Asymptotic Profile Of Solutions Research Articles

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions II: Asymptotic profiles of solutions and radial terrace solution

This paper is a continuation of our previous paper (Kaneko-Matsuzawa-Yamada, Discrete Contin. Dyn. Syst., 2022), where we have classified all large-time behaviors of radially symmetric solutions to a free boundary problem of reaction diffusion equation ut=Δu+f(u) with positive bistable nonlinearity f in high space dimensions. The positive bistable nonlinearity means that f(u)=0 has exactly two positive stable equilibria. Among the classified solutions, we are interested in a spreading solution, that is a solution (u(t,|x|),h(t)) for x∈RN with free boundary |x|=h(t) such that, as t→∞, |x|≤h(t) expands to the whole space RN and u(t,⋅) converges to a positive stable equilibrium for f(u) uniformly in any compact set of RN. When we discuss whole asymptotic profiles of spreading solutions, it has been known that they are generally described with use of a semi-wave obtained from the corresponding semi-wave problem.Our main purpose is to study precise asymptotic estimates for any spreading solution whose profile accompanies a propagating terrace with two different types of propagating speeds. Such a spreading phenomenon occurs when u(t,⋅) converges to the largest equilibrium of f and the related semi-wave problem does not have a solution. We will prove that the propagating terrace consists of two functions; one is a semi-wave corresponding to a smaller positive equilibrium of f and the other is a traveling wave connecting two positive equilibria of f. In order to give sharp estimates for a radial terrace solution with the above properties, we need so called logarithmic shiftings, which are revealed by Uchiyama (1983) and Du-Matsuzawa-Zhou (2015) in higher dimensional cases. Moreover, we will see that two types of the logarithmic shiftings appear in the estimates. The proof for the higher dimensional case is very different from the one-dimensional case established by Kaneko-Matsuzawa-Yamada (2020). To derive exact logarithmic terms, we have to construct a series of upper and lower solutions to make estimates more and more accurate.

Asymptotic profiles and concentration–diffusion effects in fractional incompressible flows

In this paper, we study the behavior of solutions to the fractional dissipative incompressible Navier–Stokes equations at large distances in Rd,d=2,3. The solvability of the fractional Navier–Stokes system in weighted L∞ spaces and the asymptotic profile of solutions are investigated. Based on the asymptotic behavior of the local solution, the concentration–diffusion phenomenon is discussed by constructing suitable initial data for the fractional dissipative incompressible Navier–Stokes system.

Long-Time Behavior of a Reaction–Diffusion Model with Strong Allee Effect and Free Boundary: Effect of Protection Zone

This paper concerns the effect of the (separated/connected) protection zone for the evolution of an endangered species on the reaction–diffusion equation with the strong Allee effect and the free boundary. First, we describe the long-time dynamical behavior of the system of two types protection zones with the same length. Furthermore, the asymptotic profiles of solutions and the asymptotic spreading speed are estimated when spreading happens. Our results, together with those in previous papers Du et al. (J Differ Equ 266:7327–7356, 2019), Du and Lou (J Eur Math Soc 17:2673–2724, 2015) on two other closely related models, show that the protection zone and the free boundary play an important role in the evolution of the endangered species.

Steady states and pattern formation of the density-suppressed motility model

Abstract This paper considers the stationary problem of density-suppressed motility models proposed in Fu et al. (2012) and Liu et al. (2011) in one dimension with Neumman boundary conditions. The models consist of parabolic equations with cross-diffusion and degeneracy. We employ the global bifurcation theory and Helly compactness theorem to explore the conditions under which non-constant stationary (pattern) solutions exist and asymptotic profiles of solutions as some parameter value is small. When the cell growth is not considered, we are able to show the monotonicity of solutions and hence achieve a global bifurcation diagram by treating the chemical diffusion rate as a bifurcation parameter. Furthermore, we show that the solutions have boundary spikes as the chemical diffusion rate tends to zero and identify the conditions for the non-existence of non-constant solutions. When transformed to specific motility functions, our results indeed give sharp conditions on the existence of non-constant stationary solutions. While with the cell growth, the structure of global bifurcation diagram is much more complicated and in particular the solution loses the monotonicity property. By treating the growth rate as a bifurcation parameter, we identify a minimum range of growth rate in which non-constant stationary solutions are warranted, while a global bifurcation diagram can still be attained in a special situation. We use numerical simulations to test our analytical results and illustrate that patterns can be very intricate and stable stationary solutions may not exist when the parameter value is outside the minimal range identified in our paper.

Persistence property and analyticity for a shallow-water model with the coriolis effect in weighted spaces

In this paper, we consider an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force is derived from the governing equation in two dimensional flows. Motivated by the eariler works (Brandolese in Int Math Res Not 22:5161–5181, 2012; Escauriaza et al. in J Funct Anal 244:504–535, 2007; Himonas et al. in Commun Math Phys 271:511–522, 2007; Himonas and Misiolek in Math Ann 327:575–584, 2003; Kohlmann in Z Angew Math Mech 94:264–272, 2014), we demonstrate the persistence results for the solution in weighted $$L^p$$ spaces for a large classs of moderate weights. We also discuss the spatial asymptotic profiles of solutions to this model equation. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.

Blow-up analysis and spatial asymptotic profiles of solutions to a modified two-component hyperelastic rod system

In this paper, we consider a modified two-component hyperelastic rod system, which is a generalization of the Camassa–Holm equation modeling shallow water waves moving over a linear shear flow. We show blow-up analysis and exact spatial asymptotic profiles of solutions to the system. Especially, for the special case $$\gamma =1$$ , we present the exact blow-up rate of the breaking-wave solution.

