Small Initial Data Research Articles

Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime

We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent p satisfies 2≤p<1+2, we establish the sharp upper bound of lifespan estimate, while for the most delicate critical power p=1+2, we show that the lifespan satisfiesT(ε)≤exp⁡(Cε−(2+2)), the optimality of which remains to be proved. The key novelty is that the compact support of the initial data can be close to the event horizon. By combining the global existence result for p>1+2 obtained by Lindblad et al. (2014) [25], we then give a positive answer to the interesting question posed by Dafermos and Rodnianski (2005) [22, the end of the first paragraph in page 1151]: p=1+2 is exactly the critical power of p separating stability and blow-up.

Stability of the 3D incompressible Navier–Stokes equations with fractional horizontal dissipation

The stability problem of Navier–Stokes equations (N–S equations) with fractional dissipation is one of the new areas in Mathematical research. The generalized N–S equations are the equations resulting from replacing −Δ in the N–S equations by (−Δ)α. It has previously been shown that any classical solution of the d-dimensional generalized N–S equations with α≥12+d4 is always global in time. This paper considers the stability problem on the 3D N–S equations with only fractional horizontal dissipation (−Δh)α, where Δh:=∂x12+∂x22. We show that, for any α∈(12,1], the solution corresponding to any sufficiently small initial data in H3(R3) is always global in time and stable in H3(R3). There are many important results on the case when α=1. Our result relaxes this requirement and allows α to go below 1.

A semigroup approach to a reaction–diffusion system with cross-diffusion

In this paper, we prove existence, uniqueness and special properties of the solution to a parabolic–elliptic reaction–diffusion system with cross-diffusion in an open bounded domain in Rd, as well as in Rd,d=1, 3, for various classes of the system parameters. In the case of a bounded domain and of Neumann boundary conditions we develop an approach based on a semigroup technique in the dual space (H1(Ω))′. This gives the possibility of considering singular initial data, distributions or functionals that may represent relevant physical data. The influence of the system parameters upon the qualitative behavior of the solution is investigated and some sufficient conditions for the existence of global or local solutions, and for global solutions in the case of small initial data are provided. In the case of the whole space, the problem is studied for a singular diffusion expressed by a multivalued operator. Existence is proved by passing to the limit in an approximating problem for which existence follows via a semigroup approach in H−1(Rd), d=1, 3. An argument for the uniqueness is developed in Beppo Levi spaces. Finally, the results are illustrated by a model exhibiting a self-organized criticality.

Well-posedness and Scattering for the Generalized Boussinesq Equation

In this paper, we show the local well-posedness of the generalized Boussinesq equation (gBQ) in and obtain the global well-posedness, finite-time blowup and small initial data scattering of gBQ in energy space . Moreover, we obtain the large radial initial data scattering of the defocusing case for by using the method of Dodson–Murphy [Proc. Amer. Math. Soc., 145 (2017), pp. 4859–4867].

Well-Posedness for the Incompressible Hall-MHD System with Initial Magnetic Field Belonging to $$H^{\frac{3}{2}}({\mathbb {R}}^3)$$

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $$(u_0,B_0)\in H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)$$ with $$\sigma \in (0,2)$$ . In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $$\sigma $$ to 0 with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31–65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Moreover, we establish the global well-posedness for small initial data in $$H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)$$ with $$\sigma \in (0,2)$$ , and get the optimal time-decay rates of solutions.

On the Cauchy Problem for the Cutoff Boltzmann Equation with Small Initial Data

We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model $$-1\le \gamma < 0$$ with the small initial data in three dimensional space. Thus our result fixes the gap for the case $$\gamma =-1$$ in three dimensional space in the authors’ previous work (He and Jiang in J Stat Phys 168(2):470–481, 2017) where the estimate for the loss term was improperly used. The other gap in He and Jiang (2017) for the case $$\gamma =0$$ in two dimensional space is recently fixed by Chen et al. (Arch Ration Mech Anal 240:327–381, 2021). The initial data $$f_{0}$$ is non-negative and satisfies that $$\Vert \langle v \rangle ^{\ell _{\gamma }} f_{0}(x,v)\Vert _{L^{3}_{x,v}}\ll 1$$ and $$\Vert \langle v \rangle ^{\ell _{\gamma }} f_0\Vert _{L^{15/8}_{x,v}}<\infty $$ where $$\ell _{\gamma }=0$$ when $$\gamma =-1$$ and $$\ell _{\gamma }=(1+\gamma )^{+}$$ when $$-1<\gamma <0$$ . We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model $$-1<\gamma <0$$ when applying the Strichartz estimates.

On the stability to Keller–Segel system coupled with Navier–Stokes equations in Besov–Morrey spaces

In this paper, we consider the Cauchy problem of Keller–Segel–Navier–Stokes equations in RN for N≥2. Under the small initial data assumption, we prove the asymptotic stability for this system in Besov–Morrey spaces by the structure of the model and semigroup estimates.

Analysis of a chemotaxis-SIS epidemic model with unbounded infection force

A susceptible–infected–susceptible (SIS) model is considered which includes both taxis type cross-diffusive motion of susceptibles and a mass action infection mechanism. An associated no-flux initial–boundary value problem in one-dimensional bounded intervals is shown to admit globally bounded classical solutions for initial data of arbitrary size. For a higher-dimensional counterpart, a result on global classical solvability in the presence of suitably small initial data is derived. This is complemented by a statement on large time stabilization of small-data trajectories toward spatially constant and infection-free equilibria under an assumption on strict positivity of the considered space-dependent recovery rate.

Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the scale-invariant case and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($ \frac{\mu}{\sqrt{1+|x|^2}}u_t $), we show that in higher dimensions the blow-up region is given by $ p \in (1, p_G(N+\mu)] $ where $ p_G(N) $ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $ \mu $ given by $ p\in (1, 1+\frac{2}{N}), $ for appropriate initial data in the energy space with noncompact support.

Global existence for quasilinear degenerate two-species chemotaxis system with small initial data

Global existence for quasilinear degenerate two-species chemotaxis system with small initial data

Decay estimates for Schrödinger systems with time-dependent potentials in 2D

&lt;abstract&gt;&lt;p&gt;We consider the Cauchy problem for systems of nonlinear Schrödinger equations with time-dependent potentials in 2D. Under assumptions about mass resonances and potentials, we prove the global existence of the nonlinear Schrödinger systems with small initial data. In particular, by analyzing the operator $ \Delta $ and time-dependent potentials $ {V_{j}} $ separately, we show that the small global solutions satisfy time decay estimates of order $ O((t\log{t})^{-1}) $ when $ p = 2 $, and the small global solutions satisfy time decay estimates of order $ O({t}^{-1}) $ when $ p &amp;gt; 2 $.&lt;/p&gt;&lt;/abstract&gt;

Global well-posedness of perturbed Navier–Stokes system around Landau solutions

In this paper, we consider the perturbed Navier–Stokes equations around the Landau solution and investigate the global well-posedness results of the perturbed system with the small initial data in the X−1 space, where X−1=f∈D′R3:∫R3|ξ|−1|f̂|dξ&amp;lt;∞.

GLOBAL EXISTENCE AND BLOW-UP FOR ONE-DIMENSIONAL WAVE EQUATION WITH WEIGHTED EXPONENTIAL NONLINEARITY

We consider the initial value problem for a one-dimensional wave equation with weighted exponential nonlinearity. We show global existence for small amplitude initial data. We also prove that blow-up in finite time occurs if the initial data are localized and the initial velocity being on the average positive.

On the fractional chemotaxis Navier-Stokes system in the critical spaces

&lt;p style='text-indent:20px;'&gt;We consider the fractional chemotaxis Navier-Stokes equations which are the fractional Keller-Segel model coupled with the Navier-Stokes fluid in the whole space, and prove the existence of global mild solutions with the small critical initial data in Besov-Morrey spaces. Our results enable us to obtain the self-similar solutions provided the initial data are homogeneous functions with small norms and considering the case of chemical attractant without degradation rate. Moreover, we show the asymptotic stability of solutions as the time goes to infinity and obtain a class of asymptotically self-similar ones.&lt;/p&gt;

GLOBAL WELL-POSEDNESS OF A CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION WITH MEMORY

In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources.

Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients

&lt;p style='text-indent:20px;'&gt;This paper concerns the asymptotic behavior of solutions to one-dimensional coupled semilinear degenerate parabolic equations with superlinear reaction terms both in bounded and unbounded intervals. The equations are degenerate at a lateral boundary point and the diffusion coefficients are general functions. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large ones in the case that the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. If the degeneracy of the equations at the lateral boundary point is strong enough, it is shown that any nontrivial solution must blow up in a finite time. If the degeneracy of the equations at the lateral boundary point is not strong, it is proved that the critical Fujita curve is determined by the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is also considered.&lt;/p&gt;

Small solutions of the Einstein–Boltzmann-scalar field system with Bianchi symmetry

We show that small homogeneous solutions to the Einstein–Boltzmann-scalar field system exist globally toward the future and tend to the de Sitter solution in a suitable sense. More specifically, we assume that the spacetime is of Bianchi type I–VIII, that the matter is described by Israel particles and that there exists a scalar field with a potential which has a positive lower bound. This represents a generalization of the work [H. Lee and E. Nungesser, Classical Quantum Gravity 35, 025001 (2018)], where a cosmological constant was considered, and a generalization of [H. Lee and J. Lee, J. Math. Phys. 63, 031502 (2022)], where a spatially flat FLRW spacetime was considered. We obtain the global existence and asymptotic behavior of classical solutions to the Einstein–Boltzmann-scalar field system for small initial data.

On global smooth small data solutions of 3-D quasilinear Klein-Gordon equations on $ \mathbb{R}^{2}\times \mathbb{T} $

In this paper, for the small initial data in the suitably weighted Sobolev spaces on the product space $ \mathbb{R}^2\times\mathbb{T} $, we establish the global existence of smooth solutions to the 3-D quasilinear Klein-Gordon equations with quadratic nonlinearities. It is noted that the topic on the well-posedness of the nonlinear hyperbolic equations on product spaces arises from the studies for the propagation of waves along infinite homogeneous waveguides and the Kaluza-Klein theory. Our main result is based on the method of normal form including the parameters $ n\in\Bbb Z $ and the continuous induction method. In addition, the free profile of the solution is obtained by applying the weighted energy estimates and the temporal decay estimates.

A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations

In this manuscript, lifespan of solutions to semilinear classical damped wave equations is investigated when the sum of initial position and speed is $ 0 $ pointwisely. The results in the existing literature are considered mainly in the case when the sum is positive. Here, it is shown that lifespan is extended with appropriate initial data with modified ODE approach. The result implies that the blowup mechanics for semilinear damped wave equations cannot be directly described by using the scaling property and the corresponding weak form approach. The results in this work leave different directions for further study of the lifespan behavior of the solutions to semilinear damped wave equations.

Qualitative analysis of a mechanical system of coupled nonlinear oscillators

In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to + ∞ , provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations.