Asymptotic Profile Research Articles

Double- and single-layered sign-changing solutions to a singularly perturbed elliptic equation concentrating at a single sphere

ABSTRACTWe exhibit new concentration phenomena for the problemas 𝜀→0, in domains of the formwhere m≥1, n≥3 and Θ is a bounded smooth domain in ℝn with , 𝜀>0, and . For particular choices of Θ we establish the existence of sign-changing solutions which concentrate on spheres that converge to a single m-dimensional sphere contained in the boundary of Ω, developing, either a positive and a negative layer, or a single sign-changing layer. In the first case, the asymptotic profile of each layer, in the transversal direction to its concentration sphere, is a rescaling of the positive or the negative ground state to the limit problemwhereas, in the second case, it is a rescaling of a nonradial sign-changing solution to this last problem.

Asymptotic behavior for the filtration equation in domains with noncompact boundary

ABSTRACTWe consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.

Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$

In this paper, we investigate the initial boundary value problem for the linearized double dispersion equation on the half space \begin{document}$\mathbb{R}^{n}_{+}$\end{document} . We convert the initial boundary value problem into the initial value problem by odd reflection. The asymptotic profile of solutions to the initial boundary value problem is derived by establishing the asymptotic profile of solutions to the initial value problem. More precisely, the asymptotic profile of solutions is associated with the convolution of the partial derivative of the fundamental solutions of heat equation and the fundamental solutions of free wave equation.

Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model

We give a complete description of the long-time asymptotic profile of the solution to a free boundary model considered recently in [ 10 ]. This model describes the spreading of an invasive species in an environment which shifts with a constant speed, and the research of [ 10 ] indicates that the species may vanish, or spread successfully, or fall in a borderline case.In the case of successful spreading, the long-time behavior of the population is not completely understood in [ 10 ].Here we show that the spreading of the species is governed by two traveling waves, one has the speed of the shifting environment, giving the profile of the retreating tail of the population, while the other has a faster speed determined by a semi-wave, representing the profile of the advancing front of the population.

Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign‐changing non‐linearity

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ and $\mu> 0$. When $\mu=0$ and $s=0$, the profile at the singularity $0$ was fully described by Caffarelli-Gidas-Spruck. We prove that when $\mu>0$ and $s>0$, besides this profile, two new profiles might occur. We provide a full description of all the singular profiles. Special attention is accorded to solutions such that $\liminf_{x\to 0}|x|^{\frac{n-2}{2}}u(x)=0$ and $\limsup_{x\to 0}|x|^{\frac{n-2}{2}}u(x)\in (0,+\infty)$. The particular case $q=(n+2)/(n-2)$ requires a separate analysis which we also perform.

Existence and Asymptotic Profile of Nodal Solutions to Supercritical Problems

Abstract We establish the existence of nodal solutions to the supercritical problem - Δ ⁢ u = | u | p - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $-\Delta u=|u|^{p-2}u\quad\text{in }\Omega,\qquad u=0\quad\text{on }\partial\Omega,$ in a symmetric bounded smooth domain Ω of ℝ N ${\mathbb{R}^{N}}$ , N ≥ 3 ${N\geq 3}$ , for p &gt; 2 N ∗ := 2 ⁢ N N - 2 ${p&gt;2_{N}^{\ast}:=\frac{2N}{N-2}}$ , up to some range which depends on the symmetries, and we study their asymptotic behavior as p → 2 N ∗ ${p\rightarrow 2_{N}^{\ast}}$ . We exhibit solutions u p ${u_{p}}$ to this problem in symmetric domains with a shrinking hole, which concentrate at a single point as the hole shrinks and p approaches 2 N ∗ ${2_{N}^{\ast}}$ from above, and whose limit profile is a rescaling of a nonradial sign-changing solution to the limit problem - Δ ⁢ u = | u | 2 N ∗ - 2 ⁢ u , u ∈ D 1 , 2 ⁢ ( ℝ N ) . $-\Delta u=|u|^{2_{N}^{\ast}-2}u,\quad u\in D^{1,2}(\mathbb{R}^{N}).$

Asymptotic profile of positive solutions of Lane–Emden problems in dimension two

We consider families \(u_p\) of solutions to the problem Open image in new window where \(p>1\) and \(\Omega \) is a smooth bounded domain of \(\mathbb {R}^2\). Under the condition Open image in new window we give a complete description of the asymptotic behavior of \(u_p\) as \(p\rightarrow +\infty \).

