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We prove the –to the best knowledge of the authors– first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by a recent answer to a long standing conjecture concerning the classification of global solutions of the obstacle problem. [ABSTRACT FROM AUTHOR]

For the obstacle problem with a nonlinear operator, we characterize the space of global solutions with compact contact sets. This is achieved by constructing a bijection onto a class of quadratic polynomials describing the asymptotic behavior of solutions. [ABSTRACT FROM AUTHOR]

We study global solutions to the thin obstacle problem with at most quadratic growth at infinity. We show that every ellipsoid can be realized as the contact set of such a solution. On the other hand, if such a solution has a compact contact set, we show that it must be an ellipsoid. [ABSTRACT FROM AUTHOR]

This paper tries to understand how a drift influences the wave propagation of a bistable reaction-diffusion equation in straight cylinders in dimension n ≥ 3. We derive a quantitative condition under which a compactly supported drift term blocks the propagation of a traveling wave. We assume for this that the drift has a potential. [ABSTRACT FROM AUTHOR]

In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side. [ABSTRACT FROM AUTHOR]