k\geq 2$. The case $k\in(1,2)$ differs by the method of constructing Galerkin approximations that for an isotropic model equation was proposed by E.R. Andriyanova and F.Kh. Mukminov.