An integro-PDE model for evolution of random dispersal

We consider an integro-PDE model for a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We focus on the asymptotic profile of positive steady state solutions. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Hastings [16] and Dockery et al. [14] showed that for two competing species in spatially heterogeneous but temporally constant environment, the slower diffuser always prevails, if all other things are held equal. Our result suggests that their findings may well hold for arbitrarily many or even a continuum of traits.

Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations

In this note we construct mixed dimensional infinite soliton trains, which are solutions of nonlinear Schrödinger equations whose asymptotic profiles at time infinity consist of infinitely many solitons of multiple dimensions. For example infinite line-point soliton trains in 2D space, and infinite plane-line-point soliton trains in 3D space. This note extends the works of Le Coz, Li and Tsai [6,7], where single dimensional trains are considered. In our approach, spatial L∞ bounds for lower dimensional trains play an essential role.

A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations

In this paper, we find the critical exponent for global small data solutions to the Cauchy problem in Rn, for dissipative evolution equations with power nonlinearities |u|p or |ut|p,utt+(−Δ)δut+(−Δ)σu={|u|p,|ut|p. Here σ,δ∈N∖{0}, with 2δ≤σ. We show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation, which are described by the diffusion equations: vt+(−Δ)σ−δv=0,wt+(−Δ)δw=0. The nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method (under suitable sign assumptions on the initial data), and lifespan estimates are obtained. By assuming small initial data in Sobolev spaces, we prove the existence of global solutions in the supercritical case, up to some maximum space dimension n̄, and we derive Lq estimates for the solution, for q∈(1,∞). For σ=2δ, the result holds in any space dimension n≥1. The existence result also remains valid if σ and/or δ are fractional.

Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension N=5∂ttu=Δu+|u|4/3u, with radial data. It is known [7] that a solution (u,∂tu) which blows up at t=0 in a neighborhood (in the energy norm) of the family of solitons Wλ, decomposes in the energy space as(u(t),∂tu(t))=(Wλ(t)+u0⁎,u1⁎)+o(1), where limt→0⁡λ(t)/t=0 and (u0⁎,u1⁎)∈H˙1×L2. We construct a blow-up solution of this type such that the asymptotic profile (u0⁎,u1⁎) is any pair of sufficiently regular functions with u0⁎(0)>0. For these solutions the concentration rate is λ(t)∼t4. We also provide examples of solutions with concentration rate λ(t)∼tν+1 for ν>8, related to the behavior of the asymptotic profile near the origin.

Sharp thresholds between finite spread and uniform convergence for a reaction–diffusion equation with oscillating initial data

We investigate the large-time dynamics of solutions of multi-dimensional reaction–diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which asymptotically oscillate around the ignition threshold. We show that, as time goes to infinity, any solution either converges uniformly in space to a constant state, or spreads with a finite speed uniformly in all directions. Furthermore, the transition between these two behaviors is sharp with respect to the period vector of the asymptotic profile of the initial data. We also show the convergence to planar fronts when the initial data are asymptotically periodic in one direction.

Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients

We study the asymptotic behavior of solutions for the semilinear damped wave equation with variable coefficients. We prove that if the damping is effective, and the nonlinearity and other lower order terms can be regarded as perturbations, then the solution is approximated by the scaled Gaussian of the corresponding linear parabolic problem. The proof is based on the scaling variables and energy estimates.

Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model

This paper performs qualitative analysis on an SIS epidemic reaction–diffusion system with a linear source in spatially heterogeneous environment. The main feature of our model lies in that its total population number varies, compared to its counterpart proposed by Allen et al. [2]. The uniform bounds of solutions are derived, based on which, the threshold dynamics in terms of the basic reproduction number is established and the global stability of the unique endemic equilibrium is discussed when spatial environment is homogeneous. In particular, the asymptotic profile of endemic equilibria is determined if the diffusion rate of the susceptible or infected population is small or large. The theoretical results show that a varying total population can enhance persistence of infectious disease, and therefore the disease becomes more threatening and harder to control.

On an integrable Camassa–Holm type equation with cubic nonlinearity

We discuss an integrable Camassa–Holm type equation with cubic nonlinearity. Asymptotic profile has been shown in the sense that strong solutions arising from algebraic decaying initial data will keep this property at infinity in the spatial variable in its lifespan. Moreover, for a global solution, measure of potential support is presented.

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.

The Burgers equation with periodic boundary conditions on an interval

We study the asymptotic profile of the solutions of the Burgers equation on a finite interval with a periodic perturbation on the boundary. The equation describes a dissipative medium, and the initial constant profile therefore passes into a wave with a decreasing amplitude. In the low-viscosity case, the asymptotic profile looks like a sawtooth wave (with periodic breaks of the derivative), similar to the known Fay solution on the half-line, but it has some new properties.

A Class of Random Walks in Reversible Dynamic Environments: Antisymmetry and Applications to the East Model

We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in http://arxiv.org/abs/1602.06322. As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical interfaces between empty and occupied regions. We study the asymptotic velocity and density profile for the environment seen from the walker. In particular, we determine the sign of the velocity when the density of the underlying East process is not 1/2, and we discuss the appearance of a drift in the balanced setting given by density 1/2.

Limiting classification on linearized eigenvalue problems for 1-dimensional Allen–Cahn equation II — Asymptotic profiles of eigenfunctions

This paper is a continuation of a previous paper by the authors. We are interested in the asymptotic behavior of eigenpairs on one dimensional linearized eigenvalue problem for Allen–Cahn equations as the diffusion coefficient tends to zero. We obtain the asymptotic profiles of all eigenfunctions by using the asymptotic formulas of corresponding eigenvalues, which have been obtained in the previous paper. Our results lead us to the concept of the classification of limiting eigenfunctions. In the case of Allen–Cahn equation it is provided by three special eigenfunctions, which correspond to the solutions of rescaled spectral problems on the whole line.

Numerical aspects of large-time optimal control of Burgers equation

In this paper, we discuss the efficiency of various numerical methods for the inverse design of the Burgers equation, both in the viscous and in the inviscid case, in long time-horizons. Roughly, the problem consists in, given a final desired target, to identify the initial datum that leads to it along the Burgers dynamics. This constitutes an ill-posed backward problem. We highlight the importance of employing a proper discretization scheme in the numerical approximation of the equation under consideration to obtain an accurate approximation of the optimal control problem. Convergence in the classical sense of numerical analysis does not suffice since numerical schemes can alter the dynamics of the underlying continuous system in long time intervals. As we shall see, this may end up affecting the efficiency on the numerical approximation of the inverse design, that could be polluted by spurious high frequency numerical oscillations. To illustrate this, two well-known numerical schemes are employed: the modified Lax−Friedrichs scheme (MLF) and the Engquist−Osher (EO) one. It is by now well-known that the MLF scheme, as time tends to infinity, leads to asymptotic profiles with an excess of viscosity, while EO captures the correct asymptotic dynamics. We solve the inverse design problem by means of a gradient descent method and show that EO performs robustly, reaching efficiently a good approximation of the minimizer, while MLF shows a very strong sensitivity to the selection of cell and time-step sizes, due to excess of numerical viscosity. The achieved numerical results are confirmed by numerical experiments run with the open source nonlinear optimization package (IPOPT).