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This paper concerns with the one dimensional compressible isentropic Navier–Stokes equations with a free boundary separating fluid and vacuum when the viscosity coefficient depends on the density. Precisely, the pressure P and the viscosity coefficient μ are assumed to be proportional to ρ γ and ρ θ respectively, where ρ is the density, and γ and θ are constants. We establish the unique solvability in the framework of global classical solutions for this problem when γ ≥ θ > 1. Since the previous results on this topic are limited to the case when θ ∈ (0 , 1 ] , the result in this paper fills in the gap for θ > 1. Note that the key estimate is to show that the density has a positive lower bound and the new ingredient of the proof relies on the study of the quasilinear parabolic equation for the viscosity coefficient by reducing the nonlocal terms in order to apply the comparison principle. [ABSTRACT FROM AUTHOR]

Global existence and boundedness of solutions to the Cauchy problem for the four dimensional fully parabolic chemotaxis system with indirect signal production are studied. We prove that solutions with initial mass below (8 π) 2 exist globally in time. This value (8 π) 2 is known as the four dimensional threshold value of the initial mass determining whether blow-up of solutions occurs or not. Furthermore, some condition on the initial mass guaranteeing that the solution remains uniformly bounded is also obtained. [ABSTRACT FROM AUTHOR]

We study a fourth‐order hyperbolic equation involving Kirchhoff type p$$ p $$‐Laplacian and superlinear source, subject to zero Navier boundary condition, utt+Δ2u−a+b‖∇u‖ppdiv(|∇u|p−2∇u)=c|u|q−2u,(x,t)∈Ω×(0,T),$$ {u}_{tt}&#x0002B;{\Delta}&#x0005E;2u-\left(a&#x0002B;b{\left\Vert \nabla u\right\Vert}_p&#x0005E;p\right)\operatorname{div}\left({\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;{p-2}\nabla u\right)&#x0003D;c{\left&#x0007C;u\right&#x0007C;}&#x0005E;{q-2}u,\left(x,t\right)\in \Omega \times \left(0,T\right), $$where Ω$$ \Omega $$ is an open bounded domain in ℝn$$ {\mathrm{\mathbb{R}}}&#x0005E;n $$ with n≥3$$ n\ge 3 $$; T(≤+∞)$$ T\left(\le &#x0002B;\infty \right) $$ denotes the maximal existence time; and a,b>0$$ a,b>0 $$ and p,q>2$$ p,q>2 $$ are constants. For q<2p$$ q<2p $$, using auxiliary function method and Sobolev inequality, we prove that there are only global solutions. For q≥2p$$ q\ge 2p $$, we obtain the optimal classification of initial energy and Nehari energy, which guarantees the existence of blow‐up solutions and global solutions. In the critical case q=2p$$ q&#x0003D;2p $$, we find out that the coefficients of the Kirchhoff term and the superlinear source play important role in separating out the property of weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a food chain model in which the primary predator moves directly toward areas of high prey density. Simultaneously, the primary predator, which serves as the prey for the secondary predator, indirectly influences the directional movements of the secondary predator through cues such as chemical signals, scents, or excretions. We investigate whether the distinct influences of direct taxis and indirect taxis, as observed in prey–predator dynamics, are also manifested in the proposed food chain model. Our study demonstrates that the model, which incorporates both direct and indirect prey‐taxis, possesses bounded and global solutions up to three‐dimensional space. [ABSTRACT FROM AUTHOR]

