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Abstract We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975Commun. Pure Appl. Math.28323–31). In particular, we present ‘random outcome’ and ‘random threshold’ voting models that yield any polynomial nonlinearityfsatisfying $f\left(0\right)=f\left(1\right)=0$and a ‘recursive up the tree’ model that allows to go beyond this restriction onf. We compute several examples of particular interest; for example, we obtain a curious interpretation of the heat equation in terms of a nontrivial voting model and a ‘group-based’ voting rule that leads to a probabilistic view of the pushed-pulled transition for a class of nonlinearities introduced by Ebert and van Saarloos. 

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength\beta\in\mathbb{R}. This equation exhibits a transition from pulled to pushed front behavior at\beta_{c}=2. We prove convergence of the solutions to a traveling wave in a reference frame centered at a positionm_{\beta}(t)and study the asymptotics of the front locationm_{\beta}(t). When\beta < 2, it has the same form as for the standard Fisher-KPP equation established by Bramson:m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1)ast\to\infty. This form is typical of pulled fronts. When\beta > 2, the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1)with c_{*}(\beta)=\beta/2+2/\beta, which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2, the expansion changes tom_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1), reflecting the “pushmi-pullyu” nature of the front. The arguments for\beta<2rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case \beta>2relies on standard pushed front techniques. The proof in the case\beta=\beta_{c}is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at\beta_{c}=2and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights. 

We propose a novel method for establishing the convergence rates of solutions to reaction–diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in An et. al. [Arch. Ration. Mech. Anal. 247 (2023), no. 5, article no. 88]. It turns out that the convergence rate is controlled by the distance between thephantom front locationfor the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave has a pulled nature, regardless of whether the traveling wave itself is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach dramatically simplifies the proof in the Fisher–KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher–KPP case and the exponential rates in the pushed case. 

Abstract We consider the long time behaviour of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers in a random environment. The model is characterized by convolution with a kernel R and an L 2 inner product. In one spatial dimension, we extend a previous result of the authors (arXiv: 2002.02799 ), where only the case R = δ was considered; in particular, we show that solutions spread according to a 2 / 3 power law consistent with the KPZ scaling conjectured for directed polymers. In the special case when R = δ , we find the exact profile of the solution in the rescaled coordinates. We also consider the behaviour in higher dimensions. When the dimension is three or larger, we show that the long-time behaviour is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution.