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

Abstract

This work is devoted to a class of parabolic equations with 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 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 compactly supported initial function precise estimates the rate of decay 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's approximations that for an isotropic model equation was proposed by E. R. Andriyanova and F. Kh. Mukminov.