Small Data Solutions Research Articles

On global existence and blowup of solutions of Stochastic Keller–Segel type equation

In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.

On the existence theory for nonlinear plate equations

This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild solutions for small initial data in $L^{1}(\mathbb{R}^{n})\cap H^s(\mathbb{R}^{n})$-spaces. We also prove the existence and uniqueness of local and global solutions in the framework of Bessel-potential spaces $H^s_p(\mathbb{R}^n)=(I-\Delta)^{s/2}L^p(\mathbb{R}^n).$ In order to derive the existence results we develop new time decay estimates of the solution of the corresponding linear problem.

Global existence for one-dimensional hyperbolic equation with power type nonlinearity

In this paper, we consider the Cauchy problem for one-dimensional damped wave equations with variable coefficients $a(x)$ having power type nonlinearity of order $p$. We discuss the global existence of solutions for small initial data and investigate the relation between the range of $a(x)$ and the order $p$.

The critical exponent for semilinear [formula omitted]-evolution equations with a strong non-effective damping

In this paper, we find the critical exponent for the existence of global small data solutions to: utt+(−Δ)σu+(−Δ)θ2ut=f(u,ut),t≥0,x∈Rn,(u,ut)(0,x)=(0,u1(x)),in the case of so-called non-effective damping, θ∈(σ,2σ], where σ≠1 and f=|u|α or f=|ut|α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α>α̃ and do not exist, in general, for subcritical powers 1<α<α̃. Assuming initial data to be small in L1 or in some other Lp space, p∈(1,2), in addition to the energy space, the critical exponent only depends on the ratio n/(σp). We also prove the global existence of small data solutions in high space dimension for α>ᾱ, but we leave open to determine if a counterpart nonexistence result for α<ᾱ holds or not.

The interplay of critical regularity of nonlinearities in a weakly coupled system of semi-linear damped wave equations

We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the p − q plane. The main goal of this paper is to find out sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of (even) small data solutions.

The Keller-Segel system of parabolic-parabolic type in homogeneous Besov spaces framework

We show the existence and uniqueness of local strong solutions of Keller-Segel system of parabolic-parabolic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also construct global strong solutions for small initial data, where the solutions belong to the Lorentz space in time direction. The proof is based on the maximal Lorentz regularity theorem of heat equations.

Concentration versus absorption for the Vlasov–Navier–Stokes system on bounded domains

We study the large time behaviour of small data solutions to the Vlasov–Navier–Stokes system set on , for a smooth bounded domain Ω of , with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to 0 while the distribution function concentrates towards a Dirac mass in velocity centred at 0, with an exponential rate. The proof, which follows the methods introduced in Han-Kwan et al (2020 Arch. Ration. Mech. Anal. 236 1273–323), requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviours for the kinetic density, from total absorption to no absorption at all.

Global small data solutions for semilinear waves with two dissipative terms

In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity |u|^p or nonlinearity of derivative type |u_t|^p, in any space dimension ngeqslant 1, for supercritical powers p>{bar{p}}. The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive L^r-L^q long time decay estimates for the solution in the full range 1leqslant rleqslant qleqslant infty . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers p<{bar{p}}.

Stability of Vacuum for the Boltzmann Equation with Moderately Soft Potentials

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter $s\in (0,1)$, i.e. with $\gamma+2s\in(0,2]$ on the whole space $\mathbb{R}^3$. We prove that if the initial data $f_{\text{in}}$ are close to the vacuum solution $f_{\text{vac}}=0$ in an appropriate weighted norm then the solution $f$ remains regular globally in time and approaches a solution to a linear transport equation. Our proof uses $L^2$ estimates and we prove a multitude of new estimates involving the Boltzmann kernel without angular cut-off. Moreover, we rely on various previous works including those of Gressman--Strain, Henderson--Snelson--Tarfulea and Silverstre. From the point of view of the long time behavior we treat the Boltzmann collisional operator perturbatively. Thus an important challenge of this problem is to exploit the dispersive properties of the transport operator to prove integrable time decay of the collisional operator. This requires the most care and to successfully overcome this difficulty we draw inspiration from Luk's work [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11] and that of Smulevici [Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2(2):Art. 11, 55, 2016]. In particular, to get at least integrable time decay we need to consolidate the decay coming from the space-time weights and the decay coming from commuting vector fields.

The global well-posedness of the compressible fluid model of Korteweg type for the critical case

In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals $0$ at a given constant state. We show that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. Consequently, we also prove the decay estimates of the solutions to the nonlinear problem. To obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for low frequencies.

