A general min-max principle established by Ghoussoub is extended to the case of functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. Some topological properties of the min-max-generated critical points in such a framework are then pointed out.

The Navier--Stokes equations on the two-dimensional rotating sphere with a family of forcing terms, whose stream functions are the Legendre polynomials $P_s$, are considered. The stability and instability properties of the corresponding generalized Kolmogorov flows are studied both analytically and numerically. Logarithmically sharp lower bounds for the dimension of the global attractor are obtained. The effect of rotation on the stability properties of the Kolmogorov flows is discussed.

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear $2m$-th order parabolic equation $$ u_t = -(-\Delta)^m u + |u|^p \quad {\rm in} \,\,\, {{\bf R}^N} \times \mathbb R_+, $$ where $m>1$, $p>1$, with bounded integrable initial data $u_0$. We prove that, in the supercritical Fujita range $p > p_F = 1+2m/N$, any small global solution with nonnegative initial mass, $\int u_0 dx \ge 0$, exhibits as $t \to \infty$ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $p \in (1,p_F]$ where solutions blow-up for any arbitrarily small nonnegative nontrivial initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents $\{p_l= 1+2m/(l+N), \, l=0,1,2,...\}$, where $p_0=p_F$, are discussed.

We present several existence and uniqueness results for the equations satisfied by the three-dimensional non-Newtonian mixture of the Oldroyd kind. We study the coupling of the constitutive law, the Navier-Stokes equations and the Cahn-Hilliard equation which stands for a model of a multiphase non-Newtonian fluid. We prove that a strong local solution exists, but also that a global solution exists if the data are small enough. This last result is established even if the fluids are strongly elastic.

We give a simple and very complete proof of the existence of strong solutions to the nonhomogeneous problem (1.1) under the nonhomogeneous boundary conditions (1.5). See Theorem 1.2. See also the pioneering paper [42] by V.A. Solonnikov and V.E. Ščadilov, and [6].

In this paper we consider the Von Kármán system with thermal effect. The constitutive assumptions we use for the propagation of temperature were proposed by Gurtin and Pipkin. In this case the heat conduction is independent of the present values of the temperature gradient. We show that the corresponding nonlinear system is globally well posed. Moreover, when the relaxation function which characterizes the memory effect on the temperature decays exponentially, then we show that the solution of the system also decays exponentially.

The paper deals with the existence and properties of front propagation between the stationary states $0$ and $1$ of the reaction-diffusion-advection equation $$ v_\tau + h(v)v_x = \big( D(v) v_x \big)_x + G(v), $$ where $G$ is a bistable reaction term and $D$ is a strictly positive diffusive process. We show that the additional transport term $h$ can cause the disappearance of such wavefronts and prove that their existence depends both on the local behavior of $G$ and $h$ near the unstable equilibrium and on a suitable sign condition on $h$ in $[0,1]$. We also provide an estimate of the wave speed, which can be negative even if $\int_0^1 G(u) D(u) du>0$, unlike what happens to the mere reaction-diffusion dynamic occurring when $h\equiv 0$.

This work is concerned with the asymptotic stability of traveling waves for scalar viscous conservation laws with a convex flux function and a dispersion term. First we prove the existence of solutions locally in time of the initial-value problem for initial data near a constant solution by Fourier analysis. Using the semigroup method the local existence for initial data that are an $L^2$ perturbation of a traveling-wave profile is proved. We also obtain a regularity property of these solutions. The solution operator generated by the linearized equation plays a crucial role. Using the energy method we establish a priori estimates. These estimates, when combined with the local existence, lead to the desired global-in-time existence as well as the time-asymptotic decay of solutions with initial data close to a monotone traveling wave.

In this paper, for a bounded, smooth domain $\Omega \subset {{\mathbb {R}}}^N$, $h\in L^r(\Omega)$, $r>N/2$, $g\in L^s(\partial \Omega)$, $s>N-1$, we prove the maximum and antimaximum principle for the quasilinear boundary-value problem $- \mbox{div} ( A(u) \nabla u) +u = h$ in $\Omega$, $ A(u) \nabla u \cdot \eta = \lambda f(u) + g$, on $\partial \Omega$, where $A$ is elliptic and bounded and $f$ is asymptotically linear. The sharpness of this result ($r>N/2$ and $s>N-1$) is discussed for the linear boundary-value problem, $-\Delta u + u = h$ in $\Omega$, $\partial u/\partial\eta = \lambda u + g $ on $\partial \Omega$.