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This paper investigates the temporal decay rates of solutions to the Cauchy problem of a model, which describes the combustion of the compressible fluid. Suppose that the initial data is a small perturbation near the equilibrium state (ρ ∞ , 0 , θ ∞ , ζ) , where ρ ∞ > 0 , θ ∞ < θ I (the ignition temperature), and 0 < ζ ⩽ 1 , we first establish the global-in-time existence of strong solutions via a standard continuity argument. With the additional L 1 -integrability of the initial perturbation, we then employ the Fourier theory and the cancellation mechanism of low-medium frequent part to derive the optimal temporal decay rates of all-order derivatives of strong solutions. Our work is a natural continuation of previous result in the case of θ ∞ > θ I discussed in Wang and Wen (Sci China Math 65:1199–1228 (2022). [ABSTRACT FROM AUTHOR]

The optimality of decay properties of the one-dimensional damped wave equations with potentials belonging to a certain class is discussed. The typical ingredient is a variant of Nash inequality which involves an invariant measure for the corresponding Schrödinger semigroup. This enables us to find a sharp decay estimate from above. Moreover, the use of a test function method with the Nash-type inequality provides the decay estimate from below. The diffusion phenomena for the damped wave equations with potentials are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a velocity correction projection method for the Micropolar Navier-Stokes Equations. The velocity correction method are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. At the same time, the optimal error estimate using the technique of dual norm are obtained. In this way, the divergence free of the velocity u can be conserved. Finally, the numerical results show the method has an optimal convergence order. The numerical results are consistent with our theoretical analysis, and our method is effective. [ABSTRACT FROM AUTHOR]

The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space RN$\mathbb {R}^N$ with N≥3$N\ge 3$. In our work, we suppose the initial condition v0$v_0$ stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution u∈L∞((0,T);Hs(RN)),∇u∈L2((0,T);Hs(RN)),$$\begin{equation*} \hspace*{60pt}u\in L^\infty ((0,T);H^s(\mathbb {R}^N)),\quad \nabla u\in L^2((0,T);H^s(\mathbb {R}^N)), \end{equation*}$$∇v∈L∞((0,T);Hs(RN)),∇2v∈L2((0,T);Hs(RN))$$\begin{equation*} \hspace*{55pt}\nabla v\in L^\infty ((0,T);H^s(\mathbb {R}^N)),\quad \nabla ^2v\in L^2((0,T);H^s(\mathbb {R}^N)) \end{equation*}$$for s>N2−1$s>\frac{N}{2}-1$, asking low regularity assumptions on u0$u_0$ and v0$v_0$. [ABSTRACT FROM AUTHOR]

In this article, we study the inviscid limit of the solution to the Cauchy problem of a one-dimensional viscous conservation law, where the second-order term is nonlinear. Under the assumption that the inviscid equation admits a piecewise smooth solution with two noninteracting entropy shocks, we prove that the solution of the viscous equation converges uniformly to the piecewise smooth inviscid solution away from the shocks, even the strength of shocks is not small.