In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is sufficiently small. Moreover, despite the absence of zeroth-order dissipation for the magnetic field, we are able to derive exponential decay estimates for solutions in higher-order regularity space. This is achieved by leveraging the higher-order dissipation properties of the magnetic field, which are deduced from the Maxwell equation. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear thermo -viscoelastic system with logarithmic nonlinearity of the form u tt - Δ u + ∫ 0 t g (t - s) Δ u (s) d s + v = | u | p - 2 u ln | u | , v t - Δ v = u t , on a bounded domain Ω ⊂ R n . By using the modified potential-well method, we first show the global existence and the finite-time blow-up results of solutions. Further, we give explicit and general decay rates of the energy functional associated with the global solution under a general class of relaxation function g which includes exponential, logarithmic, and polynomial rates. Besides, the lower and upper bounds for the blow-up time are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial-boundary value problem for finitely degenerate pseudo-parabolic equation associated with the Hörmander type operator. We first specify the conditions under which the weak solutions of the equation exist globally and finite time blow up when the initial energy J (u 0) > d . Especially, the life span of the blow-up solution and the blow-up result with arbitrarily high initial energy level are derived. Additionally, we show the solution of degenerate pseudo-parabolic problem will converge to the steady state solution. Finally, we obtain the existence of ground state solution to the stationary problem. We prove that there exists initial value such that the instability occurs as a blow-up in finite time and the corresponding solution exists globally, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the time-periodic solutions of one-dimensional compressible Euler equations with linear damping $ \alpha(x)\rho u $. The boundary conditions have no dissipative structure on one side, which involves some physical reality. By using a delicately designed iteration scheme, we establish the existence and stability of time-periodic solutions on the perturbation of a subsonic Fanno flow. Moreover, if the boundary has a higher regularity, we show that the time-periodic solution has the same higher regularity and stability. Our results are also valid for Euler equations without damping, in that case the steady subsonic background solution is a constant state. Both isentropic and isothermal polytropic gases are considered. [ABSTRACT FROM AUTHOR]

In this paper, we delve into an initial-boundary value problem of the dispersive-dissipative wave equation having strong damping, $ u_{tt}-\Delta u-\Delta u_{t}+\Delta^{2} u-\Delta u_{tt} = a|u|^{p-1} u-b|u|^{q-1} u \quad \text{in}\ \Omega\times(0, T_{\max}), $where $ \Omega $ is an open bounded domain in $ \mathbb{R}^{n} $ with smooth boundary; $ T_{\max} $ is the maximal existence time; $ a, b>0 $, $ p, q>1 $ are constants. Through the combination of the contraction mapping principle and the Faedo-Galerkin method, we achieve the local existence and uniqueness of the weak solution. For $ p>q $ or $ p = q $ and $ a>b $, we obtain the existence of the global weak solution, and the energy decay estimate. Moreover, based on the auxiliary function approach, we demonstrate that the weak solutions blow up in finite time under distinct conditions related to the initial energy. Furthermore, we succeed in obtaining lower and upper bounds for blow-up time (life span). For $ p

While much literature on chemotaxis systems focuses on bounded domains, this paper emphasizes the global existence of classical solutions for three primary chemotaxis systems with a logistic source on R n. We present a unified proof demonstrating global existence of solutions can be deduced from their locally uniform boundedness in L p (R n) for some p > max ⁡ { 1 , n 2 }. We then provide sufficient conditions for the global existence and boundedness of classical solutions. Notably, our findings even improve several existing results for bounded domains. [ABSTRACT FROM AUTHOR]

In this work, we deal with a fourth‐order parabolic equation with variable exponent logarithmic nonlinearity. We obtain the global existence and blowup solutions using the energy functional and potential well method. [ABSTRACT FROM AUTHOR]

This article is dedicated to the no-flux attraction-repulsion chemotaxis model { u t = Δ u − χ ∇ ⋅ (u v ∇ v) + ξ ∇ ⋅ (u w ∇ w) + h (u) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = Δ w − w + u , x ∈ Ω , t > 0 defined in a smooth and bounded domain Ω ⊂ R n (n ≥ 1). The parameters χ , ξ > 0 and the logistic source fulfilling h (s) ≤ r − μ s γ or h (s) = r s − μ s 2 with r ≥ 0 , μ > 0 and proper γ > 0 are used in certain scenarios. Our first result exhibits that the corresponding initial-boundary value problem admits a classical solution which is global if γ > 2 or γ = 2 and μ is sufficiently large. Moreover, assuming that h (s) satisfies the latter condition, the global existence of a weak solution is achieved. [ABSTRACT FROM AUTHOR]

