Global Existence Result Research Articles

Small data global existence for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions

This paper provides a small data global existence result for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions. This is an extension of the previous results by Li (2012) [25] and Ikeda–Katayama–Sunagawa (2015) [14].

Almost Global Existence for the Fractional Schrödinger Equations

We study the time of existence of the solutions of the following nonlinear Schrodinger equation (NLS) $$\begin{aligned} \hbox {i}u_t =(-\Delta +m)^su - |u|^2u \end{aligned}$$ on the finite x-interval $$[0,\pi ]$$ with Dirichlet boundary conditions $$\begin{aligned} u(t,0)=0=u(t,\pi ),\qquad -\infty< t<+\infty , \end{aligned}$$ where $$(-\Delta +m)^s$$ stands for the spectrally defined fractional Laplacian with $$0<s<1/2$$ . We prove an almost global existence result for the above fractional Schrodinger equation, which generalizes the result in Bambusi and Sire (Dyn PDE 10(2):171–176, 2013) from $$s>1/2$$ to $$0<s<1/2$$ .

Global solutions and blow-up phenomena for a generalized Degasperis–Procesi equation

In this paper, we mainly investigate global solutions and blow-up phenomena for the Cauchy problem of a generalized Degasperis–Procesi equation. We develop a new approach to consider the equation for (1−∂x)u instead of (1−∂x2)u. We first prove a new conserved quantity ‖(1−∂x)u‖L1 and give the local characterization of this conserved quantity. These properties then allow us to improve considerably a previous global existence result in [38], and establish two new blow-up results for strong solutions to the equation. In the end, based on vanishing viscosity method and using the conserved quantity we deduce the existence of global weak solutions to the equation for any initial data belonging to L1∩BV.

Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability

We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across the fixed-in-time curve ( x 1 , − 1 + f ( x 1 ) ) (x_1,-1+f(x_1)) , while the interface separating the fluid from the vacuum region is given by the time-dependent curve ( x 1 , h ( x 1 , t ) ) (x_1,h(x_1,t)) . Our estimates are based on a new methodology that relies upon a careful study of the PDE system, coupling Darcy’s law and incompressibility of the fluid, rather than the analysis of the singular integral contour equation for the interface function h h . We are able to develop an existence theory for any initial interface given by h 0 ∈ H 2 h_0 \in H^2 and any permeability curve-of-discontinuity that is given by f ∈ H 2.5 f \in H^{2.5} . In particular, our method allows for both curves to have (pointwise) unbounded curvature. In the case that the permeability discontinuity is the set f = 0 f=0 , we prove global existence and decay to equilibrium for small initial data. This decay is obtained using a new energy-energy dissipation inequality that couples tangential derivatives of the velocity in the bulk of the fluid with the curvature of the interface. To the best of our knowledge, this is the first global existence result for the Muskat problem with discontinuous permeability.

Normalized standing waves for the Hartree equations

In this paper, we study the existence and stability/instability of standing waves with a prescribed L2-norm for the following Hartree equation(0.1)iut+Δu−a|x|2u+(|x|−λ⁎|u|2)u=0,(t,x)∈R+×RN, where t is time, N≥3, λ∈(0,N) and a≥0 is a constant. To get such solutions, we look for normalized solutions of the associated stationary equation(0.2)−Δu+a|x|2u+ωu−(|x|−λ⁎|u|2)u=0,x∈RN. Two cases are considered: (i) the system is free (a=0) and (ii) a harmonic trapping potential is added in the system (a>0). In case (i), by constructing a suitable submanifold of a L2-sphere, we prove the existence of a normalized solution for problem (0.2) with least energy in the L2-sphere, which corresponds to a normalized ground state standing wave of problem (0.1). Then, we show that each normalized ground state of problem (0.2) coincides a ground state of problem (0.2) in the usual sense. Furthermore, we give a global existence result for problem (0.1) and also show that any normalized ground state standing wave is strongly unstable. In case (ii), for small a>0, we prove the existence of two different standing waves, one corresponds to a topological local minimizer of the constrained energy functional, the other corresponds to a mountain pass type solution of problem (0.2) with its energy strictly larger than that of the normalized ground state. Moreover, we show that the set of normalized ground state solutions for problem (0.2) is stable under the associated flow. Finally, to connect case (i) and case (ii), we analyze the asymptotic behavior of the two solutions to problem (0.2) obtained in case (ii) as the trapping frequency a vanishes.

Global existence for the periodic dispersive Hunter–Saxton equation

In this paper, we study an integrable dispersive Hunter–Saxton equation in periodic domain. Firstly, we establish the local well-posedness of the Cauchy problem of the equation in $$H^s ({\mathbb {S}}), s \ge 2,$$ by applying the Kato method. Then, based on a sign-preserve property, we obtain a global existence result for the equation. Moreover, we extend the obtained result to some periodic nonlinear partial differential equations of second order of the general form.

Compressible fluids and active potentials

We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.

Global Existence Results for Neutral Functional Differential Inclusions with State-Dependent Delay

We study the existence of global solutions for a class of neutral functional differential inclusions with state-dependent delay. The proof of the main result is based on the semigroup theory and the Bohnenblust–Karlin fixed-point theorem.

A global existence result for a semilinear scale‐invariant wave equation in even dimension

In the present article a, semilinear scale‐invariant wave equation with damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension n is proved by using weighted L∞ − L∞ estimates, under the assumption that the multiplicative constants, which appear in the coefficients of damping and of mass terms, fulfill an interplay condition, which yields somehow a “wave‐like” model. In particular, combining this existence result with a recently proved blow‐up result, a suitable shift of Strauss exponent is proved to be the critical exponent for the considered model. Moreover, the still open part of a conjecture done by D'Abbicco‐Lucente‐Reissig is proved to be true in the massless case.

