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This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $$u_t+(-\Delta +1)\beta \left(u\right)+G\left(u\right)=g\text{in}\Omega \times (0,T)$$ in an unbounded domain $\Omega \subset {ℝ}^N$ with smooth bounded boundary, where $N\in \mathbb{N}$, $T>0$, $\beta$, is a single-valued maximal monotone function on $ℝ$, e.g., $$\beta \left(r\right)={\left|r\right|}^{q-1}r(q>0,q\ne 1)$$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${\left(H^1\left(\Omega \right)\right)}^{*}$ to ${\left(H^1\left(\Omega \right)\right)}^{*}$. The present work establishes existence and estimates for the above problem.

This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\left(1\right)}_t+u·1=𝔻1-{\chi }_1·\left(1c\right)+{\mu }_{1}1(1-1-a_{1}2)\hfill & \text{in}\times (0,\infty ),\hfill \\ {\left(2\right)}_t+u·2=𝔻2-{\chi }_2·\left(2c\right)+{\mu }_{2}2(1-a_{2}1-2)\hfill & \text{in}\times (0,\infty ),\hfill \\ c_t+u·c=𝔻c-(\alpha 1+\beta 2)c\hfill & \text{in}\times (0,\infty ),\hfill \\ u_t+(u·)u=𝔻u+\nabla P+(\gamma 1+2)\Phi ,·u=0\hfill & \text{in}\times (0,\infty )\hfill \end{array}\right.\end{array}$$ under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can...