A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space

TL;DR

The paper studies diffusion on a moving half-space with an inward-shifting boundary, modeled by $\\partial_t u = d \\\\partial_{zz} u$ for $z\\ge b(t)$ and a Robin boundary at $z=b(t)$ that enforces mass conservation. By transforming to a fixed half-space via $v(t,x)=u(t,x+b(t))$, the authors analyze the resulting diffusion-advective equation with a time-dependent boundary flux, focusing on algebraic boundary motion $b(t)=c[(1+t)^{\\beta}-1]$ with $\\beta\\in[0,1]$. The main contributions are: (i) an explicit solution and exponential $L^1$-convergence to a stationary profile for the linear regime $\\beta=1$; (ii) a comprehensive self-similar asymptotic classification for all $\\beta\in[0,1]$, with Gaussian self-similar profiles in the subcritical and critical cases and exponential self-similar profiles in the supercritical case; (iii) precise convergence rates: $O((1+t)^{-(1/4-\\beta/2)})$ for $0<\\beta<\\tfrac{1}{2}$, $O((1+t)^{-1/2})$ for $\\beta=\\tfrac{1}{2}$, and $O(\log(1+t)/(1+t)^{\\beta-1/2})$ for $\\tfrac{1}{2}<\\beta\le 1$, with the subcritical/critical regimes treated via entropy methods and the supercritical regime via Duhamel’s principle. The work highlights how external boundary motion counteracts diffusion, producing a spectrum of self-similar states and providing a foundation for extending to nonlinear and reaction-diffusion settings in receding environments.

Abstract

To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line {z $\ge$ b(t)}, where the boundary position b(t) is a given nondecreasing function of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall-a ''piston''-that sweeps the individuals it encounters. Our analysis focuses on the cases where b(t) $\sim$ ct^$β$ with $β$ $\in$ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When $β$ goes through the critical value 1/2, the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when $β$ = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary.

Theorems & Definitions (3)

• Theorem 2.1: Fundamental solution and asymptotic behavior for $\beta=1$
• Proposition 2.2: Asymptotic behavior for $\beta=0$
• Theorem 2.3: Asymptotic behavior for $\beta\in(0,1$] Assume that $b(t) = c[(1+t)^{\beta} - 1]$ with $c>0$ and $\beta\in (0,1]$. Let $v$ be the unique solution to \ref{['v-eqn']} starting from bounded, compactly supported, and nonnegative initial condition $v_{0}$. Then the asymptotic behavior of $v$ is characterized by convergence to self-similar profiles, according to the three following regimes: If $0 < \beta<1/2$ (sub-critical regime), then there exists $\ell>0$ such that $\left\Vert v(t,x) - \frac{1}{\sqrt{1+t}} W(\frac{x}{\sqrt{1+t}}) \right\Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})} \leq {\frac{\ell}{(1+t)^{\frac{1}{4}-\frac{\beta}{2}}}}, \qquad \forall t>0,$ where $W(y) = W(0) \exp\!\left(-\frac{y^{2}}{4d}\right)$ for any $y\in\mathbb{R}_{+}$.If $\beta=1/2$ (critical regime), then there exists $\ell>0$ such that $\left\Vert v(t,x) - \frac{1}{\sqrt{1+t}} W(\frac{x}{\sqrt{1+t}}) \right\Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})} \leq {\frac{\ell}{\sqrt{1+t}}}, \qquad \forall t>0,$ where $W(y) = W(0) \exp\!\left(-\frac{y^{2}}{4d} - \frac{cy}{2d}\right)$ for any $y\in\mathbb{R}_{+}$.If ${1/2<\beta\leq 1}$ (super-critical regime), then there exists $\ell>0$ such that $\left\Vert v(t,x) - \frac{b'(t)}{c\beta} W(\frac{b'(t)}{c\beta}\,x) \right\Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})} \leq {\frac{\ell \log(1+t)}{(1+t)^{\beta-\frac{1}{2}}}}, \qquad \forall t>1,$ where $W(y) = W(0) \exp\!\left(-\frac{c\beta}{d} y\right)$ for any $y\in\mathbb{R}_{+}$. In each case above, $W(0)\geq 0$ is the uniquely determined constant such that the total mass of the self-similar profile $W$ equals $M$. \ref{['TH_asymptotic_behavior_solutions']} reveals three distinct regimes, governed by the value of the parameter $\beta$, as further illustrated in \ref{['FIG_2']}. One of the main ingredients of its proof is to introduce a general self-similar transformation of the form $v(t,x) = f(t)\cdot w(g(t),f(t)x),$ where the scaling functions $f(t)$ and $g(t)$ are determined according to the different regimes, see \ref{['S3_diffusive_regime']} and \ref{['S5_super_critical_case']}. Under such scaling, the problem satisfied by $w$ becomes a Fokker--Planck type equation: $\partial_{\tau} w = \partial_{y}\left[ D \partial_{y} w + \partial_{y}\Phi(\tau,y) w \right],\,\tau > 0,y>0,$ where the time-dependent potential $\Phi(\tau,y)$ stabilizes over time toward a stationary potential $\Phi_{\infty}(y)$. Depending on the parameter $\beta$, this stationary potential $\Phi_{\infty}$ is either quadratic or linear in $y$, and the stationary solutions of this resulting problem \ref{['EQ_Fokker_Planck']} are given by: $W(y) = W(0) e^{-\frac{1}{D}\Phi_{\infty}(y)},$ consequently leading to Gaussian, exponential, or mixed profiles, according to the precise form of $\Phi_{\infty}$. It is worth noting that $f(t)\equiv 1$ and $g(t)=t$ in the linear regime $\beta=1$, so that the self-similar transformation \ref{['EQ_RESCALING']} reduces to the identity and the unknown $v$ and $w$ coincide. As anticipated by the stochastic perspective introduced in \ref{['S1_intro']}, the model displays a sharp transition at the critical value $\beta=1/2$. At this threshold, the boundary drift is of magnitude $\pazocal{O}(\sqrt{t})$, exactly matching the mean displacement of a reflected Brownian particle on $\mathbb{R}_{+}$. This alignment marks the onset of significant boundary effects. In particular, it triggers a self-similar intermediate profile, characterized by a Gaussian shape modulated by an exponential factor, reflecting the emergence of a competition between diffusion and boundary-driven accumulation. In the sub-critical regime $\beta \in(0,\frac{1}{2})$, the self-similar profiles adopt Gaussian shapes---exactly as for the solutions to the classical heat equation on $\mathbb{R}_{+}$ with Neumann boundary condition, see \ref{['PROPO_BETA_0']}. This point is consistent with the weak interactions between the particles and the boundary discussed in \ref{['S1_intro']}, see \ref{['FIG_Brownian_1']}. By contrast, in the super-critical regime $\beta\in(\frac{1}{2},1]$, the boundary outpaces the diffusive scaling, giving rise to exponential self-similar profiles with a noticeably heavier tail. This highlights the strong interaction between the moving boundary and the diffusion process, see \ref{['FIG_Brownian_1']}. We also point out that, when $\beta=1$, \ref{['TH_asymptotic_behavior_solutions']} provides a polynomial convergence rate, which is significantly less optimal than the sharp exponential convergence exhibited in \ref{['TH_fundamental_sol_for_beta_1']}. When $\beta=1/2$, the convergence rate we obtain coincides with that of the Neumann heat equation (see \ref{['PROPO_BETA_0']}), for which the entropy method is known to yield the sharp decay rate (see \ref{['REM_beta=0']}). Since our proof strategy relies on the same type of entropy arguments---involving only the logarithmic Sobolev and Csiszár--Kullback inequalities, without any crude estimates---it is reasonable to believe that the resulting decay rate is also sharp in this critical case. For $\beta<1/2$, we do not expect better rates with our technique, as explained in \ref{['SS33_beta_lower_1_2']}---see \ref{['EQ_entropy_diff_yet_sharp']}-\ref{['3']}. Lastly, let us emphasize that, for $0 \leq \beta < 1$, the