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The current paper is to explore traveling waves in general time dependent bistable equations. In order to do so, it first introduces a notion of traveling wave solutions in general time dependent equations, which is a natural extension of classical traveling wave solutions. The basic point of view in the paper is that traveling wave solutions are certain limits of wave-like solutions. It then introduces a notion of wave-like solutions and shows in terms of certain backward-forward limits that the existence of wave-like solutions in general time dependent equations implies the existence of traveling wave solutions. It is shown that wave-like solutions exist in time dependent bistable equations and hence traveling wave solutions exist in such equations. Moreover, it is shown that traveling wave solutions in a time dependent bistable equation are stable and unique. The results obtained in the paper extend many of the results on traveling wave solutions of time independent (periodic, almost periodic) bistable equations.
In this paper we study periodic problems driven by the scalar ordinary $p$-Laplacian and with a nonsmooth potential. Using degree theoretic methods based on a fixed-point index for nonconvex-valued multifunctions, we prove two existence theorems. In the first we employ nonuniform nonresonance conditions between two successive eigenvalues of the negative $p$-Laplacian with periodic boundary conditions. In the second we use Landesman-Lazer conditions.
In this paper we consider the Euler equations of an incompressible fluid in a $3D$ channel with permeable walls; a portion of the boundary is standing an inflow and another an outflow. We prove the existence, uniqueness and regularity of solutions, locally in time, in various function spaces of Hölder type.
We investigate blow-up solutions of the equation $\Delta u=u^p+g(u)$ in a bounded smooth domain $\Omega$. If $p>1$ and if $g$ satisfies appropriate growth conditions (compared with the growth of $t^p$) as $t$ goes to infinity we find optimal asymptotic estimates of the solution $u(x)$ in terms of the distance of $x$ from the boundary $\partial\Omega$.