On Long-Time Dynamics of the Solution of Doubly Nonlinear Equation

The subject of this investigation is the long time behavior of the positive solutions of the following nonhomogeneous equation $$\begin{aligned} \rho \left( \left| x\right| \right) \frac{\partial \beta \left( v\right) }{\partial t}-\overset{N}{\underset{i=1}{\sum }}D_{i}(|D_{i}v|^{ \lambda -1}D_{i}v)+\rho \left( \left| x\right| \right) g\left( \beta \left( v\right) \right) +l\beta ^{1+m}\left( v\right) =f\left( x\right) \qquad \end{aligned}$$ (1) in unbounded domain \( \mathbb {R} _{+}\times \mathbb {R} ^{N},\) where the term \(g\left( s\right) \) is supposed to satisfy a condition \(g^{\prime }\left( s\right) >-l_{1}\) and \(D_{i}=\partial _{x_{i}}\) . The existence of the global attractor for the Eq. (1) in \(L^{1+\theta }\left( \mathbb {R} ^{N},\rho \right) =\left\{ v;v\rho ^{1/(1+\theta )}\in L^{1+\theta }\left( \mathbb {R} ^{N}\right) \right\} \) is proved.