This paper deals with global compactness and the multiplicity of positive solutions to problems of the type −Δ픹Nu − λu = a(x)|u|2∗−2u + f(x)in 픹N, u ∈ H1(픹N), where 픹N denotes the ball model of the hyperbolic space of dimension N ≥ 4, 2∗ = 2N N−2, N(N−2) 4 < λ < (N−1)2 4 and f ∈ H−1(픹N) (f≢0) is a nonnegative functional in the dual space of H1(픹N). The potential a ∈ L∞(픹N) is assumed to be strictly positive, such that limd(x,0)→∞a(x) = 1, where d(x, 0) denotes the geodesic distance. We establish profile decomposition of the functional associated with the above equation. We show that concentration occurs along two different profiles, namely, hyperbolic bubbles and localized Aubin–Talenti bubbles. For f = 0 and a ≡ 1, profile decomposition was studied in Bhakta and Sandeep [Poincaré–Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations 44 (2012) 247–269]. However, due to the presence of the potential a(.), an extension of profile decomposition to the present set-up is highly nontrivial and requires several delicate estimates and geometric arguments concerning the isometry group (Möbius group) of the hyperbolic space. Further, using the decomposition result, we derive various energy estimates involving the interacting hyperbolic bubbles and hyperbolic bubbles with localized Aubin–Talenti bubbles. Finally, combining these estimates with topological and variational arguments, we establish a multiplicity of positive solutions in the cases: a ≥ 1 and a < 1 separately. The equation studied in this paper can be thought of as a variant of a scalar-field equation with a critical exponent in the hyperbolic space, although such a critical exponent problem in the Euclidean space ℝN has only a trivial solution when f ≡ 0, a(x) ≡ 1 and λ < 0. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-dimensional full hydrodynamic model in a bounded domain with insulating and adiabatic boundary. The model takes the form of nonisentropic Euler-Poisson system and incorporates recombination/generation terms, describing the bipolar transport of hot carriers in semiconductor optoelectronic devices. Of particular concern are the existence, uniqueness and exponential stability of thermal equilibrium states to the model, since these mathematical results are rendered useful in numerical simulation and physical theory of semiconductors. They are rigorously proved by the perturbation argument and energy method. [ABSTRACT FROM AUTHOR]

This text discusses the behavior of solutions and the energy stability within Schwarzschild spacetimes, with a particular emphasis on the behavior of massless scalar fields under the influence of a non-rotating and spherically symmetric black hole. The stability of solutions in the proximity of the event horizon of black holes in general relativity remains an open question, especially given the difficulties introduced by minor perturbations that may resemble Kerr solutions. To address this, this work explores a simplified model, including massless scalar fields, to better understand perturbation behaviors around black holes under the Schwarzschild approach. We depart from Richard Price's work in connection with how scalar, electromagnetic, and gravitational fields behave. The tortoise coordinate transformation is considered to set the stage for numerical solutions to the wave equations. Afterward, we explore energy estimates, which are used to gauge stability and wave behavior over time. Our analysis reveals that the time evolution of the energy does not exceed twice its initial value. Further and under the assumption of initial conditions in L2-spaces, we obtain an exponential decreasing behavior in the energy time evolution. A question to continue exploring is how perturbations in L2 in the initial conditions that introduce Kerr solutions as a second-order effect in the linearized equations perturb this obtained exponential decay. [ABSTRACT FROM AUTHOR]

In this paper, we rigorously derive the compressible one-fluid Navier–Stokes equations from the scaled compressible two-fluid Navier–Stokes–Maxwell equations under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by using the elaborate energy estimates. [ABSTRACT FROM AUTHOR]

A class of fourth-order strictly pseudohyperbolic operators with variable coefficients is considered. Energy estimates are established under certain conditions on the coefficients. These estimates imply the uniqueness of the solution to the Cauchy problem and a priori estimates. [ABSTRACT FROM AUTHOR]

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Nernst-Planck-Poisson-Navier-Stokes system in ℝ³. First, we obtain a priori estimates by using energy estimates. Then, we construct an iterative solution sequence by solving the approximate problem and obtaining the local existence and uniqueness of the classical solution. Finally, combining the local existence with a priori estimates, the global existence and uniqueness of the classical solution with small initial data are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Nernst-Planck-Poisson-Navier-Stokes system in $ \mathbb{R}^{3} $. First, we obtain a priori estimates by using energy estimates. Then, we construct an iterative solution sequence by solving the approximate problem and obtaining the local existence and uniqueness of the classical solution. Finally, combining the local existence with a priori estimates, the global existence and uniqueness of the classical solution with small initial data are obtained.