On potential wells to a semilinear hyperbolic equation with damping and conical singularity

This article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by Alimohammady et al. (2017) [1], where the weak solution with low initial energy (I(0)<d,d is the potential well depth) was considered. We extend the previous results in the following three aspects. Firstly, we consider the vacuum isolating behavior of the solution for initial energy I(0)≤0 and 0<I(0)<d, respectively. We find that there are two explicit vacuum regions: an annulus and a ball. Moreover, we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time, and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained. Secondly, we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results. Finally, the weak solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the weak solution exists globally and blows up in finite time are given.

Global existence of the harmonic map heat flow into Lorentzian manifolds

We investigate a parabolic–elliptic system for maps (u,v) from a compact Riemann surface M into a Lorentzian manifold N×R with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the v-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic–elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data.

Regularity results for a class of generalized surface quasi-geostrophic equations

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high frequencies and inner/outer spatial scales to pass to the limit; while the proof of both the global regularity result and the eventual regularity for the supercritical diffusion are essentially based on the use of the so-called modulus of continuity method.

Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition

We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature of the new proof lies in that it never uses the Lorentz boost operator in the energy integral argument. The proof presented here has an advantage over the former one in that the assumption of compactness of the support of data can be eliminated and the amount of regularity of data can be lowered in a straightforward manner. A recent result of Zha for the scalar unknowns is also refined.

Boundedness in a fully parabolic chemotaxis system with logistic-type source and nonlinear production

In this paper we study the fully parabolic chemotaxis system with logistic-type source and nonlinear production: ut=Δu−χ∇⋅(u∇v)+f(u), vt=Δv−v+g(u), subject to the non-flux boundary conditions with a smooth and bounded domain Ω⊂Rn (n≥1), and the nonnegative initial data u0∈C0(Ω̄) and v0∈W1,∞(Ω), where the sensitivity χ>0, the logistic-type source f(s)≤s−μsα for s≥0 with α>1, μ>0, and the nonlinear production g(s)≤s(s+1)β−1 for s≥0 with β>0. It is obtained that the problem possesses a globally bounded classical solution if the self-restriction mechanism of the logistic-type source dominates the nonlinear production that β<α−1, or β=α−1 with μ>0 sufficiently large. This extends the current global existence result by Nakaguchi and Osaki (2018). The situation with the nonlinear production controlled by the linear diffusion that β∈(0,2n) is considered as well, which is parallel to the model with nonlinear diffusion and subcritical chemotactic sensitivity without logistic source studied by Tao and Winkler (2012).

Global dynamics of delayed intraguild predation model with intraspecific competition

A delayed intraguild predation (IGP) model with intraspecific competition is considered. It is shown that the delay has a destabilizing effect and induces oscillations. The global existence results of periodic solutions bifurcating from the positive equilibrium are established. It is shown that there exists at least one nontrival periodic solution when the delay passes through a certain critical value. Numerical simulations are performed to illustrate our theoretical results and show that intraspecific competition can also affect the stability of the positive equilibrium of the system.

On the Cauchy problem for a generalized Degasperis-Procesi equation

ABSTRACTWe first establish the local well-posedness for a generalized Degasperis-Procesi equation in nonhomogeneous Besov spaces. Then we present a global existence result for the equation. Moreover, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

Space dependent adhesion forces mediated by transient elastic linkages: New convergence and global existence results

In the first part of this work we show the convergence with respect to an asymptotic parameter ε of a delayed heat equation. It represents a mathematical extension of works considered previously by the authors [14–16]. Namely, this is the first result involving delay operators approximating protein linkages coupled with a spatial elliptic second order operator. For the sake of simplicity we choose the Laplace operator, although more general results could be derived. The main arguments are (i) new energy estimates and (ii) a stability result extended from the previous work to this more involved context. They allow to prove convergence of the delay operator to a friction term together with the Laplace operator in the same asymptotic regime considered without the space dependence in [14]. In a second part we extend fixed-point results for the fully non-linear model introduced in [16] and prove global existence in time. This shows that the blow-up scenario observed previously does not occur. Since the latter result was interpreted as a rupture of adhesion forces, we discuss the possibility of bond breaking both from the analytic and numerical point of view.

Viability for stochastic functional differential equations in Hilbert spaces driven by fractional Brownian motion

In this paper, we consider a class of stochastic functional differential equations in Hilbert spaces driven by a fractional Brownian motion with Hurst parameter 1/2 < H < 1. By using pathwise approach, we prove a global existence and uniqueness result of the mild solution for the equations considered under some local Lipschitz conditions. Subsequently, by establishing some new estimates, we also prove some viability results to the stochastic systems under investigation.

Global weak solutions for a generalized Camassa–Holm equation

AbstractIn this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data.

An integrable example of gradient flow based on optimal transport of differential forms

Optimal transport theory has been a powerful tool for the analysis of parabolic equations viewed as gradient flows of volume forms according to suitable transportation metrics. In this paper, we present an example of gradient flows for closed (d-1)-forms in the Euclidean space R^d. In spite of its apparent complexity, the resulting very degenerate parabolic system is fully integrable and can be viewed as the Eulerian version of the heat equation for curves in the Euclidean space. We analyze this system in terms of ``relative entropy" and ``dissipative solutions" and provide global existence and weak-strong uniqueness results.