solution $v$ tends to vanish as a result of the decaying boundary speed $b'(t)$. Varying $\beta$ from $0$ to $1$ thus provides a clear depiction of how the influence of the moving boundary counteracts diffusion up to the limiting case $\beta = 1$, where the balance is achieved between the dispersive effect of diffusion and the aggregative effect of the moving boundary. Figure III --- long-time behavior of the solutions to \ref{['EQ_global_problem']} regarding different regimes of $\beta$. We recall that the moving boundary $b(t)$ (depicted by the purple vertical lines) behaves asymptotically like $ct^{\beta}$ as $t\to \infty$. A transition in the shape of the asymptotic profiles occurs at $\beta=1/2$, shifting from Gaussian to exponential. This threshold also marks a change in the $L^{\infty}$ decay rate, achieved by $u(t,b(t))=v(t,0)$, shifting from $\pazocal{O}(t^{-1/2})$ for $\beta\in[0,\frac{1}{2}]$ to $\pazocal{O}(t^{\beta-1})$ for $\beta\in[\frac{1}{2},1]$. As a complement to this macroscopic viewpoint, typical individual trajectories are illustrated in \ref{['FIG_Brownian_1']} for the cases $\beta \in \lbrace0,\frac{1}{4},\frac{3}{4}\rbrace$, highlighting the increasing influence of the moving boundary at the microscopic scale. It is worth noting that the super-critical self-similar rescaling \ref{['EQ_rescaling_super_sqrt']} remains valid for $\beta>1$, and that the Cauchy problem \ref{['EQ_for_w_beta_larger_1/2']} still governs the resulting function $w$ in this case. Stationary solutions with exponential decay therefore persist, exactly as in the intermediate regime $\beta\in(\frac{1}{2},1]$. Numerical experiments further suggest that, in this superlinear setting, $w(t,)$ converges to such an exponentially decaying steady state. This behavior suggests that the function $v$ increasingly concentrates near the moving boundary over time, yet without finite-time blow-up. Specifically, we anticipate the following approximate long-time behavior: $\Vert v(t,) \Vert_{L^{\infty}(\mathbb{R}_{+})} \approx v(t,0) \approx \frac{b'(t)}{c\beta} = (1+t)^{\beta-1}.$ A key point is that the coefficient $\eta(\tau)$ appearing in problem \ref{['EQ_for_w_beta_larger_1/2']}, and given by \ref{['EQ_eta']}: $\eta(\tau) = \frac{\beta-1}{1 + (2\beta-1)\tau},$ changes sign as $\beta$ crosses the value $1$. This sign change drastically alters the behavior of the drift term in \ref{['EQ_for_w_beta_larger_1/2']}, and introduces new analytical challenges. To be more specific, when $\beta \leq 1$, the drift $c\beta-\eta(\tau)y$ consistently drives the population toward the boundary, thereby promoting mass concentration near $y=0$. In contrast, for $\beta>1$, the drift points toward the boundary only for small values of $y$ (where $c\beta-\eta(\tau)y$ remains nonnegative). As $y$ increases, the influence of $-\eta(\tau)y$ eventually dominates, causing the drift to reverse and effectively driving the population away from the boundary. This subtle interplay between boundary accumulation and long-range repulsion significantly complicates the analysis. In particular, a major obstacle lies in controlling the first moment of the function $w$, for which the standard techniques used for $\beta\in(\frac{1}{2},1]$ (such as the comparison principle and the construction of supersolutions) fail, since the reversed sign of $\eta(\tau)$ turns our supersolutions into subsolutions. As the first moment directly impacts the upper bound of the convolution term in \ref{['EQ_Duhamel_1']}, it is natural to first explore whether the strategy developed for $\beta\in(\frac{1}{2},1]$ can be adapted to recover a suitable bound on that moment. Otherwise, a completely different approach will be required. Nonetheless, we believe that the first moment is decisive for the long-time dynamics, as clearly evidenced in related studies by Calvez et al. CalvezAnalysis12 and Lepoutre et al. LepoutreCell14. In those works, explicit evolution equations for the first moment turned out to be significantly useful for the analysis of convergence and potential blow-up behaviors. In our model, however, the drift varies in both space and time, which prevents the direct application of these earlier strategies. Although the analytical framework developed here provides a solid foundation, a refined analysis is still needed to tackle the sign-changing drift and its influence on the first-moment dynamics. We leave this challenging question for future work, with the expectation that sharp bounds on the first moment will ultimately yield a complete description of the super-linear regime. The proof of \ref{['TH_fundamental_sol_for_beta_1']} is based on the explicit construction of the solution via an extension method: we extend the initial data $v_{0}$ to the whole real line, choosing the extension so that, under the evolution governed by $\partial_t = d \partial_{xx} + c \partial_x$, the resulting solution satisfies the Robin boundary condition at $x = 0$ for all $t > 0$. We then recover the desired solution by restricting the extended solution to the positive half-line $\mathbb{R}_{+}$, and a detailed manipulation of integrals reveals the explicit form of the fundamental solution $H(t,x,\xi)$, as given in \ref{['EQ_heat_kernel_linear']}. In turn, the explicit representation of the solution allows for a precise analysis of its long-time behavior, as stated in \ref{['TH_fundamental_sol_for_beta_1']}. By carefully analyzing the structure of the fundamental solution, we derive $L^{1}$-estimates for $v(t,)-V$ demonstrating exponential convergence toward the stationary profile. As already explained, \ref{['TH_asymptotic_behavior_solutions']} is grounded on a self-similar change of variables---see \ref{['EQ_RESCALING']}. When $b(t)$ is of order $\pazocal{O}(\sqrt{t})$, namely $\beta \leq 1/2$, we use the parabolic scaling $\tau=\log\sqrt{1+t}$ and $y=\frac{x}{\sqrt{1+t}}$, and study the rescaled density $w$ satisfying $v(t,x)= \frac{1}{\sqrt{1+t}} \; w ( \log\sqrt{1+t}, \frac{x}{\sqrt{1+t}} ).$ The distance between $w$ and the associated stationary profile $W$ is then measured by means of the Boltzmann relative entropy VazquezAsymptotic18 defined by $\pazocal{H}(w|W):=\int_0^\infty w(\tau,y)\log\frac{w(\tau,y)}{W(y)}\, \mathrm{d} y.$ As in the case of the heat equation, this functional can be differentiated with respect to $\tau$. The strict uniform convexity of the quadratic potential $\Phi(\tau,y)$ (see \ref{['EQ_Fokker_Planck']}) then allows us to apply logarithmic Sobolev (\ref{['Lem_logSobolev']}) and Csiszár--Kullback (\ref{['Lem_Csiszar_Kullback']}) inequalities to estimate the entropy dissipation ${\mathrm{d}\pazocal{H}}/{\mathrm{d}\tau}$ and establish convergence. When $\beta>1/2$, the change of variables is more delicate: $v(t,x) : = \frac{b'(t)}{c\beta} \, w ( \int_{s=0}^{t} [\frac{b'(s)}{c\beta}]^{2} ds , \; \frac{b'(t)}{c\beta}x ).