This paper concerns initial boundary value problem for 3-dimensional compressible bipolar Navier-Stokes-Poisson equations with density-dependent viscosities. When the initial data is large, discontinuous, and spherically symmetric, we prove the global existence of the weak solution.

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipative term. We establish some sufficient conditions to guarantee finite time blow-up of strong solutions.

In this paper a variable-coefficient reaction-diffusion equation is studied. We classify the equation into three kinds by different restraints imposed on the variable coefficient $b(x)$ in the process of solving the determining equations of Lie groups. Then, for each kind, the conservation laws corresponding to the symmetries obtained are considered. Finally, some exact solutions are constructed.

We consider the nonlinear pseudoparabolic equation with a memory term $u_t-\Delta u-\Delta u_t+{\int }_0^t\lambda \left(t-\tau \right)\Delta u\left(\tau \right)d\tau =\text{d}\text{i}\text{v}\left({\left|\nabla u\right|}^{p-2}u\right)+u^{1+\alpha }$, $x\in \mathrm{\Omega }$, $t>0$, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions on p, $\alpha$, and the relaxation function $\lambda (t)$, we prove a finite-time blow-up result by using the concavity method.

This paper is concerned with the optimal convergence rates for solutions of the monopolar non-Newtonian flows. By using the energy methods, the perturbed weak solution of perturbed system asymptotically converges to the solution of the original system with the optimal rates $(1+t{)}^{-1/4}$.

This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial data $u_0\in L^2({ℝ}^3)$. Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior as $O({(1+t)}^{-3/8})$.

We obtained the algebraic $L^2$ time decay rate for weak solutions of the nonlinear heat equations with the nonlinear term ${\left|\nabla u\right|}^{2}u$ in whole space ${\mathbf{R}}^3$. The methods are based on energy methods and Fourier analysis technique.

This paper is concerned with the large time behavior of the weak solutions for three-dimensional globally modified Navier-Stokes equations. With the aid of energy methods and auxiliary decay estimates together with $L^p-L^q$ estimates of heat semigroup, we derive the optimal upper and lower decay estimates of the weak solutions for the globally modified Navier-Stokes equations as $C_{\mathrm{1}}(\mathrm{1}+t{)}^{-\mathrm{3}/\mathrm{4}}\le {∥u∥}_{L^2}\le C_{\mathrm{2}}(\mathrm{1}+t{)}^{-\mathrm{3}/\mathrm{4}},\mathrm{}t>\mathrm{1}.$ The decay rate is optimal since it coincides with that of heat equation.