Let a > 0 , b > 0 and V (x) ≥ 0 be a coercive function in R 2 . We study the solutions with normalized L 2 -norm for the following Kirchhoff type equation - a + b ∫ R 2 | ∇ u | 2 d x Δ u + V (x) u = β | u | 2 u + λ u on a suitable weighted Sobolev space H = u ∈ H 1 (R 2) : ∫ R 2 V (x) u 2 d x < ∞. Our aim is to investigate the limit behaviors of the solutions with normalized L 2 -norm for this equation as (a , b) → (0 , 0) . Moreover, the uniqueness of the solution with normalized L 2 -norm for this equation is also discussed for a, b close to 0 [ABSTRACT FROM AUTHOR]

We study the relaxation problem for a two-fluid Euler-Poisson system. We prove the global-in-time convergence of the system for smooth solutions near the constant equilibrium states. The limit system is the two-fluid drift-diffusion system as the relaxation time tends to zero. In the proof, we establish uniform energy estimates of smooth solutions for all the parameters and the time. These estimates allow us to pass to the limit in the system to obtain the limit system. Moreover, the global convergence rate of the solutions is obtained by stream function techniques. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of generalized viscoelastic nonlinear equations with Dirichlet-Neumann boundary conditions. We establish the local and uniqueness of weak solutions results in Sobolev spaces with variable exponents. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space. [ABSTRACT FROM AUTHOR]

This paper focuses on the existence of solutions and energy estimates for a specific type of periodic boundary value problem. By precisely determining the parameter's localization, we demonstrate the existence of non-zero solutions and infer the presence of multiple solutions for positive parameter values. This analysis necessitates the sublinearity of the nonlinear component at both the origin and infinity. Lastly, we present a multiplicity result along with an example. The proof relies on a local minimum theorem for differentiable functionals. [ABSTRACT FROM AUTHOR]

We prove exact controllability for quasi-linear Hamiltonian Schrödinger equations on tori of dimension greater or equal then two. The result holds true for sufficiently small initial conditions satisfying natural minimal regularity assumptions, provided that the region of control satisfies the geometric control condition. [ABSTRACT FROM AUTHOR]

This paper is a part of a series devoted to the Euclidean-hyperboloidal foliation method introduced by the authors for investigating the global existence problem associated with nonlinear systems of coupled wave-Klein–Gordon equations with small data. This method was developed especially for investigating the initial value problem for the Einstein-massive field system in wave gauge. Here, we study the (fourth-order) field equations of f(R) modified gravity and investigate the global dynamical behavior of the gravitational field in the near-Minkowski regime. We establish the existence of a globally hyperbolic Cauchy development approaching Minkowski spacetime (in spacelike, null, and timelike directions), when the initial data set is sufficiently close to an asymptotically Euclidean and spacelike hypersurface in Minkowski spacetime. We cast the (fourth-order) f(R)-field equations in the form of a second-order wave-Klein–Gordon system, which has an analogous structure to the Einstein-massive field system but, in addition, involves a (possibly small) effective mass parameter. We establish the nonlinear stability of the Minkowski spacetime in the context of f(R) gravity, when the integrand f(R) in the action functional can be taken to be arbitrarily close to the integrand R of the standard Hilbert–Einstein action. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the well-posedness of a diffuse interface model of non-Newtonian two-phase flows in ℝ 3 . First, we prove the local existence of a unique strong solution provided that the initial velocity and initial concentration of two phases are sufficiently regular. Then, by using the energy estimates and standard continuity argument, we show that there exists a unique global strong solution provided that the initial velocity and initial concentration are sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic properties of solutions for the constrained minimization problems. d b p (p) : = inf { u ∈ H V s (R 2) : ∫ R 2 | u | 2 d x = 1 } I p , b p (u) , where s ∈ (1 2 , 1) , p ∈ (0 , 2 s) , b p > 0 and I p , b p (u) : = 1 2 ∫ R 2 | (- Δ) s 2 u | 2 + V (x) | u | 2 d x - b p p + 2 ∫ R 2 | u | p + 2 d x , u ∈ H V s (R 2). First, when lim p ↗ 2 s b p = b < b ∗ , the set of minimizers of d b p (p) is compact in a suitable space as p ↗ 2 s . In addition, when lim p ↗ 2 s b p = b ≥ b ∗ , by developing suitable trial functions for some fine energy estimates, we prove that all minimizers must blow up and give decay properties of minimizers. [ABSTRACT FROM AUTHOR]