$ In this super-critical regime, our entropy approach breaks down, since the potential $\Phi$ in the Fokker--Planck equation \ref{['EQ_Fokker_Planck']} loses its strict convexity as $\tau\to\infty$ and becomes linear, thus preventing the use of the logarithmic Sobolev inequality. Instead, our approach is based on the observation that the Fokker--Planck problem \ref{['EQ_for_w_beta_larger_1/2']} governing $w$ can be regarded as the equation associated with $\beta=1$ plus a vanishing perturbation in the interior equation, which we treat as a source term using Duhamel’s principle GigaNonlinear10. The main difficulty then lies in showing that the convolution term in \ref{['EQ_Duhamel_1']}, arising from the perturbative source in Duhamel’s formula, vanishes in $L^{1}$ as $\tau\to\infty$. While this method ultimately allows us to establish the convergence of $w$ toward $W$, it crucially relies on an a priori control of the first moment of the solution---which becomes the main obstruction when $\beta>1$ (see \ref{['R_beta_greater_1']}). In this paper, we focus specifically on the purely diffusive mechanism in the presence of a moving boundary. This simple setting exhibits a surprisingly rich variety of behaviors---as concisely illustrated in \ref{['FIG_2']}---and paves the way toward the study of more complex nonlinear and density-dependent models, such as reaction--diffusion equations: $\left\lbrace \partial_{t}u = d\partial_{zz} u + f(u),\qquadt>0,z>b(t),-d\partial_{z}u = b'(t)u ,\qquadt>0,z=b(t). \right .$ For instance, understanding the $L^{\infty}$ decay rate of the diffusive solutions is essential for studying the viability of biological populations when the reaction term takes the degenerate monostable form AronsonMultidimensional78FujitaBlowing66 $f(u) = u^{1+p}(1 - u), \qquad p\geq 0,$ This reaction function $f$ models a well-known ecological phenomenon of growth difficulty at low densities, known as the Allee effect AlleeAnimal31. In the context of our dynamic domain, the speed of the moving boundary is expected to play a crucial role in determining population survival. In particular, it may induce ecological rescues in the presence of an Allee effect. In such cases, populations that would otherwise become extinct in static domains could survive by aggregating against the moving boundary, thus benefiting from enhanced growth conditions. Conversely, a fast-moving boundary may instead become harmful to the population, amplifying intraspecific competition. This phenomenon, captured by the negative term $-u^{2+p}$ in \ref{['EQ_reaction_function']}, may ultimately drive the population to extinction, as a result of excessive aggregation of individuals in the vicinity of the boundary. Importantly, this risk persists even in the classical KPP case $f(u) = u(1 - u)$, i.e., when $p=0$ in \ref{['EQ_reaction_function']}, in sharp contrast with the well-known hair-trigger effect---namely the success of invasion in unbounded domains. As a result, the relationship between boundary speed and population persistence may become non-monotonic, since extinction may occur at both very low and very high speeds, while intermediate speeds could allow for survival. It is worth mentioning that similar non-monotonic phenomena between initial fragmentation and success rate of invasion in the presence of an Allee effect have already been documented in previous works GarnierSuccess12AlfaroPropagation24. Beyond deterministic dynamics, adopting a stochastic perspective on the underlying microscopic diffusive process could be highly valuable for better understanding the evolution of the first moment of the solution, i.e., the mean position of an individual. Such an approach would complement our deterministic analysis and may offer new insights into the dynamics, particularly in the superlinear regime $\beta > 1$---whose analytical understanding remains an open challenge (see \ref{['R_beta_greater_1']}). These directions naturally emerge from our analysis and will be the subject of continued investigation. We begin with the critical and sub-critical regimes $\beta \leq 1/2$ in \ref{['S3_diffusive_regime']}, where we prove the convergence of the solution $v$ toward Gaussian and mixed self-similar profiles via an entropy method. Then, \ref{['S4_linear_regime']} is devoted to the linear regime $\beta = 1$, giving the explicit expression of the solution and establishing the asymptotic behavior of the solution $v$. Lastly, we address in \ref{['S5_super_critical_case']} the super-critical regime $\beta > 1/2$, proving the convergence of the solution $v$ toward exponential self-similar profiles using Duhamel's principle and appropriate estimates on the arising convolution term. In this section, we focus on the diffusive scale $b(t)=\pazocal{O}(\sqrt{t})$. We prove that the solution converges at a polynomial rate to some self-similar profile, based on entropy dissipation arguments. This corresponds to \ref{['TH_asymptotic_behavior_solutions']} \ref{['TH_asymptotic_behavior_solutions']} and \ref{['TH_asymptotic_behavior_solutions']} that are treated in \ref{['SS33_beta_lower_1_2']} and \ref{['SS32_beta_1_2']}, respectively. We make the change of variable $\tau:=\log\sqrt{1+t}, \qquad y:=\frac{x}{\sqrt{1+t}}, \qquad v(t,x)= \frac{1}{\sqrt{1+t}} \; w ( \log\sqrt{1+t}, \frac{x}{\sqrt{1+t}} ).$ Then the function $w$ satisfies the following problem $\partial_{\tau}w = 2d \partial_{yy} w + \partial_{y} ((y+\psi(\tau)) w), \qquad\tau>0,\;y>0,- 2d \partial_{y} w(\tau,0) = \psi(\tau) w(\tau,0) , \qquad\tau>0,$ with $w_0=v_0$, where $\psi(\tau):= 2c\beta e^{\tau(2\beta-1)}.$ We notice that when $\beta=1/2$, we have $\psi(\tau)\equiv c$, and the associated stationary problem reads $\left\lbrace 2d \partial_{yy} W + \partial_{y}((y + c)W) = 0,\qquady > 0,-2d \partial_{y}W(0) = cW(0). \right .$ When $\beta\in[0,\frac{1}{2})$, we find that $\psi\equiv0$ if $\beta=0$, and $\psi(\tau)= 2c\beta e^{\tau(2\beta-1)}$ vanishes as $\tau\to\infty$ if $\beta<1/2$. Therefore, the corresponding stationary problem writes in this case $\left\lbrace 2d \partial_{yy} W + \partial_{y}(yW) = 0,\qquady > 0,\partial_{y}W(0) = 0. \right .$ It then follows that for $y\ge 0$, $W(y) = W(0) \exp(\!\!-\dfrac{y^{2}}{4d} - \dfrac{cy}{2d}),\text{when } \beta = {1}/{2},W(0) \exp(\!\!-\dfrac{y^{2}}{4d}),\text{when } \beta \in [0, {1}/{2}),$ where in each case $W(0)\geq 0$ is a uniquely determined constant such that the mass of the function $W$ equals $M$, i.e., $\int_{0}^{\infty}W(y)\mathrm{d} y=M$. The main strategy in dealing with the diffusive regime is the entropy approach, together with the logarithmic Sobolev inequality GrossLogarithmic75BakryAnalysis14 and the Csiszár--Kullback inequality CsiszarInformationtype67CoverElements06, which we recall below. Let $F(y)=\exp(-\Phi(y))$ be a smooth density on $\mathbb{R}_+$ and assume that there is a constant $\rho$ such that $\Phi"(y)\ge \rho>0$ for any $y>0$. Then for any nonnegative $f\in L^{1}_{y}(\mathbb{R}_{+})$ satisfying $\int_{0}^{\infty}f(y)\mathrm{d} y=\int_{0}^{\infty}F(y)\mathrm{d} y$, we have $\int_0^\infty f(y)\log(\frac{f(y)}{F(y)})\mathrm{d} y \le \frac{1}{2\rho}\int_0^\infty f(y)\left(\partial_y\log(\frac{f(y)}{F(y)})\right)^2\!\mathrm{d} y.$ For any nonnegative functions $f,F\in L^1_{y}(\mathbb{R}_+)$ such that $\int_{0}^{\infty}f(y)\mathrm{d} y=\int_{0}^{\infty}F(y)\mathrm{d} y=M$, we have that $\Vert f-F\Vert_{L^1_{y}(\mathbb{R}_{+})}^2\le 2M \int_0^\infty f(y)\log(\frac{f(y)}{F(y)}) \, \mathrm{d} y.$ Proof of Theorem \ref{['TH_asymptotic_behavior_solutions']} ii. Since $\beta=1/2$, equation \ref{['w(tau) beta<=1/2']} can be reduced to $\left\lbrace \partial_{\tau}w = 2d \partial_{yy} w + \partial_{y} ((y+c)w),\qquad\tau>0,y>0,- 2d \partial_{y} w(\tau,0) = c w(\tau,0) ,\qquad\tau>0, \right .