On the large time asymptotics of Schrödinger type equations with general data

Wright–Fisher–type equations for opinion formation, large time behavior and weighted logarithmic-Sobolev inequalities

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev–Petviashvili (KP) equations $$$$ where σ= 1 or σ=− 1. When ρ= 2 and σ=− 1, (KP) is known as the KPI equation, while ρ= 2, σ=+ 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case ρ= 3, σ=− 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if ρ≥ 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: $$$$ for all t∈R, where κ= 1 if ρ= 3 and κ= 0 if ρ≥ 4. We also find the large time asymptotics for the solution.

We consider solutions to the Einstein-massless-scalar field system with a positive cosmological constant, arising from sufficiently regular, near-FLRW (Friedmann-Lemaître-Robertson-Walker), initial data. We establish global existence in the future direction and derive their precise asymptotic behavior toward infinity. As a corollary, we infer that, unlike the FLRW background, the perturbed solutions do not describe a regular irrotational stiff fluid with a linear equation of state p = ρ for general asymptotic data at infinity. The reason for the breakdown of this interpretation is that the gradient of the scalar field stops being timelike at large times, eventually becoming null and then spacelike near infinity. Our results hold for open sets of initial data in Sobolev spaces without symmetries.

Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition

This chapter describes some constructive approaches to the study of the asymptotic nature of solutions of some nonlinear partial differential equations of parabolic type: travelling waves, and self-similar or more general solutions obtained by the so-called balancing argument and its extensions. This is accomplished by first constructing these special solutions and hence showing their asymptotic nature analytically, numerically, or both. Sometimes it becomes possible to obtain asymptotic solutions in terms of power series in time with coefficient functions depending on the similarity variable. This approach is adopted to obtain solutions more general than self-similar or travelling waves. The analysis for such problems requires solutions of an infinite system of nonlinear ODEs. This class of solutions either ‘nonlinearise’ the linear solution of the given problem and/or embed in them limiting behaviour such as inviscid forms for convective–diffusive equations.KeywordsLarge TimeAsymptotic SolutionTravel Wave SolutionBurger EquationNonlinear DiffusionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the higher-order dispersive equations of Korteweg-de Vries type with weak nonlinearity (wKdV): $$ u_t + \partial_x f(u) + \mathcal{K} u = 0, $$ where $x, t \in \mathbf {R}$, $f(u) = |u|^{\rho -1}u$ if $\rho > \nu$ or $f(u) = u^{\rho }$ if $\rho > \nu$ is integer, the operator $\mathcal{K} $ is a pseudodifferential operator with a homogeneous and conservative symbol $K(p)$ of order $\nu > 3$, namely, $\mathcal{K}u = \mathcal{F}^{-1} K(p) \hat u(p), K(p) = - \frac{i}{\nu} |p|^{\nu - 1} p, $ $\mathcal{F}\phi$ or $\hat \phi$ is the Fourier transformation of $\phi$ and $\mathcal{F}^{-1} \phi $ is the inverse Fourier transformation of $\phi$. If the power $\rho$ of the nonlinearity is greater than $\nu$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solution $u(t)$ satisfies the decay estimate $$ \|u(t)\|_{L^\beta} \le C(1 + t)^{-\frac{1}{\nu}(1-\frac{1}\beta)} \quad \text{for} \quad \beta \in (\frac{2\nu -2}{\nu - 2},\infty], $$ $$ \|uu_x(t)\|_{L^\infty} \le Ct^{-2/\nu}(1+t)^{-1/\nu} $$ and using these estimates we prove the existence of the scattering state $u_+\in L^2$ such that $$ \|u(t) - U(t)u_+\|_{L^2} \le Ct^{-\frac{\rho -\nu}{\nu}} \quad \text{and} \quad \|u(t) - U(t)u_+\|_{L^\infty} \le Ct^{-\frac{1+\rho -\nu}{\nu}} $$ for any small initial data belonging to the weighted Sobolev space $H^{1,1} = \{ \phi \in L^2; \|(1+|x|^2)^{1/2}(1-\partial_x^2)^{1/2} \phi\|_{L^2}<\infty\}$, where $U(t)$ is the free evolution group, associated with corresponding linear equation.

Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space $ L^2(0,1)^2 $ according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the $ L^m $-norms for all $ m\in [1,+\infty] $ and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

We study large time asymptotics of solutions to the Korteweg–de Vries–Burgers equation $$ u_{t} + uu_{x} - u_{{xx}} + u_{{xxx}} = 0,x \in {\text{R}},t > 0. $$ We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u 0 ∈ H s (R) ∩ L 1 (R) , where \( s > - \frac{1} {2}, \) then there exists a unique solution u (t, x) ∈ C ∞ ((0,∞) ;H ∞ (R)) to the Cauchy problem for the Korteweg–de Vries–Burgers equation, which has asymptotics $$ u{\left( t \right)} = t^{{ - \frac{1} {2}}} f_{M} {\left( {{\left( \cdot \right)}t^{{ - \frac{1} {2}}} } \right)} + o{\left( {t^{{ - \frac{1} {2}}} } \right)} $$ as t → ∞, where f M is the self–similar solution for the Burgers equation. Moreover if xu 0 (x) ∈ L 1 (R) , then the asymptotics are true $$ u{\left( t \right)} = t^{{ - \frac{1} {2}}} f_{M} {\left( {{\left( \cdot \right)}t^{{ - \frac{1} {2}}} } \right)} + O{\left( {t^{{ - \frac{1} {2} - \gamma }} } \right)}, $$ where \( \gamma \in {\left( {0,\frac{1} {2}} \right)}. \)

The cytochrome bc complexes found in mitochondria, chloroplasts and many bacteria play critical roles in their respective electron transport chains. The quinol oxidase (Q(o)) site in this complex oxidizes a hydroquinone (quinol), reducing two one-electron carriers, a low potential cytochrome b heme and the "Rieske" iron-sulfur cluster. The overall electron transfer reactions are coupled to transmembrane translocation of protons via a "Q-cycle" mechanism, which generates proton motive force for ATP synthesis. Since semiquinone intermediates of quinol oxidation are generally highly reactive, one of the key questions in this field is: how does the Q(o) site oxidize quinol without the production of deleterious side reactions including superoxide production? We attempt to test three possible general models to account for this behavior: 1) The Q(o) site semiquinone (or quinol-imidazolate complex) is unstable and thus occurs at a very low steady-state concentration, limiting O(2) reduction; 2) the Q(o) site semiquinone is highly stabilized making it unreactive toward oxygen; and 3) the Q(o) site catalyzes a quantum mechanically coupled two-electron/two-proton transfer without a semiquinone intermediate. Enthalpies of activation were found to be almost identical between the uninhibited Q-cycle and superoxide production in the presence of antimycin A in wild type. This behavior was also preserved in a series of mutants with altered driving forces for quinol oxidation. Overall, the data support models where the rate-limiting step for both Q-cycle and superoxide production is essentially identical, consistent with model 1 but requiring modifications to models 2 and 3.

We consider a class of time-dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker–Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics.

Estimates for solutions of Burgers type equations and some applications

We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of theheight function of the associated growth model along any space-like path are described by the Airy 1 and Airy 2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

On the large time behavior of solutions of the alpha Navier–Stokes equations

In the last decade, a number of authors have studied extensively the large time behavior of solutions of first-order Hamilton–Jacobi equations. Several convergence results have been established. The first general theorem in this direction was proven by Namah and Roquejoffre in [37], under the assumptions: $$\displaystyle{p\mapsto H(x,p) \text{is convex},H(x,p) \geq H(x,0) \text{for all} (x,p)\,\in \,\mathbb{T}^{n} \times \mathbb{R}^{n}, \max _{ x\,\in \,\mathbb{T}^{n}}H(x,0) = 0.}$$ We will first discuss this setting in Sect. 5.2. In this setting, as the Hamiltonian has a simple structure, we are able to find an explicit subset of \(\mathbb{T}^{n}\) which has the monotonicity of solutions and the property of the uniqueness set. Therefore, we can relatively easily get a convergence result of the type ( 4.5), that is, $$\displaystyle{u(x,t) - (v(x) - ct) \rightarrow 0\quad \text{uniformly for} x\,\in \,\mathbb{T}^{n},}$$ where u is the solution of the initial value problem and (v, c) is a solution to the associated ergodic problem.