Asymptotic profiles of solutions for regularity-loss-type generalized thermoelastic plate equations and their applications

In this paper, we consider generalized thermoelastic plate equations with Fourier's law of heat conduction. By introducing a threshold for decay properties of regularity-loss, we investigate decay estimates of solutions with/without regularity-loss in a framework of weighted $L^1$ spaces. Furthermore, asymptotic profiles of solutions are obtained by using representations of solutions in the Fourier space, which are derived by employing WKB analysis. Next, we study generalized thermoelastic plate equations with additional structural damping, and analysis the influence of structural damping on decay properties and asymptotic profiles of solutions. We find that the regularity-loss structure is destroyed by structural damping. Finally, we give some applications of our results on thermoelastic plate equations and damped Moore-Gibson-Thompson equation.

Asymptotic Profiles of Solutions and Propagating Terrace for a Free Boundary Problem of Nonlinear Diffusion Equation with Positive Bistable Nonlinearity

We will study a free boundary problem of nonlinear diffusion equation $u_t=u_{xx}+f(u),t>0, 0<x<h(t)$, with free boundary $h(t)$, which describes the spreading of a new or invasive species. When $f...

A note on optimal L 2 -estimates of solutions to some strongly damped σ-evolution equations

We consider the Cauchy problem in R n for the so-called σ-evolution equations with damping terms. We derive asymptotic profiles of solutions with weighted L 1 , 1 ( R n ) initial data, and investigates the optimality of estimates of solutions in L 2 -sense. The obtained results will generalize and compensate those already known in (J. Math. Anal. Appl. 478 (2019) 476–498, J. Diff. Eqns 257 (2014) 2159–2177, Diff. Int. Eqns 30 (2017), 505–520).

Decay estimate and asymptotic profile for a plate equation with memory

In this paper, the initial value problem for a plate equation with a memory is considered. Under weaker assumptions on the memory function, we obtain the decay estimates of solutions which is of the regularity-loss type by applying the energy method in Fourier space. Moreover, if the memory kernel is an exponential function, we derive the large-time behavior and asymptotic profile of solutions in the one-dimensional space by virtue of the technique of spectral analysis.

Lie symmetry reductions and dynamics of solitary wave solutions of breaking soliton equation

In this paper, some new invariant solutions of breaking soliton (BS) equation have been derived by using similarity transformations method. The system represents the interaction of Riemann waves propagating along [Formula: see text]-axis and long waves along [Formula: see text]-axis. The commutative relation and symmetry analysis of BS equation are derived using Lie group theory. Meanwhile, the method reduces the number of independent variables by one in each step. A repeated application of similarity transformations method reduces the BS equation into overdetermined equations, which provide invariant solutions. The derived results are more general than previous findings. The obtained solutions are supplemented by numerical simulation, which makes this research physically meaningful. Eventually, doubly soliton, multisoliton and asymptotic profiles of solutions are analyzed in the analysis and discussion section.

Spatial Asymptotic Profiles of Solutions to the Navier-Stokes System in a Rotating Frame with Fast Decaying Data

The nonstationary Navier-Stokes system for a viscous, incompressible fluid influenced by a Coriolis force in the whole space ${\mathbb R}^3$ is considered at large distances. The solvability of the corresponding integral equations of these equations in weighted $L^\infty$-spaces is established. Furthermore, the leading terms of the asymptotic profile of the solution at fixed time $t \gt 0$ for $|x| \gt t$ and far from the axis of rotation are investigated.

On diffusion phenomena for the linear wave equation with space-dependent damping

We study the diffusion phenomenon associated with the linear wave equation with space-dependent damping, and give a proof that the asymptotic profile of solutions is represented by solutions to the associated heat equation, in the L2sense.

High energy rotation type solutions of the forced pendulum equation

In this article we study the existence and asymptotic profiles of high-energy rotation type solutions of the singularly perturbed forced pendulum equationWe prove that the asymptotic profile of these solutions is described in terms of an energy function which satisfy an integro-differential equation. Also we show that given a suitable energy function E satisfying the integro-differential equation, a family of solutions of the pendulum equation above exists having E as the asymptotic profile, when ε → 0.

Asymptotic profile of solutions for 1-D wave equation with time-dependent damping and absorbing semilinear term

We consider the Cauchy problem for the wave equation with time-dependent damping and absorbing semilinear term � utt − Δu + b(t)ut + |u| ρ−1 u = 0, (t, x) ∈ R+ × R N ,

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved L2 decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.

Blow-up in a degenerate parabolic equation

This paper investigates the asymptotic behavior of positive blow-up solutions to the Dirichlet problem for u t = u p (Δu + u), p ≥ 2, in bounded domains Ω ⊂ R n . It is proved that for any initial datum, blow-up occurs essentially faster than at the ODE rate (T - t) -1/p , and that the subset of Ω of points of such fast blow-up has positive measure. In the one-dimensional case, asymptotic profiles of solutions are discussed. Concerning the corresponding Cauchy problem, which has explicit spatially homogeneous solutions blowing up exactly at the ODE rate, it is shown that even arbitrarily slow spatial decay of the initial datum can enforce fast blow-up.

Uniqueness and profile of solutions for a superlinear elliptic equation

The uniqueness and profile of solutions for a Matukuma-type equation on the whole space as the power of the nonlinear term tends to one are discussed. By using information on a linear eigenvalue problem and Lyapunov-Schmidt-like reduction, we obtain the uniqueness and asymptotic profile of solutions. We also consider the case of bounded domain in the same manner.