$ with $w_0=v_0$, for which the corresponding stationary solution $W$ is given in \ref{['W-beta <= 1/2']}. The goal is to establish the convergence of the solution $w$ to problem \ref{['w-eqn:beta=1/2']} toward $W$. By introducing the relative entropy associated with \ref{['w-eqn:beta=1/2']} as $\pazocal{H}(w|W) : =\int_0^\infty w(\tau,y)\log\left(\frac{w(\tau,y)}{W(y)}\right) \mathrm{d} y \geq 0, \qquad \forall\tau\geq 0,$ which is nonnegative and vanishes if and only if $w=W$ (due to Jensen's inequality), we find that \frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W)=\int_0^\infty \partial_{\tau}w(\tau,y)(\log (w(\tau,y))+\frac{1}{4d}y^{2} + \frac{c}{2d}y) \, \mathrm{d} y=\int_0^\infty\partial_y( 2dw_{y}(\tau,y)+(y+c) w(\tau,y) ) (\log (w(\tau,y))+\frac{1}{4d}y^{2} + \frac{c}{2d}y) \, \mathrm{d} y=-\int_0^\infty( 2dw_{y}(\tau,y)+(y+c) w(\tau,y) ) (\frac{ \partial_{y}w(\tau,y)}{w(\tau,y)}+\frac{1}{2d} (y+c) ) \, \mathrm{d} y=-\int_0^\infty \left[2d\frac{ \partial_{y}w(\tau,y)^2}{w(\tau,y)}+2(y+c) \partial_{y}w(\tau,y)+\frac{1}{d}(y+c)^2w(\tau,y)\right] \, \mathrm{d} y=-\int_0^\infty w(\tau,y)( \sqrt{2d}\partial_y\log (w(\tau,y)) +\frac{1}{\sqrt{2d}}(y+c))^2 \, \mathrm{d} y=-2d \pazocal{I}(w|W), where $\pazocal{I}(w|W)$ is called the Fisher information, given by $\pazocal{I}(w|W) : =\int_0^\infty w(\tau,y)\left(\partial_y\log \left(\frac{w(\tau,y)}{W(y)}\right)\right)^2 \mathrm{d} y \qquad \forall \tau\geq 0.$ Notice that $W"(y)=\frac{1}{2d}$, then it follows from the Logarithmic Sobolev inequality that $\pazocal{H}(w|W)\le d \pazocal{I}(w|W).$ Therefore, we infer from \ref{['1']} and \ref{['2']} that \frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W)+2\pazocal{H}(w|W)=-2d \pazocal{I}(w|W)+2\pazocal{H}(w|W)\le 0, which implies from Grönwall's lemma that $\pazocal{H}(w|W)\le \pazocal{H}(w_0|W)e^{-2\tau}.$ Using the Csiszár--Kullback inequality yields that $\Vert w(\tau,y)-W(y)\Vert_{L^1_y(\mathbb{R}_{+})}\le \sqrt{2M\pazocal{H}(w_0|W)}e^{-\tau}, \qquad \forall\tau\geq 0.$ Turning back to function $v$ using \ref{['self sim 2']}, we finally obtain \ref{['beta=1/2: entropy conclu']} with $\ell := \sqrt{2M\pazocal{H}(w_0|W)}$. This concludes the proof of \ref{['TH_asymptotic_behavior_solutions']}. ∎ Proof of Theorem \ref{['TH_asymptotic_behavior_solutions']} i. Similar to the case of $\beta=1/2$, again we introduce for any $\tau\geq 0$ the relative entropy: $\pazocal{H}(w|W) =\int_0^\infty w(\tau,y)\log\left(\frac{w(\tau,y)}{W(y)}\right) \mathrm{d} y,$ and the Fisher information: \pazocal{I}(w|W)=\int_0^\infty w(\tau,y)\left(\partial_y\log \left(\frac{w(\tau,y)}{W(y)}\right)\right)^2 \, \mathrm{d} y=-\int_0^\infty \left[\frac{ \partial_{y}w(\tau,y)^2}{w(\tau,y)}+\frac{y}{d} \partial_{y}w(\tau,y)+\frac{y^2}{4d^2}w(\tau,y)\right] \mathrm{d} y. It follows that \frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W)=\int_0^\infty \partial_{\tau}w(\tau,y)(\log (w(\tau,y))+\frac{1}{4d}y^{2}) \, \mathrm{d} y\space=\int_0^\infty\partial_{y}( 2dw_{y}(\tau,y)+(y+\psi(\tau)) w(\tau,y) ) (\log (w(\tau,y))+\frac{1}{4d}y^{2}) \, \mathrm{d} y\space=-\int_0^\infty( 2dw_{y}(\tau,y)+(y+\psi(\tau)) w(\tau,y) ) (\frac{ \partial_{y}w(\tau,y)}{w(\tau,y)}+\frac{y}{2d}) \, \mathrm{d} y\space=-\int_0^\infty \left[2d\frac{ \partial_{y}w(\tau,y)^2}{w(\tau,y)}+(2y+\psi(\tau)) \partial_{y}w(\tau,y)+\frac{y}{2d}(y+\psi(\tau))w(\tau,y)\right] \mathrm{d} y\space=-2d\int_0^\infty\left( \frac{ \partial_{y}w(\tau,y)^2}{w(\tau,y)}+\frac{y}{d} \partial_{y}w(\tau,y)+\frac{y^2}{4d^2}w(\tau,y)\right)+\psi(\tau)\left(\partial_{y}w(\tau,y)+\frac{y}{2d}w(\tau,y)\right)\mathrm{d} y\space=-2d \pazocal{I}(w|W)+\psi(\tau)\left(w(\tau,0)-\frac{1}{2d}\int_0^\infty yw(\tau,y) \, \mathrm{d} y\right). In the case $\beta=0$, \ref{['EQ_entropy_diff_yet_sharp']} simply reduces to $\frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W)= -2d \pazocal{I}(w|W)$ by noticing that $\psi(\tau)\equiv 0$. Following the lines in the case $\beta=1/2$, we obtain in this case that $\Vert w(\tau,y)-W(y)\Vert_{L^1_y(\mathbb{R}_{+})}\le \sqrt{2M\pazocal{H}(w_0|W)}e^{-\tau}, \qquad \forall\tau\geq 0.$ This produces the optimal convergence rate $\pazocal{O}(t^{-1/2})$ as stated in \ref{['PROPO_BETA_0']} for the Neumann heat equation. Let us look at the case $\beta\in(0,\frac{1}{2})$. Since the precise behavior of the quantity $w(\tau,0)-\frac{1}{2d}\int_0^\infty y w(\tau,y)\mathrm{d} y$ in \ref{['EQ_entropy_diff_yet_sharp']} remains unknown, we make a rough estimate by considering the worst-case scenario, relying on the nonnegativity of both $\psi(\tau)$ and the first moment of $w$. This leads to the inequality: $\frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W)\le -2d \pazocal{I}(w|W)+\psi(\tau)w(\tau,0).$ The Logarithmic Sobolev inequality leads to $\pazocal{H}(w|W)\le d\pazocal{I}(w|W)$ which together with \ref{['3']} gives $\frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W) \le -2\pazocal{H}(w|W)+\psi(\tau)w(\tau,0).$ Before proceeding, let us give a control of $w(\tau,0)$. Assume that $\beta\in(0,\frac{1}{2})$ and that $w_0$ is bounded, compactly supported, and nonnegative in $\mathbb{R}_+$, then there exists $\Lambda>0$ such that $w(\tau,0)<\Lambda$ for every $\tau\ge 0$. Proof of \ref{['lem5']}. The proof follows from a comparison argument. Define $\overline w(\tau,y):=\exp\left(-\frac{y^2}{4d}-\frac{\psi(\tau)y}{2d}\right), \qquad \forall\tau\ge 0,\;\, \forall y\ge 0,$ then we can easily check that, for any $\tau>0$ and $y>0$, \partial_{\tau}w - 2d \partial_{yy} w - \partial_{y} ((y+\psi(\tau) )w)=-\frac{\psi'(\tau)}{2d}y\overline w(\tau,y)=\frac{2c\beta(1-2\beta)}{2d}e^{\tau(2\beta-1)}y\overline w\ge 0, and $- 2d \partial_{y} w(\tau,0) - \psi(\tau) w(\tau,0)=0.$ In addition, we can choose $\Lambda>0$ large such that $\Lambda\overline w_0(y)\ge w_0(y)=v_0(y)$ for $y\in\mathbb{R}_+$. We then conclude that $\Lambda\overline w$ is a supersolution to problem \ref{['w(tau) beta<=1/2']} in $[0,\infty)\times\mathbb{R}_{+}$ and the comparison principle implies that $0\le w(\tau,y)\le \Lambda\overline w(\tau,y)$ in $[0,\infty)\times\mathbb{R}_{+}$. This implies in particular that $0\le w(\tau,0)\le \Lambda\overline w(\tau,0)=\Lambda$ for every $\tau\ge 0$. ∎ Now we turn back to \ref{['4']} and obtain from \ref{['lem5']} that $\frac{\mathrm{d}}{\mathrm{d} \tau}\pazocal{H}(w|W)\le -2\pazocal{H}(w|W)+\psi(\tau)\Lambda.$ This ordinary differential inequality can be solved by the method of variation of parameters: $\pazocal{H}(w|W)\le \pazocal{H}(v_0|W)e^{-2\tau}+\Lambda e^{-2\tau}\int_0^\tau \psi(s)e^{2s} \, \mathrm{d} s,$ which together with \ref{['psi_beta<=1/2']} implies that \pazocal{H}(w|W)\le \pazocal{H}(v_0|W)e^{-2\tau}+2c\beta\Lambda e^{-2\tau}\int_0^\tau e^{s(2\beta-1)} e^{2s} \, \mathrm{d} s\le \pazocal{H}(v_0|W)e^{-2\tau}+\frac{2c\beta\Lambda }{2\beta+1}\left(e^{(2\beta-1)\tau}-e^{-2\tau}\right)=\left(\pazocal{H}(v_0|W)-\frac{2c\beta\Lambda }{2\beta+1}\right)e^{-2\tau}+\frac{2c\beta\Lambda }{2\beta+1}e^{(2\beta-1)\tau}\le k e^{(2\beta-1)\tau}, with $k:=\pazocal{H}(v_0|W)+\frac{2c\beta\Lambda }{2\beta+1}>0$. It then follows from the Csiszár--Kullback inequality that $\Vert w(\tau,y)-W(y)\Vert_{L^1_y(\mathbb{R}_{+})}\le \sqrt{2Mk}e^{(\beta-\frac{1}{2})\tau}, \qquad \forall\tau>0.$ Therefore, turning back to the function $v$ using \ref{['self sim 2']}, we obtain \ref{['beta<1/2: entropy + Duhamel']} with $\ell : = \sqrt{2Mk}$. This completes the proof of \ref{['TH_asymptotic_behavior_solutions']}. ∎ We consider here the linear case $\beta=1$, corresponding to the moving boundary $b(t)=ct$. In this regime, equation \ref{['v-eqn']}$|_{b(t)=ct}$ simplifies to $\left\lbrace \partial_{t}v = d \partial_{xx}v+c\partial_x v,\qquadt>0,x>0,-d\partial_{x}v = cv ,\qquadt>0,x=0. \right .