We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f (u) , u ∈ D 1,2 (R N) , where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c0 if N = 5, 6 and smaller than 10c0 if N ≥ 7, where c0 is the ground state energy. [ABSTRACT FROM AUTHOR]

This text discusses the behavior of solutions and the energy stability within Schwarzschild spacetimes, with a particular emphasis on the behavior of massless scalar fields under the influence of a non-rotating and spherically symmetric black hole. The stability of solutions in the proximity of the event horizon of black holes in general relativity remains an open question, especially given the difficulties introduced by minor perturbations that may resemble Kerr solutions. To address this, this work explores a simplified model, including massless scalar fields, to better understand perturbation behaviors around black holes under the Schwarzschild approach. We depart from Richard Price’s work in connection with how scalar, electromagnetic, and gravitational fields behave. The tortoise coordinate transformation is considered to set the stage for numerical solutions to the wave equations. Afterward, we explore energy estimates, which are used to gauge stability and wave behavior over time. Our analysis reveals that the time evolution of the energy does not exceed twice its initial value. Further and under the assumption of initial conditions in L2−spaces, we obtain an exponential decreasing behavior in the energy time evolution. A question to continue exploring is how perturbations in L2 in the initial conditions that introduce Kerr solutions as a second-order effect in the linearized equations perturb this obtained exponential decay.

In this paper, we studied the stability of traveling wave solutions of a two-species Lotka-Volterra competition model in the form of a coupled system of reaction diffusion equations with nonlocal intraspecific and interspecific competitions in space at times. First, the uniform upper bounds for the solutions of the model was proved. By using the anti-weighted method and the energy estimates, the asymptotic stability of traveling waves with large wave speeds of the system was established.

This paper concerns a two-fluid Euler-Maxwell system for plasmas with two small parameters. We study the non-relativistic quasi-neutral limit for smooth solutions of the system in whole space R 3. When the initial data are smooth and sufficiently close to constant equilibrium states, we give a rigorous justification of the limit for all time. The limit system is governed by a compressible Euler system. The proof is based on uniform energy estimates and various dissipative estimates of smooth solutions with respect to the parameters and time. A key step in these estimates is to control the quasi-neutrality of the velocities by using a projection operator. [ABSTRACT FROM AUTHOR]

We study the small vibrations of an axially travelling string with a dashpot damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed $ \mathbf{v} $ v . For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower exponentially decaying estimates for the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case $ \mathbf{v}=0 $ v = 0. [ABSTRACT FROM AUTHOR]

We consider a constrained minimization problem arising in fractional NLS equation with inhomogeneous perturbation 0 < f (x) ≤ 1 . By exploring some delicate energy estimates and studying decay properties of solution sequences, we obtain the concentration behavior of each minimizer of the fractional NLS energy functional when a ↗ a ∗ : = ‖ Q ‖ 2 2 s , where Q is the unique positive radial solution of (- Δ) s u + s u - | u | 2 s u = 0 in R 2 . Moreover, the uniqueness of the minimizers is also discussed when a > 0 is small enough. Note that we generalize the results of papers (Du et al. in Proc R Soc Edinb Sect A 149:617–653, 2019; Deng et al. in Calc Var Partial Differ Equ 54:99–118, 2015; Guo and Seiringer in Lett Math Phys 104:141–156, 2013) in different senses. [ABSTRACT FROM AUTHOR]

In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction 1≤n≤4$$ 1\le n\le 4 $$. The method we use is based on a finite‐dimensional approach by combining the Faedo‐Galerkin method with suitable energy estimates and multiplier techniques. [ABSTRACT FROM AUTHOR]

