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By analysing the uniform attractor for multi-valued processes, we study the long-time behaviour of the solutions of a model of non-autonomous porous-medium equations. The result is obtained by using the <i>a priori</i> estimates and suitable compactness arguments.

G-equations are well-known front propagation models in combustion and are HamiltonJacobi type equations with convex but non-coercive Hamiltonians. Viscous G-equations arise from numerical discretization or modeling dissipative mechanisms. Although viscosity helps to overcome non-coercivity, we prove homogenization of an inviscid G-equation based on approximate correctors and attainability of controlled flow trajectories. We verify the attainability for two-dimensional mean zero incompressible flows, and demonstrate asymptotically and numerically that viscosity reduces the homogenized Hamiltonian in cellular flows. In the case of one-dimensional compressible flows, we found an explicit formula of homogenized Hamiltonians, as well as necessary and sufficient conditions for wave trapping (effective Hamiltonian vanishes identically). Viscosity restores coercivity and wave propagation.

We consider a family \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} of \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} -periodic Schrdinger operators with \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} -interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} surfaces. We show that in the limit when \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} and the interactions strengths are appropriately scaled, \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} has at most \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} gaps within finite intervals, and moreover, the limiting behavior of the first \{\mathcal {H}^arepsilon\} _ {arepsilon&gt; 0} gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

We consider a model semilinear reaction-diffusion system with cubic nonlinear reaction terms and small spatially decaying initial data on R 1. The model system is motivated by the thermal-diffusive system in combustion, and it reduces to a scalar reaction-diffusion equation with Zeldovich nonlinearity when the Lewis number is one and proper initial data are prescribed. For scalar equations of similar type it is well known that while a nonlinearity of degree greater than three (supercritical case) has no effect for large times a cubic nonlinearity qualitatively changes the long time behaviour. The latter case has been treated in the literature by a rescaling method under the additional assumption of smallness of the nonlinearity. Although for our system the cubic nonlinearity is also critical we establish large time behaviour when the nonlinearity is not necessarily small which essentially differs from the supercritical case. This