$ This section is dedicated to the proof of \ref{['TH_fundamental_sol_for_beta_1']} which splits in two parts. The first concerns the explicit construction of the solution to \ref{['EQ_linear_problem_on_v']}, relying on suitable extension arguments. The second establishes the asymptotic $L^{1}$-convergence of $v(t,)$ toward the exponential stationary profile $V$, based on a detailed analysis of the fundamental solution. These two aspects are treated in \ref{['SS41_fundamental_sol']} and \ref{['SS42_asymptotic_behavior']}, respectively. Proof of Theorem \ref{['TH_fundamental_sol_for_beta_1']} i. We anticipate an extension $\widetilde{v}_{0}$ of $v_{0}$ in $\lbrace x<0\rbrace$, such that the solution $\widetilde{v}$ to $\left\lbrace \partial_{t}\widetilde{v} = d\partial_{xx} \widetilde{v}+c\partial_{x} \widetilde{v},\qquadt>0,x\in\mathbb{R},\widetilde{v}|_{t=0} = \widetilde{v}_{0} ,\qquadt=0,x\in \mathbb{R}, \right .$ satisfies $\widetilde{v}(t,x)|_{x>0}= v(t,x)$. To achieve this, we need to check that the flux condition $-d\partial_{x}\widetilde{v} = c \widetilde{v}$ is verified at $x=0$ for any positive time---this recovers the Robin boundary condition, namely second line in \ref{['EQ_linear_problem_on_v']}. Explaining the solution $\widetilde{v}$ to \ref{['EQ_tilde_v']} by convolving $v_{0}$ with the biased heat kernel $G(t,x+ct)$, this condition reads $\int_{\xi=0}^{\infty} G(t,ct+\xi)[c \widetilde{v}_{0}(-\xi) + d \partial_{x} \widetilde{v}_{0}(-\xi) ] + G(t,ct-\xi)[c v_{0}(\xi) + d \partial_{x} v_{0}(\xi) ] \, \mathrm{d} \xi = 0,$ and can be recast, thanks to the algebraic trick $G(t,ct+\xi) = G(t,ct-\xi)e^{-\frac{c}{d}\xi}$, as $\int_{\xi=0}^{\infty} G(t,ct-\xi) \underbrace{[ e^{-\frac{c}{d}\xi} (c \widetilde{v}_{0}(-\xi) + d \partial_{x} \widetilde{v}_{0}(-\xi)) + (c v_{0}(\xi) + d \partial_{x} v_{0}(\xi)) ]}_{\text{We ask then the vanishing of this quantity for all $\xi>0$.}} \mathrm{d} \xi = 0.$ Therefore, for $x<0$, $\widetilde{v}_{0}$ should solve the following linear ODE: $\left\lbrace d\partial_{x} \widetilde{v}_{0}(x) + c \widetilde{v}_{0}(x) = - e^{-\frac{c}{d}x} [d\partial_{x}v_{0}(-x) + c v_{0}(-x)],\qquadx<0,\widetilde{v}_{0}(0) = v_{0}(0) ,\qquadx=0, \right .$ where the second line ensures the compatibility of $\widetilde{v}_{0}$ regarding the Robin boundary condition at $x=0$. As a result, the extended initial data $\widetilde{v}_{0}$ is given by $\widetilde{v}_{0}(x) = v_{0}(x),\quad \text{if }x\geq 0,e^{-\frac{c}{d}x}[v_{0}(-x) + \frac{c}{d}\int_{0}^{-x}v_{0}(\xi) \, \mathrm{d} \xi],\quad \text{if }x\leq 0.$ We can now provide a first expression for $v$: v(t,x)= \int_{\xi=0}^{\infty} G(t,x+ct-\xi)v_{0}(\xi) \, \mathrm{d} \xi + \int_{\xi=0}^{\infty} G(t,x+ct+\xi)e^{\frac{c}{d}\xi}v_{0}(\xi) \, \mathrm{d} \xi\space + \frac{c}{d} \int_{\xi=0}^{\infty} G(t,x+ct+\xi) e^{\frac{c}{d}\xi} \int_{\omega=0}^{\xi} v_{0}(\omega) \mathrm{d} \omega \, \mathrm{d} \xi, which can be recast into \ref{['EQ_fundamental_sol_v_linear']} with the kernel $H$ given by \ref{['EQ_heat_kernel_linear']} by using Fubini's theorem on \ref{['EQ_almost_kernel_v_2']}. This concludes the proof of \ref{['TH_fundamental_sol_for_beta_1']}. ∎ Proof of Theorem \ref{['TH_fundamental_sol_for_beta_1']} ii. This proof is divided into two steps. We show first that there exist some positive constants $\tilde{\ell}$, $\tilde{k}$ and $t_{0}$ such that $\Vert v(t,x) - V(x) \Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})} \leq \tilde{\ell} e^{-\tilde{k} t}, \qquad \forall t>t_{0}.$ We prove then the uniform boundedness of $v$ which enables, combined with \ref{['EQ_cv_to_V_when_beta_equals_1_first']}, to reach \ref{['EQ_cv_to_V_when_beta_equals_1']} for any $t>0$. As a stationary solution to \ref{['EQ_linear_problem_on_v']}, the function $V$, defined in \ref{['EQ_def_V_beta_equals_1']}, can be rewritten using \ref{['EQ_almost_kernel_v_1']}-\ref{['EQ_almost_kernel_v_2']} as V(x)= \int_{\xi=0}^{\infty} G(t,x+ct-\xi)V(\xi) \, \mathrm{d} \xi + \int_{\xi=0}^{\infty} G(t,x+ct+\xi)e^{\frac{c}{d}\xi}V(\xi) \, \mathrm{d} \xi\space + \frac{c}{d} \int_{\xi=0}^{\infty} G(t,x+ct+\xi) e^{\frac{c}{d}\xi} \int_{\omega=0}^{\xi} V(\omega) \mathrm{d} \omega \, \mathrm{d} \xi, for any $t>0$. This equality, combined with \ref{['EQ_almost_kernel_v_1']}-\ref{['EQ_almost_kernel_v_2']}, allows us to write $v(t,x)-V(x)$ as v(t,x)-V(x)= \int_{\xi=0}^{\infty} G(t,x+ct-\xi)(v_{0}(\xi)-V(\xi)) \, \mathrm{d} \xi\space + \int_{\xi=0}^{\infty} G(t,x+ct+\xi)e^{\frac{c}{d}\xi}(v_{0}(\xi)-V(\xi)) \, \mathrm{d} \xi\space - \frac{c}{d} \int_{\xi=0}^{\infty} G(t,x+ct+\xi) e^{\frac{c}{d}\xi} \int_{\omega=\xi}^{\infty} (v_{0}(\omega)-V(\omega)) \mathrm{d} \omega \, \mathrm{d} \xi. Notice, in the last integral above, we replaced $\int_{\omega=0}^{\xi} (v_{0}-V) \qquad \text{by} \qquad -\int_{\omega=\xi}^{\infty} (v_{0}-V),$ owing to the fact that $\int_{0}^{\infty} v_{0} = \int_{0}^{\infty} V$. This is the only point of the proof where we use this assumption. Taking now the $L^{1}$-norm of $v(t,)-V$, we have then \Vert v(t,x)-V(x) \Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})}\leq \underbrace{\int_{\xi=0}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert \int_{x=ct}^{\infty} G(t,x -\xi) \mathrm{d} x \, \mathrm{d} \xi}_{=:I_{1}(t)}\space + \underbrace{\int_{\xi=0}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert e^{\frac{c}{d}\xi} \int_{x=ct}^{\infty} G(t,x +\xi) \mathrm{d} x \, \mathrm{d} \xi}_{=:I_{2}(t)}\space + \frac{c}{d} \underbrace{\int_{\xi=0}^{\infty} \int_{\omega=\xi}^{\infty} \left\vert v_{0}(\omega)-V(\omega) \right\vert \mathrm{d} \omega \, e^{\frac{c}{d}\xi} \int_{x=ct}^{\infty} G(t,x +\xi) \mathrm{d} x \, \mathrm{d} \xi,}_{=:I_{3}(t)} where we have performed the change of variable $\bar{x} = x+ct$, and then renamed $\bar{x}$ back to $x$, and applied Fubini's theorem to interchange the integration orders. Our objective is now to bound $I_{1}$, $I_{2}$ and $I_{3}$ from above. Before proceeding, let us state the following lemma which we shall rely on in the next steps. For any $t>0$, there holds $\space\int_{x=ct}^{\infty}G(t,x -\xi)\, \mathrm{d} x \leq \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ], \qquad \forall \xi \in (0,ct),$ $e^{\frac{c}{d}\xi} \int_{x=ct}^{\infty}G(t,x +\xi)\, \mathrm{d} x \leq \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ], \qquad \forall \xi \in (0,\infty).$ Proof of \ref{['LE_beta_equals_1']}. Expanding the integral in the left-hand side of \ref{['EQ_lemma_beta_equals_1_a']}, we have, for any $\xi \in (0,ct)$, \int_{x=ct}^{\infty} G(t,x-\xi) \, \mathrm{d} x=\frac{1}{\sqrt{4\pi dt}} \int_{x=ct}^{\infty} \exp [ -\left(\frac{(x - \xi)^{2}}{4dt}\right) ] \, \mathrm{d} x=\frac{1}{2} \, \text{erfc}\left(\frac{c}{2\sqrt{d}}\sqrt{t}-\frac{\xi}{2\sqrt{dt}}\right), where we performed the change of variable $\bar{x} = (x-\xi)/(2\sqrt{dt})$, and then renamed $\bar{x}$ back to $x$, to transition from the first line to the second. Now, since $\frac{c}{2\sqrt{d}}\sqrt{t}-\frac{\xi}{2\sqrt{dt}}$ is positive (as $0 < \xi < ct$), and using the fact that $\frac{1}{2}\text{erfc}(\gamma) \leq \exp(-\gamma^{2})$ for any $\gamma>0$, we finally reach \ref{['EQ_lemma_beta_equals_1_a']}. Similarly to the proof of \ref{['EQ_lemma_beta_equals_1_a']}, we have e^{\frac{c}{d}\xi}\int_{x=ct}^{\infty} G(t,x+\xi) \, \mathrm{d} x= \frac{e^{\frac{c}{d}\xi}}{\sqrt{4\pi dt}} \int_{x=ct}^{\infty} \exp [ -\left(\frac{(x + \xi)^{2}}{4dt}\right) ] \, \mathrm{d} x=\frac{e^{\frac{c}{d}\xi}}{2} \, \text{erfc}\left(\frac{c}{2\sqrt{d}}\sqrt{t}+\frac{\xi}{2\sqrt{dt}}\right)\leq