Qualitative properties of solutions to mixed-diffusion bistable equations

We consider a fourth-order extension of the Allen-Cahn model with mixed-diffusion and Navier boundary conditions. Using variational and bifurcation methods, we prove results on existence, uniqueness, positivity, stability, a priori estimates, and symmetry of solutions. As an application, we construct a nontrivial bounded saddle solution in the plane.Comment: New version with minor change

Rayleigh-B\'enard convection with stochastic forcing localised near the bottom

We prove stochastic stability of the three-dimensional Rayleigh-B\'enard convection in the infinite Prandtl number regime for any pair of temperatures maintained on the top and the bottom. Assuming that the non-degenerate random perturbation acts in a thin layer adjacent to the bottom of the domain, we prove that the random flow periodic in the two infinite directions stabilises to a unique stationary measure, provided that there is at least one point accessible from any initial state. We also discuss sufficient conditions ensuring the validity of the latter hypothesis.Comment: 34 page

Hydrodynamic stability in the presence of a stochastic forcing:a case study in convection

We investigate the stability of a statistically stationary conductive state for Rayleigh-B\'enard convection between stress-free plates that arises due to a bulk stochastic internal heating. This setup may be seen as a generalization to a stochastic setting of the seminal 1916 study of Lord Rayleigh. Our results indicate that stochastic forcing at small magnitude has a stabilizing effect, while strong stochastic forcing has a destabilizing effect. The methodology put forth in this article, which combines rigorous analysis with careful computation, also provides an approach to hydrodynamic stability for a variety of systems subject to a large scale stochastic forcing

Convergence to a steady state for asymptotically autonomous semilinear heat equations on RN

AbstractWe consider parabolic equations of the formut=Δu+f(u)+h(x,t),(x,t)∈RN×(0,∞), where f is a C1 function with f(0)=0, f′(0)<0, and h is a suitable function on RN×[0,∞) which decays to zero as t→∞ (hence the equation is asymptotically autonomous). We show that, as t→∞, each bounded localized solution u⩾0 approaches a set of steady states of the limit autonomous equation ut=Δu+f(u). Moreover, if the decay of h is exponential, then u converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations

Singular radial solutions for the Lin-Ni-Takagi equation

We study singular radially symmetric solution to the Lin-Ni-Takagi equation for a supercritical power non-linearity in dimension N >= 3. It is shown that for any ball and any k >= 0, there is a singular solution that satisfies Neumann boundary condition and oscillates at leastktimes around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph-Lundgren exponent.Peer reviewe