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting. [ABSTRACT FROM AUTHOR]

This paper addresses the low Mach number limit for two-dimensional Navier–Stokes–Korteweg systems. The primary purpose is to investigate the relevance of the capillarity tensor for the analysis. For the sake of a concise exposition, our considerations focus on the case of the quantum Navier-Stokes (QNS) equations. An outline for a subsequent generalization to general viscosity and capillarity tensors is provided. Our main result proves the convergence of finite energy weak solutions of QNS to the unique Leray-Hopf weak solutions of the incompressible Navier-Stokes equations, for general initial data without additional smallness or regularity assumptions. We rely on the compactness properties stemming from energy and BD-entropy estimates. Strong convergence of acoustic waves is proven by means of refined Strichartz estimates that take into account the alteration of the dispersion relation due to the capillarity tensor. For both steps, the presence of a suitable capillarity tensor is pivotal.

In this paper, we are concerned with the nonlinear stability of traveling wave solutions for a conservation laws arising from the well‐known chemotaxis model with logarithmic sensitivity. By constructing appropriate "ansatz," we shall prove the nonlinear stability of traveling waves even though the perturbations oscillate at the far fields. More precisely, it is shown that as time tends to infinite, the solutions converge to arbitrarily large amplitude viscous shock wave solutions with a shift, which are partially determined by the periodic oscillations. [ABSTRACT FROM AUTHOR]

We study the small vibrations of axially moving strings described by a wave equation in an interval with two endpoints moving in the same direction with a constant speed. The solution is expressed by a series formula where the coefficients are explicitly computed in function of the initial data. We also define an energy expression for the solution that is conserved in time. Then, we establish boundary observability inequalities with explicit constants. [ABSTRACT FROM AUTHOR]

The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by ε, goes to zero, they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L∞(L2) ∩ L2(H1) sense on a uniform time interval independent of the small parameter ε. The proof consists of two parts: One is the construction of a suitable approximate solution and another is the estimate of the error functions in Sobolev spaces. Besides the careful energy estimates, a spectral estimate of the linearized operator for the incompressible Navier-Stokes and Allen-Cahn coupled system around the approximate solution is essentially used to derive the uniform estimates of the error functions. The convergence of the velocity is well expected due to the fact that the layer of the velocity across the diffuse interfacial zone is relatively weak. [ABSTRACT FROM AUTHOR]

In this paper, we study a fractional NLS system with trapping potentials in R 2 . By constructing a constrained minimization problem, we show that minimizers exist for the minimization problem if and only if the attractive interaction strength a i < a ∗ : = ‖ Q ‖ 2 2 s , where i = 1 , 2 and Q is the unique positive radial solution of (- Δ) s u + s u - | u | 2 s u = 0 in R 2 , s ∈ (0 , 1) . Moreover, by analyzing some precise energy estimates, we obtain the concentration and blow-up behavior for the minimizers of the minimization problem as (a 1 , a 2) ↗ (a ∗ , a ∗) . Comparing to the NLS system and fractional NLS equation, we encounter some new difficulties because of the nonlocal nature of the fractional Laplace. One of the main difficulties is that the energy functional is changed, we have to develop a suitable trial function to do some precise integral computation for the energy of minimization problem. Another difficulty is given by the fact that the Q(x) is polynomially decay at infinity, which is in contrast to the fact that the ground state exponentially decays at infinity in s = 1 , we need to give a more detailed proof to establish the best estimate of the trial function. The last major difficulty lies in the decay estimates of the sequences of solution to the nonlocal problem at infinity are different from those in the case of the classical local problem, we must build decay estimates for nonlocal operators. [ABSTRACT FROM AUTHOR]

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri-Giusti to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions. [ABSTRACT FROM AUTHOR]