e^{\frac{c}{d}\xi} \, \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} + \frac{\xi}{2\sqrt{dt}}\right)^{2} ]=\exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ], which establishes \ref{['EQ_lemma_beta_equals_1_b']}. ∎ We are now ready to begin estimating $I_1$, $I_2$, and $I_3$. $\bullet$ Control of $I_{1}(t)$. Splitting the integral $\int_{\xi=0}^{\infty}$ into $\int_{\xi=0}^{\frac{c}{2}t}+\int_{\xi=\frac{c}{2}t}^{\infty}$ in the expression of $I_{1}$, we have I_{1}(t)= \int_{\xi=0}^{\frac{c}{2}t} \left\vert v_{0}(\xi)-V(\xi) \right\vert \int_{x=ct}^{\infty} G(t,x -\xi) \mathrm{d} x \, \mathrm{d} \xi\space + \int_{\xi=\frac{c}{2}t}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert \int_{x=ct}^{\infty} G(t,x -\xi) \mathrm{d} x \, \mathrm{d} \xi. Now substituting \ref{['EQ_lemma_beta_equals_1_a']} into \ref{['EQ_control_I1_a']}, and noticing that $\int_{x=ct}^{\infty}G(t,x-\xi)\mathrm{d} x\leq 1$ in \ref{['EQ_control_I1_b']} yields \hbox{$I_{1}(t)~$}\hbox{$\leq \int_{\xi=0}^{\frac{c}{2}t} \left\vert v_{0}(\xi)-V(\xi) \right\vert \times \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ] \, \mathrm{d} \xi + \int_{\xi=\frac{c}{2}t}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert \, \mathrm{d} \xi$}\hbox{$\leq \exp [ -\frac{c^{2}}{16d}t ] \int_{\xi=0}^{\frac{c}{2}t} \left\vert v_{0}(\xi)-V(\xi) \right\vert \, \mathrm{d} \xi + \int_{\xi=\frac{c}{2}t}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert \, \mathrm{d} \xi.$} Remember at this point that $v_{0}$ is compactly supported. Hence, for $t_{0}>0$ chosen sufficiently large so that $\text{supp}(v_{0})\cap\{\xi>\frac{c}{2}t_{0}\} = \varnothing$, we have, for any $t>t_{0}$, $\left\vert v_{0}(\xi)-V(\xi) \right\vert = \left\vert V(\xi) \right\vert =\left\vert \int_{\omega=0}^{\infty}v_{0}(\omega)\mathrm{d} \omega \right\vert \frac{c}{d} \, e^{-\frac{c}{d}\xi}, \qquad \forall \xi > \frac{c}{2}t.$ As a consequence, I_{1}(t)\leq \Vert v_{0}-V \Vert_{L(\mathbb{R}_{+}^{})} \times e^{-\frac{c^{2}}{16d}t} + \Vert v_{0} \Vert_{L^{1}} \times \int_{\xi = \frac{c}{2}t}^{\infty} \frac{c}{d} \, e^{-\frac{c}{d}\xi} \, \mathrm{d} \xi= \Vert v_{0}-V \Vert_{L(\mathbb{R}_{+}^{})} \times e^{-\frac{c^{2}}{16d}t} + \Vert v_{0} \Vert_{L^{1}} \times e^{-\frac{c^{2}}{2d}t}\leq \tilde{\ell}_{1}e^{-\tilde{k}_{1}t}, for some positive constants $\tilde{\ell}_{1}$ and $\tilde{k}_{1}$. $\bullet$ Control of $I_{2}(t)$. From \ref{['EQ_lemma_beta_equals_1_b']} in \ref{['LE_beta_equals_1']}, we have \hbox{$I_{2}(t) ~$}\hbox{$\leq \int_{\xi=0}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert \times \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ] \, \mathrm{d} \xi,$}\hbox{$\leq \int_{\xi=0}^{\frac{c}{2}t} \left\vert v_{0}(\xi)-V(\xi) \right\vert \times \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ] \, \mathrm{d} \xi + \int_{\xi=\frac{c}{2}t}^{\infty} \left\vert v_{0}(\xi)-V(\xi) \right\vert \, \mathrm{d} \xi$}, which is precisely the right-hand-side of \ref{['EQ_reuse_I1']}. Following then the same computation that allowed to reach \ref{['EQ_upper_bound_beta_equals_1_a']} in the control of $I_{1}$, we get $I_{2}(t)\leq \tilde{\ell}_{2}e^{-\tilde{k}_{2}t},$ with $(\tilde{\ell}_{2} , \tilde{k}_{2}) = (\tilde{\ell}_{1} , \tilde{k}_{1})$. $\bullet$ Control of $I_{3}(t)$. Similar arguments as in the controls of $I_{1}$ and $I_{2}$ yield I_{3}(t)= \int_{\xi=0}^{\infty} \int_{\omega=\xi}^{\infty} \left\vert v_{0}(\omega)-V(\omega) \right\vert \mathrm{d} \omega \, e^{\frac{c}{d}\xi} \int_{x=ct}^{\infty} G(t,x +\xi) \, \mathrm{d} x \, \mathrm{d} \xi\leq \int_{\xi=0}^{\infty} \int_{\omega=\xi}^{\infty} \left\vert v_{0}(\omega)-V(\omega) \right\vert \mathrm{d} \omega \times \exp [ -\left(\frac{c}{2\sqrt{d}}\sqrt{t} - \frac{\xi}{2\sqrt{dt}}\right)^{2} ] \, \mathrm{d} \xi\leq e^{-\frac{c^{2}}{16d}t} \int_{\xi=0}^{\frac{c}{2}t} \int_{\omega = \xi}^{\infty} \left\vert v_{0}(\omega)-V(\omega) \right\vert \mathrm{d} \omega\,\mathrm{d} \xi + \Vert v_{0} \Vert_{L^{1}} \int_{\xi=\frac{c}{2}t}^{\infty} \int_{\omega = \xi}^{\infty} \frac{c}{d}e^{-\frac{c}{d}\omega} \, \mathrm{d} \omega \, \mathrm{d} \xi\leq e^{-\frac{c^{2}}{16d}t} \int_{\xi=0}^{\infty} \int_{\omega = \xi}^{\infty} \left\vert v_{0}(\omega)-V(\omega) \right\vert \mathrm{d} \omega\,\mathrm{d} \xi + \frac{d}{c} \Vert v_{0} \Vert_{L^{1}} e^{-\frac{c^{2}}{2d}t}= e^{-\frac{c^{2}}{16d}t} \int_{\omega=0}^{\infty} \omega \left\vert v_{0}(\omega)-V(\omega) \right\vert \, \mathrm{d} \omega + \frac{d}{c} \Vert v_{0} \Vert_{L^{1}} e^{-\frac{c^{2}}{2d}t}, where we applied Fubini's theorem to derive the last line. It remains to observe that the integral in \ref{['EQ_moment']} is bounded, since both $\left\vert v_{0} \right\vert$ and $\left\vert V \right\vert$ have finite first moment. This results in $I_{3}(t) \leq \tilde{\ell}_{1}e^{-\tilde{k}_{1}t},$ for some positive constants $\tilde{\ell}_{3}$ and $\tilde{k}_{3}$. By gathering \ref{['EQ_upper_bound_beta_equals_1_a']}, \ref{['EQ_upper_bound_beta_equals_1_b']} and \ref{['EQ_upper_bound_beta_equals_1_c']}, we can eventually find some positive constants $\tilde{\ell}$, $\tilde{k}$ and $t_{0}$ so that \ref{['EQ_cv_to_V_when_beta_equals_1_first']} holds for any $t>t_{0}$. We now focus on the boundedness of the solution $v$ to extend \ref{['EQ_cv_to_V_when_beta_equals_1_first']} for any $t>0$ (up to change $(\tilde{\ell},\tilde{k})$ into $({\ell},{k})$). Owing to the comparison principle, there are two real constants $\underline{\lambda}\leq \overline{\lambda}$ such that $v$ is sandwiched between the two stationary solutions $\underline{\lambda} \, \frac{c}{d}e^{-\frac{c}{d}x}$ and $\overline{\lambda} \, \frac{c}{d}e^{-\frac{c}{d}x}$, namely $\underline{\lambda} \, \frac{c}{d}e^{-\frac{c}{d}x} \leq v(t,x) \leq \overline{\lambda} \, \frac{c}{d}e^{-\frac{c}{d}x}, \qquad \forall t>0, \, \forall x>0.$ Hence, $(\underline{\lambda}-M) \, \frac{c}{d}e^{-\frac{c}{d}x} \leq v(t,x) -V(x) \leq (\overline{\lambda}-M) \, \frac{c}{d}e^{-\frac{c}{d}x}, \qquad \forall t>0, \, \forall x>0,$ and therefore, for $\Lambda : = \max({\vert{\underline{\lambda}-M}\vert , \vert{\overline{\lambda}-M}\vert})$, $\left\vert v(t,x)-V(x) \right\vert \leq \Lambda \, \frac{c}{d}e^{-\frac{c}{d}x} \qquad \forall t>0, \, \forall x>0.$ Integrating the inequality \ref{['EQ_ready_to_integrate']} over $\mathbb{R}_{+}$ then yields $\Vert v(t,x) \Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})} \leq \Lambda, \qquad\forall t>0.$ Finally, we only need to combine \ref{['EQ_cv_to_V_when_beta_equals_1_first']} and \ref{['EQ_cv_to_V_when_beta_equals_1_second']} to eventually establish \ref{['EQ_cv_to_V_when_beta_equals_1']} for any $t>0$. This concludes the proof of \ref{['TH_fundamental_sol_for_beta_1']}. ∎ In this last section, we address the super-critical regime $b(t)\sim ct^{\beta}$ with $\beta > 1/2$. We prove that the solution converges at a polynomial rate to some self-similar profile, based on Duhamel's principle and refined kernel estimates. This corresponds to point \ref{['TH_asymptotic_behavior_solutions']} of \ref{['TH_asymptotic_behavior_solutions']}. We make the change of variable $v(t,x) : = \frac{b'(t)}{c\beta} \, w (\, \overbrace{\int_{s=0}^{t} [\frac{b'(s)}{c\beta}]^{2} ds}^{=:\tau}, \; \overbrace{\frac{b'(t)}{c\beta}x\newline}^{=:y} \,),$ where we recall that the function $b$ is defined in \ref{['EQ_def_algebraic_b']}, and that the function $v$ satisfies the equations given in \ref{['v-eqn']}. The relation between the variables $t$ and $\tau$ writes then $\tau = \frac{1}{2\beta-1} [(1+t)^{2\beta-1} - 1] \quad \iff \quad t = (1+(2\beta-1)\tau)^{\frac{1}{2\beta-1}} - 1,$ and the new unknown $w$ satisfies the following problem $\left\lbrace \partial_{\tau}w = d \partial_{yy} w + \partial_{y} [ (c\beta-\eta(\tau)y)w],\qquad\tau>0,y>0,- d \partial_{y} w = c \beta w ,\qquad\tau>0,y=0,w|_{\tau=0} = v_{0} ,\qquad\tau=0,y>0, \right .$ where $\eta(\tau) := \frac{\beta-1}{1 + (2\beta-1)\tau}.$ Notice that $\eta(\tau)$ has the sign of $\beta-1$ and vanishes as $\tau\to\infty$ in the regime $\beta>1/2$. As a result, the stationary problem associated to \ref{['EQ_for_w_beta_larger_1/2']} writes $\left\lbrace \partial_{yy} W = - \frac{c\beta}{d} \partial_{y} W,\qquady>0,- d \partial_{y} W|_{y=0} = c\beta W|_{y=0} ,\qquady=0, \right .$ which gives for $y>0$, $W(y) = W(0) \exp\left(-\frac{c\beta}{d}y\right),$ where $W(0)$ is uniquely determined so that the total masses of $v_{0}$ and $W$ are equal, that is $\int_{0}^{\infty}W(y)\mathrm{d} y=M$. Proof of Theorem \ref{['TH_asymptotic_behavior_solutions']} iii. Observe in the first line of \ref{['EQ_for_w_beta_larger_1/2']}, that $\eta(\tau) \partial_{y}(yw)$ is a vanishing perturbation as $\tau$ goes to $+\infty$. Hence, it is natural to consider this quantity as an additional source term to the following evolution problem $\left\lbrace \partial_{\tau}\tilde{w} = d \partial_{yy} \tilde{w} + c\beta \partial_{y}\tilde{w},\qquad\tau>0,y>0,- d \partial_{y} \tilde{w}|_{y=0} = c\beta \tilde{w}|_{y=0} ,\qquad\tau>0,y=0,\tilde{w}|_{\tau=0} = v_{0} ,\qquad\tau=0,y>0, \right .$ which has the same form as the linear case \ref{['EQ_linear_problem_on_v']}, with the constant $c$ replaced by $c\beta$ now. Therefore, it follows from \ref{['TH_fundamental_sol_for_beta_1']} that there are some positive constants $\ell$ and $k$ such that $\Vert \tilde{w}(\tau,y) - \underbrace{\frac{c\beta}{d}\left(\int_{\xi=0}^{\infty}v_{0}(\xi)\mathrm{d} \xi\right)}_{=: W(0)}e^{-\frac{c\beta}{d}y} \Vert _{L^{1}_{y}(\mathbb{R}_{+}^{})} \leq \ell e^{-k \tau}, \qquad \forall \tau>0.$ As a consequence, to establish \ref{['EQ_asymptotic_behavior_solutions_larger_half']}, we only need to bound $\left\Vert {w}(\tau,y) - \tilde{w}(\tau,y) \right\Vert_{L^{1}_{y}(\mathbb{R}_{+}^{})}$ which can be rewritten with the Duhamel's principle GigaNonlinear10 as \left\Vert {w}(\tau,y) - \tilde{w}(\tau,y) \right\Vert_{L^{1}_{y}(\mathbb{R}_{+}^{})}= \left\Vert \int_{s=0}^{\tau} {\pazocal{S}}_{s}[-\eta(\tau-s) \partial_{y}(yw(\tau-s,y))] \mathrm{d} s \right\Vert_{L^{1}_{y}(\mathbb{R}_{+}^{})} where $({\pazocal{S}}_{s})_{s>0}$ is the $C_{0}$-semigroup on $L^{1}_{y}(\mathbb{R}_{+}^{})$ associated with the linear problem \ref{['EQ_for_w_beta_larger_1/2_linear']}. Expanding the expression of this semigroup with the fundamental solution $H$ \ref{['EQ_heat_kernel_linear']} yields $\left\Vert {w}(\tau,y) - \tilde{w}(\tau,y) \right\Vert_{L^{1}_{y}} \leq \int_{s=0}^{\tau} \vert\eta(\tau-s)\vert \underbrace{ \int_{y=0}^{\infty} \vert \overbrace{ \int_{\xi=0}^{\infty} -H(s,y,\xi) \partial_{\xi}[\xi w(\tau-s,\xi)] \mathrm{d} \xi}^{=: I(\tau,s,y) }\vert \,\mathrm{d} y}_{=:J(\tau,s)} \,\mathrm{d} s.$ Integrating $I$ by parts gives $I(\tau,s,y) = \int_{\xi=0}^{\infty} \partial_{\xi}H(s,y,\xi) [\xi w(\tau-s,\xi)] \,\mathrm{d} \xi,$ where $\partial_{\xi}H$ can be computed from the expression \ref{['EQ_heat_kernel_linear']} of $H$ (with $c$ replaced by $c\beta$) : $\partial_{\xi}H(s,y,\xi) = \frac{1}{2ds\sqrt{4\pi ds}} \left[ (y+c\beta s-\xi)e^{-\frac{(y+c\beta s-\xi)^{2}}{4ds}} - (y+c\beta s+\xi)e^{-\frac{(y+c\beta s+\xi)^{2}}{4ds}}e^{\frac{c\beta }{d}\xi} \right].$ Hence, by Fubini's theorem, $\hbox{$J(\tau,s) \leq \int_{\xi=0}^{\infty} \frac{\xi w(\tau-s,\xi)}{2ds\sqrt{4\pi ds}} \underbrace{ \int_{y=0}^{\infty} \left[ \left\vert y+c\beta s-\xi \right\vert e^{-\frac{(y+c\beta s-\xi)^{2}}{4ds}} + (y+c\beta s+\xi)e^{-\frac{(y+c\beta s+\xi)^{2}}{4ds}}e^{\frac{c\beta }{d}\xi} \right] \mathrm{d} y}_{=:K(\tau,s,\xi)} \mathrm{d} \xi,$}$ where $K$ can be bounded as follows K(\tau,s,\xi)= \int_{\omega=c\beta s-\xi}^{\infty} \left\vert \omega \right\vert e^{-\frac{\omega^{2}}{4ds}} \,\mathrm{d} \omega + e^{\frac{c\beta }{d}\xi}\int_{\omega=c\beta s+\xi}^{\infty} {\omega}e^{-\frac{\omega^{2}}{4ds}} \,\mathrm{d} \omega\leq \int_{\mathbb{R}}^{} \left\vert \omega \right\vert e^{-\frac{\omega^{2}}{4ds}} \,\mathrm{d} \omega + 2ds \, e^{\frac{c\beta }{d}\xi} e^{-\frac{(c\beta s+\xi)^{2}}{4ds}}= 4ds + 2ds \, e^{-\frac{(c\beta s-\xi)^{2}}{4ds}}\leq 6ds. As a result $J(\tau,s) \leq \frac{3}{\sqrt{4\pi ds}} \int_{\xi=0}^{\infty} \xi w(\tau-s, \xi) \,\mathrm{d} \xi.$ To proceed with our analysis, we need to establish a uniform upper bound on the first moment of $w$: Assume that $\beta\in(\frac{1}{2},{1}]$ and that $v_{0}$ is bounded, compactly supported, and nonnegative in $\mathbb{R}_{+}$. Then there exists $\Lambda>0$, depending on $c$, $d$, $\beta$, $\Vert v_{0} \Vert_{L^{\infty}}$ and $\max(\text{supp}(v_{0}))$, such that $\int_{\xi=0}^{\infty} \xi w(\tau, \xi) \,\mathrm{d} \xi \leq \Lambda, \qquad \forall \tau > 0.$ Proof of Lemma \ref{['LE_upper_bound_first_moment_beta_1_2']}. The proof follows from a comparison argument. Define, for $\lambda>0$, $\overline{w}(\tau,y):= \lambda \exp\left( -\dfrac{c\beta }{d}y +\dfrac{\eta(\tau)}{2d}y^{2} \right),\quad\tau>0,y>0.$ Then we can check that, for any $\tau>0$ and $y>0$, $\pazocal{L}\overline{w} : = \partial_{t}\overline{w} - d \partial_{yy}\overline{w} - \partial_{y} [ (c\beta -\eta(\tau)y)\overline{w}] = \frac{y^{2}\eta'(t)}{2d}\overline{w},$ and for any $\tau>0$, $-d\partial_{y}\overline{w}(\tau,0) - c\beta \overline{w}(\tau,0) = 0, \qquad \text{at $y=0$}.$ In particular, $\text{sign}(\pazocal{L}\overline{w}) = \text{sign}(\eta'(\tau)) = \text{sign}(1-\beta)$, so that $\overline{w}$ is a super solution to problem \ref{['EQ_for_w_beta_larger_1/2_linear']} since $\frac{1}{2}<\beta\leq 1$. As a result, by taking $\lambda>0$ large enough so that $\overline{w}|_{\tau=0}>w|_{\tau=0}$ for any $y>0$, we have $\overline{w}>w$ for any $\tau>0$ and $y>0$. Considering that $\eta$ is nonpositive, we can finally write $\int_{\xi=0}^{\infty} \xi \, w(\tau, \xi) \,\mathrm{d} \xi \leq \int_{\xi=0}^{\infty} \xi \, \overline{w}(\tau, \xi) \,\mathrm{d} \xi \leq \lambda \int_{\xi=0}^{\infty} \xi \, e^{-\frac{c\beta }{d}\xi} \,\mathrm{d} \xi =: \Lambda.$ ∎ We now turn back to \ref{['EQ_Duhamel_2']}, and obtain from \ref{['LE_upper_bound_first_moment_beta_1_2']} and \ref{['EQ_control_J_beta_larger_1_2']} that \left\Vert {w}(\tau,y) - \tilde{w}(\tau,y) \right\Vert_{L^{1}_{y}(\mathbb{R}_{+}^{})}\leq \int_{s=0}^{\tau} \vert\eta(\tau-s)\vert \frac{3\Lambda}{\sqrt{4\pi ds}} \,\mathrm{d} s= \frac{3\Lambda(1-\beta)}{(2\beta-1)\sqrt{4\pi d}} \underbrace{\int_{s=0}^{\tau} \frac{1}{\frac{1}{2\beta-1}+\tau-s} \, \frac{1}{\sqrt{s}} \,\mathrm{d} s}_{= : L(\tau)}. It remains to control the convolution $L(\tau)$. To achieve this, let us set $s=\tau\sigma$ in the integral in $L(\tau)$. This yields L(\tau)= \int_{\sigma=0}^{1} \frac{\tau}{\frac{1}{2\beta-1}+\tau-\tau\sigma} \, \frac{1}{\sqrt{\tau\sigma}} \,\mathrm{d} \sigma.= \int_{\sigma=0}^{1/2} \frac{\tau}{\frac{1}{2\beta-1}+\tau-\tau\sigma} \, \frac{1}{\sqrt{\tau\sigma}} \,\mathrm{d} \sigma + \int_{\sigma=1/2}^{1} \frac{\tau}{\frac{1}{2\beta-1}+\tau-\tau\sigma} \, \frac{1}{\sqrt{\tau\sigma}} \,\mathrm{d} \sigma.\leq \frac{2}{\sqrt{\tau}} \int_{\sigma=0}^{1/2} \frac{\mathrm{d} \sigma}{\sqrt{\sigma}} + \frac{\sqrt{2}}{\sqrt{\tau}} \int_{\sigma=1/2}^{1} \frac{\tau}{\frac{1}{2\beta-1}+\tau-\tau\sigma} \,\mathrm{d} \sigma.