In this paper we study solutions of the quasi-linear parabolic equations $\partial u/\partial t -{\Delta} _p u = a(x) |u|^{q-1}u$ in $(0,T) \times {\Omega} $ with Robin boundary condition ${\partial} u /{\partial} \nu|\nabla u|^{p-2} = b(x) |u|^{r-1}u$ in $(0,T) \times {\partial} {\Omega}$ where $\Omega$ is a regular bounded domain in ${\mathbb R}^N$, $N \geq 3$, $q>1$, $r>1$ and $p \geq 2$. Some sufficient conditions on $a$ and $b$ are obtained for those solutions to be bounded or blowing up at a finite time. Next we give the asymptotic behavior of the solution in special cases.

Initial-boundary-value problems in three different domains are considered for quasilinear evolution partial differential equations of an odd (not less than third) order with respect to spatial variables in the multidimensional case. The nonlinearity has the divergent form and at most a quadratic rate of growth. Assumptions on the differential operator of odd order provide global estimates on solutions in $L_2$ and a local smoothing effect. Results on existence and uniqueness of global weak solutions are established. The essential part of the study is the construction of special solutions to the corresponding linear equations of the "boundary potential" type, which ensures the results under natural smoothness assumptions on initial and boundary data provided we have certain relations between the dimension and the order of the equations.

In the case $f \in C(\mathbb R,\mathbb R)$ is asymptotically linear, we give a lower estimate of number of sign-changing solutions to the problem $$-d^2 \Delta u + u =f(u)\;\;\text{in $\Omega$,}\quad u=0 \;\;\text{on $\partial\Omega$,} $$ where $\Omega$ is a bounded domain in $\\mathbb R^N$ ($N \geq 2$) and $d>0$ is an appropriate small number.

This article is focused on qualitative properties of solutions to generalized Forchheimer equations for slightly compressible fluids in porous media subject to the flux condition on the boundary. The pressure and pressure gradient are proved to depend continuously on the boundary flux and coefficients of the Forchheimer polynomial in the momentum equation. In particular, the asymptotic dependence of the shifted solution on the asymptotic behavior of the boundary data is obtained. In order to improve various a priori estimates for the pressure, its gradient and time derivative, we prove and utilize suitable Poincaré-Sobolev and nonlinear Gronwall inequalities, as well as obtain uniform Gronwall-type inequalities from a system of coupled differential inequalities. Also, additional flux-related quantities are introduced as controlling parameters of fluid flows for large time in the case of unbounded fluxes.

We investigate nonautonomous quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions. We establish local wellposedness and study the time and space regularity of the solutions. Our main results give principles of linearized (orbital) stability and instability for solutions in the vicinity of a periodic solution. Our approach relies on a detailed study of regularity properties of the linearized nonautonomous problem and its evolution family.