This paper studies the stability of weak dispersive shock profiles for a quantum hydrodynamics system in one space dimension with nonlinear viscosity and dispersive (quantum) effects due to a Bohm potential. It is shown that, if the shock amplitude is sufficiently small, then the profiles are spectrally stable. This analytical result is consistent with numerical estimations of the location of the spectrum [43]. The proof is based on energy estimates at the spectral level, on the choice of an appropriate weighted energy function for the perturbations involving both the dispersive potential and the nonlinear viscosity, and on the montonicity of the dispersive profiles in the small-amplitude regime. [ABSTRACT FROM AUTHOR]

In the present paper, we study the influence of oscillations of the time-dependent damping term b (t) u t on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation u t t + (− Δ) σ u + b (t) u t = 0 , (t , x) ∈ [ 0 , ∞) × R n , u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b (t) = β (1 + t) − 1 , with β > 0 , and we discuss the influence of oscillations of b on the energy estimates according to the size of β. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlinear stability of contact waves to the Cauchy problem of the compressible Euler-Fourier system with a reacting mixture in one dimension under the non-zero mass condition. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the decay rate of contact waves by using anti-derivative methods and elaborated energy estimates. [ABSTRACT FROM AUTHOR]

We prove the nonlinear stability of large-amplitude traveling waves under space-periodic perturbations to a parabolic-hyperbolic system derived from a singular chemotaxis model describing the initiation of tumor angiogenesis. Compared with the previous results, we are able to prove the nonlinear stability of traveling waves even though the perturbation oscillates at the far fields. [ABSTRACT FROM AUTHOR]

We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P) (− Δ) s u + V (x) u = μ u + a m (x) ∣ u ∣ 4 s N u , ∫ ℝ N ∣ u ∣ 2 d x = 1 , u ∈ H s (ℝ N) , where s ∈ (0, 1), μ ∈ ℝ, a > 0, V (x) and m(x) are L∞(ℝN) functions with N ≥ 2. We prove that there is a threshold a s ∗ > 0 such that problem (P) has a least energy solution ua (x) for each a ∈ (0 , a s ∗) and ua blows up, as a ↗ a s ∗ , at some point x0 ∈ ℝN, which makes V (x0) be the minimum and m(x0) be the maximum. Moreover, the precise blowup rates for ua are obtained under suitable conditions on V (x) and m(x). [ABSTRACT FROM AUTHOR]

We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P) (− Δ) s u + V (x) u = μ u + a m (x) ∣ u ∣ 4 s N u , ∫ ℝ N ∣ u ∣ 2 d x = 1 , u ∈ H s (ℝ N) , where s ∈ (0, 1), μ ∈ ℝ, a > 0, V (x) and m(x) are L∞(ℝN) functions with N ≥ 2. We prove that there is a threshold a s ∗ > 0 such that problem (P) has a least energy solution ua (x) for each a ∈ (0 , a s ∗) and ua blows up, as a ↗ a s ∗ , at some point x0 ∈ ℝN, which makes V (x0) be the minimum and m(x0) be the maximum. Moreover, the precise blowup rates for ua are obtained under suitable conditions on V (x) and m(x). [ABSTRACT FROM AUTHOR]

In this paper, we consider the decay rates of the higher-order spatial derivatives of solution to the Cauchy problem of 3D compressible nematic liquid crystals system. The $ \dot{H}^s\; (\frac12 < s < \frac32) $ negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.

This paper concerns the quasi-neutral limit to the Cauchy problem for a two-fluid Euler-Poisson system in several space dimensions. When the initial data are sufficiently close to constant equilibrium states, we prove the global existence of smooth solutions with uniform bounds with respect to the Debye length in Sobolev spaces. This allows us to pass to the limit in the system for all times to obtain a compressible Euler system. We also prove global error estimates between the solution of the two-fluid Euler-Poisson system and that of the compressible Euler system. These results are obtained by establishing uniform energy estimates and various dissipation estimates. A key step in the proof is the control of the quasi-neutrality of the velocities. For this purpose, an orthogonal projection operator is used. [ABSTRACT FROM AUTHOR]