= \frac{2\sqrt{2}}{\sqrt{\tau}} + \frac{\sqrt{2}\log\left(1+\frac{2\beta-1}{2}\,\tau\right)}{\sqrt{\tau}}. Now using the relation \ref{['EQ_rel_between_tau_and_t']} between the variables $\tau$ and $t$, namely $\tau = \frac{1}{2\beta-1} [(1+t)^{2\beta-1} - 1] \quad \iff \quad t = (1+(2\beta-1)\tau)^{\frac{1}{2\beta-1}} - 1,$ it can be shown that, for any $t>1$, $\frac{1}{\sqrt{\tau}} \leq \frac{\sqrt{(2\beta-1)/(1-2^{1-2\beta})}}{(1+t)^{\beta-\frac{1}{2}}} \qquad \text{and} \qquad \log\left(1+\frac{2\beta-1}{2}\,\tau\right) \leq (2\beta-1)\log(1+t).$ As a result, there exists $k>0$, depending on $c$, $d$, $\beta$, $\Vert v_{0} \Vert_{L^{\infty}}$ and $\max(\text{supp}(v_{0}))$, such that $\left\Vert {w}(\tau,y) - \tilde{w}(\tau,y) \right\Vert_{L^{1}_{y}(\mathbb{R}_{+}^{})} \leq \frac{k \log(1+t)}{(1+t)^{\beta-\frac{1}{2}}}, \qquad \forall t>1.$ Combining \ref{['EQ_almost_there_beta_ge_1_2']} and the exponential convergence of $\tilde{w}(\tau,)$ toward $W(y)=W(0) e^{-\frac{c\beta}{d}y}$ (see \ref{['TH_fundamental_sol_for_beta_1']} with $c$ is replaced by $c\beta$), we reach, up to increasing $\ell$, $\left\Vert {w}(\tau,y) - W(y) \right\Vert_{L^{1}_{y}(\mathbb{R}_{+}^{})} \leq \frac{\ell \log(1+t)}{(1+t)^{\beta-\frac{1}{2}}}, \qquad \forall t>1.$ Now making the change of variable $y=\frac{b'(t)}{c\beta}\,x$ in the left-hand-side of \ref{['EQ_almost_there_beta_ge_1_2_bis']} and recalling that $v(t,x) = \frac{b'(t)}{c\beta}w(\tau,y)$ (see \ref{['EQ_rescaling_super_sqrt']}), we finally obtain $\left\Vert {v}(t,x) - \frac{b'(t)}{c\beta} W(\frac{b'(t)}{c\beta}\,x) \right\Vert_{L^{1}_{x}(\mathbb{R}_{+}^{})} \leq \frac{\ell \log(1+t)}{(1+t)^{\frac{1}{2}}}, \qquad \forall t>1,$ which gives the control \ref{['EQ_asymptotic_behavior_solutions_larger_half']} announced in \ref{['TH_asymptotic_behavior_solutions']} and concludes the proof. ∎ S.T. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 865711). M.Z. acknowledges support from the ANR via the project ReaCh ANR-23-CE40-0023-01, and from the Occitanie region, the European Regional Development Fund (ERDF), and the French government, through the France 2030 project managed by the National Research Agency (ANR) ANR-22-EXES-0015. The authors would like to thank everyone with whom they had the opportunity to discuss this paper, in particular http://math.univ-lyon1.fr/homes-www/lepoutre/, whose insightful advice has marked significant milestones in advancing this work. M. Alfaro, H. Berestycki, and G. Raoul, The Effect of Climate Shift on a Species Submitted to Dispersion, Evolution, Growth, and Nonlocal Competition, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 562--596.M. Alfaro, F. Hamel, and L. Roques, Propagation or extinction in bistable equations: The non-monotone role of initial fragmentation, Discrete and Continuous Dynamical Systems - S, 17 (2024), pp. 1460--1484.W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, 1931.J. Allwright, Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain, Proceedings of the Edinburgh Mathematical Society, 65 (2022), pp. 53--79., Reaction--diffusion on a time-dependent interval: Refining the notion of 'critical length', Communications in Contemporary Mathematics, 25 (2023), p. 2250050.D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30 (1978), pp. 33--76.D. Bakry, I. Gentil, and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, vol. 348 of Grundlehren Der Mathematischen Wissenschaften, Springer International Publishing, Cham, 2014.H. Berestycki, L. Desvillettes, and O. Diekmann, Can climate change lead to gap formation?, Ecological Complexity, 20 (2014), pp. 264--270.H. Berestycki, O. Diekmann, C. J. Nagelkerke, and P. A. Zegeling, Can a Species Keep Pace with a Shifting Climate?, Bulletin of Mathematical Biology, 71 (2009), pp. 399--429.H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space, Discrete and Continuous Dynamical Systems, 21 (2008), pp. 41--67.J. Bouhours and T. Giletti, Spreading and Vanishing for a Monostable Reaction--Diffusion Equation with Forced Speed, Journal of Dynamics and Differential Equations, 31 (2019), pp. 247--286.V. Calvez, R. J. Hawkins, N. Meunier, and R. Voituriez, Analysis of a Nonlocal Model for Spontaneous Cell Polarization, SIAM Journal on Applied Mathematics, 72 (2012), pp. 594--622.R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Jan. 2004.M. Carrer, R. Dibona, A. L. Prendin, and M. Brunetti, Recent waning snowpack in the Alps is unprecedented in the last six centuries, Nature Climate Change, 13 (2023), pp. 155--160.T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 2nd edition ed., 2006.I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), pp. 299--318.L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence (Rhode Island), 2nd ed., 2010.S. Figueroa Iglesias and S. Mirrahimi, Long Time Evolutionary Dynamics of Phenotypically Structured Populations in Time-Periodic Environments, SIAM Journal on Mathematical Analysis, 50 (2018), pp. 5537--5568., Selection and mutation in a shifting and fluctuating environment, Communications in Mathematical Sciences, 19 (2021), pp. 1761--1798.R. A. Fisher, The Wave of Advance of Advantageous Genes, Annals of Eugenics, 7 (1937), pp. 355--369.H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = $\Delta$u + u1+$\alpha$, Journal of the Faculty of Science, University of Tokyo, 13 (1966), pp. 109--124.J. Garnier, L. Roques, and F. Hamel, Success rate of a biological invasion in terms of the spatial distribution of the founding population, Bulletin of Mathematical Biology, 74 (2012), pp. 453--473.M.-H. Giga, Y. Giga, and J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions, Springer Science & Business Media, 2010.L. Gross, Logarithmic Sobolev Inequalities, American Journal of Mathematics, 97 (1975), pp. 1061--1083.A. Kolmogorov, I. Petrovskii, and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter, and its application to a biological problem, Moscow University Mathematics Bulletin, 1 (1937), pp. 1--26.T. Lepoutre, N. Meunier, and N. Muller, Cell polarisation model: The 1D case, Journal de Mathématiques Pures et Appliquées, 101 (2014), pp. 152--171.C. Loucks, S. Barber-Meyer, M. A. A. Hossain, A. Barlow, and R. M. Chowdhury, Sea level rise and tigers: Predicted impacts to Bangladesh's Sundarbans mangroves, Climatic Change, 98 (2010), pp. 291--298.Y. Malhi, O. L. Phillips, J. Barlow, and C. A. Peres, Ecological responses to El Niño--induced surface fires in central Brazilian Amazonia: Management implications for flammable tropical forests, Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 359 (2004), pp. 367--380.S. Menon, J. Soberón, X. Li, and A. T. Peterson, Preliminary global assessment of terrestrial biodiversity consequences of sea-level rise mediated by climate change, Biodiversity and Conservation, 19 (2010), pp. 1599--1609.S. E. Moore and H. P. Huntington, Arctic marine mammals and climate change: Impacts and resilience, Ecological Applications: A Publication of the Ecological Society of America, 18 (2008), pp. s157--165.F. Otto and C. Villani, Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality, Journal of Functional Analysis, 173 (2000), pp. 361--400.A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, 1983.L. Roques, F. Patout, O. Bonnefon, and G. Martin, Adaptation in General Temporally Changing Environments, SIAM Journal on Applied Mathematics, 80 (2020), pp. 2420--2447.J. L. Vázquez, Asymptotic behaviour methods for the Heat Equation. Convergence to the Gaussian, arXiv.1706.10034, (2018).