Small data solutions for the Euler-Poisson-Darboux equation with a power nonlinearity

We study the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity:utt−uxx+μtut=tα|u|p,t>t0,x∈R, where μ>0, p>1 and α>−2. Here either t0=0 (singular problem) or t0>0 (regular problem). We show that this model may be interpreted as a semilinear wave equation with borderline dissipation: the existence of global small data solutions depends not only on the power p, but also on the parameter μ. Global small data weak solutions exist if(p−1)min⁡{1,μ,μ2+1p}>2+α. In the case of α=0, the above condition is equivalent to p>pcrit=max⁡{pStr(1+μ),3}, where pStr(k) is the critical exponent conjectured by W.A. Strauss for the semilinear wave equation without dissipation (i.e. μ=0) in space dimension k. Varying the parameter μ, there is a continuous transition from pcrit=∞ (for μ=0) to pcrit=3 (for μ≥4/3). The optimality of pcrit follows by known nonexistence counterpart results for 1<p≤pcrit (and for any p>1 if μ=0).As a corollary of our result, we obtain analogous results for generalized semilinear Tricomi equations and other models related to the Euler-Poisson-Darboux equation.

Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms

We prove the global existence of small data solution in all spaces of all dimensions ngeq 1 for weakly coupled systems of semilinear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, we assume that the nonlinearity terms f(t,u) and g(t,v) satisfy some properties of parabolic equations. We study the problem in several classes of regularity.

On the critical decay for the wave equation with a cubic convolution in 3D

<p style='text-indent:20px;'>We consider the wave equation with a cubic convolution <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_{t}^2 u-\Delta u = (|x|^{- \gamma}*u^2)u $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>in three space dimensions. Here, <inline-formula><tex-math id="M1">\begin{document}$ 0&lt; \gamma&lt;3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ * $\end{document}</tex-math></inline-formula> stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then <inline-formula><tex-math id="M3">\begin{document}$ \gamma\ge2 $\end{document}</tex-math></inline-formula> assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when <inline-formula><tex-math id="M4">\begin{document}$ 2\le \gamma&lt;3 $\end{document}</tex-math></inline-formula>. In this paper, we consider the Cauchy problem for <inline-formula><tex-math id="M5">\begin{document}$ 2\le \gamma&lt;3 $\end{document}</tex-math></inline-formula> in the space-time weighted <inline-formula><tex-math id="M6">\begin{document}$ L^ \infty $\end{document}</tex-math></inline-formula> space in which functions have critical decay rate. When <inline-formula><tex-math id="M7">\begin{document}$ \gamma = 2 $\end{document}</tex-math></inline-formula>, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in [<xref ref-type="bibr" rid="b13">13</xref>]). When <inline-formula><tex-math id="M8">\begin{document}$ 2&lt; \gamma&lt;3 $\end{document}</tex-math></inline-formula>, we also prove unique global existence of solutions for small data.

Energy decay for small solutions to semilinear wave equations with weakly dissipative structure

This article gives an energy decay result for small data solutions to a class of semilinear wave equations in two space dimensions possessing weakly dissipative structure relevant to the Agemi condition.

Global Well-Posedness for the Navier–Stokes Equations with the Coriolis Force in Function Spaces Characterized by Semigroups

We consider the initial value problem for the Navier–Stokes equations with the Coriolis force. We introduce function spaces of the Besov type characterized by the time evolution semigroup associated with the linear Stokes–Coriolis operator. Then, we show the unique existence of global in time mild solutions for small initial data belonging to our function spaces in both the scaling subcritical and critical settings.

Sharp Asymptotics for the Solutions of the Three-Dimensional Massless Vlasov\u2013Maxwell System with Small Data

This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d. We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in x or v is required on the initial data, and the decay in v is in particular initially optimal. Consistently with Proposition 8.1 of Bigorgne (Asymptotic properties of small data solutions of the Vlasov–Maxwell system in high dimensions. arXiv:1712.09698, 2017), the Vlasov field is supposed to vanish initially for small velocities. In order to deal with the slow decay rate of the solutions near the light cone and to prove that the velocity support of the particle density remains bounded away from 0, we make crucial use of the null properties of the system.

On the derivative nonlinear Schrödinger equation with weakly dissipative structure

We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equation in one space dimension. Under a suitable weakly dissipative condition on the nonlinearity, we show that the small data solution has a logarithmic time decay in $L^2$.

On the Cauchy problem for the semilinear Proca equations in the de Sitter spacetime

The Cauchy problem for the semilinear Proca equations is considered in the de Sitter spacetime. The effects of the spatial expansion on the existence of solutions of the equations are considered. The global solutions for small data are shown based on the dissipative effect caused by the spatial expansion.

A test function method for evolution equations with fractional powers of the Laplace operator

In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global-in-time existence of small data solutions to the following problem: utt+(−Δ)θut+(−Δ)σu=f(u,ut),t>0,x∈Rn,u(0,x)=u0(x),ut(0,x)=u1(x),with f=|u|p or f=|ut|p, where θ≥0 and σ>0 are fractional powers.

Global Existence of Small Data Solutions to Semi-linear Fractional $$\sigma $$-Evolution Equations with Mass and Nonlinear Memory

In this paper we study the global (in time) existence of small data solutions to semi-linear fractional $$\sigma -$$ evolution equations with mass and nonlinear memory. Our main goals was to explain on the one hand the influence of the memory term and on the other hand, the influence of higher regularity of the data on qualitative properties of solutions.