In this paper, we study the diffusive Lotka–Volterra system containing the variable coefficients. In the Sobolev‐space settings, we achieve the well‐posedness properties (existence and regularity) and large‐time asymptotics (continuation and global existence or finite‐time blowup) of the solutions in the Sobolev spaces and time‐weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

This work deals with a three-component chemotaxis system u t = Δ u - ∇ · (u χ 1 (w) ∇ w) + w - μ 1 u 2 , x ∈ Ω , t > 0 , v t = Δ v - ∇ · (v χ 2 (w) ∇ w) + w + r u v - μ 2 v 2 , x ∈ Ω , t > 0 , w t = Δ w + u + v - w , x ∈ Ω , t > 0 , describing the spatiotemporal Alopecia Areata dynamics with signal-dependent sensitivity under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω ⊂ R n (n ≥ 1) , where the parameters μ 1 , μ 2 , r are positive constants. The chemotactic function χ i (w) (i = 1 , 2) is smooth and satisfying some conditions. Given suitably regular initial data, it is shown that the corresponding initial-boundary value problem admits a unique global bounded classical solution. Furthermore, by constructing the appropriate energy function, it is proved that the unique coexistence equilibrium is globally convergent in a specific parameter regime. These results generalize and improve some well-known results in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the initial-boundary value problem of a possibly degenerate reaction-diffusion system over Ω ⊂ R n with n ≥ 1 of the following form ∂ t m i - κ Δ m i + | m i | γ - 2 m i = (∂ x i p) 2 , - ∇ · [ m ∇ p ] = S , with m = diag (m 1 , ⋯ , m n) , the diffusivity κ > 0 , the metabolic exponent γ ≥ 2 and the given function S. When κ = 0 , this system was introduced by Haskovec, Kreusser and Markowich as a continuous version of the discrete Hu-Cai model for biological transport networks. In this work, our result asserts that whenever the random fluctuations of the conductance in the medium were considered, i.e., κ > 0 , then for general large data the corresponding initial-boundary value problem possesses a global weak solution. [ABSTRACT FROM AUTHOR]

The initial boundary-value problem for a plate equation with nonlocal damping is considered. The local existence of solution is proved by the monotone operator theory with locally Lipschitz perturbation. By using the potential well theory, we get the global existence of solution. By Nakao's inequality, the decay estimate of polynomial type is obtained. We provide also the sufficient conditions of finite time blow-up of weak solutions with suitable conditions on the initial data by an ordinary differential inequality for an appropriately chosen functional. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic Kirchhoff equation with a delay term in the internal feedback. By using the Faedo–Galerkin approximation method, we prove the well posedness of the global solutions. Introducing suitable energy, we prove the general uniform decay results. [ABSTRACT FROM AUTHOR]

We consider the no-flux initial-boundary value problem for the growth-expansion model with chemotaxis in nutrient-replete environments in smoothly bounded domains Ω ⊂ ℝ n . It is shown that if n = 1 or if n = 2 under some structural assumptions on parameter functions therein, for any suitably regular initial data the problem admits a unique global bounded classical solution. For any dimensions n ≥ 3 , we also prove that the problem has a unique global classical solution which is bounded under a small assumption on the initial data. Moreover, we obtain that these solutions stabilize to a uniquely determined spatially uniform equilibrium. We also provide exponential rates of convergence of solutions in a special case. [ABSTRACT FROM AUTHOR]

We prove that the quartic Korteweg-de Vries equation is globally well-posed for real-valued initial data in H^s(\mathbb {R}), s>-1/24. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the Cauchy problem for a weakly coupled system of two semi-linear structurally damped σ k -evolution equations, where σ k ≥ 1 for k = 1 , 2 . Our first purpose involves the proof of global (in time) existence of small data energy solutions by mixing additional L m k regularity for the data, where m k ∈ [ 1 , 2) . We want to point out that in some cases of choosing suitable parameters m k , with k = 1 , 2 , the obtained lower bound of one exponent p or q related to power nonlinearities on the right-hand sides is really smaller than the critical exponent, the so-called modified Fujita exponent. The second aim of this paper is to discuss a blow-up result for Sobolev solutions with any different fractional values of σ k ≥ 1 when m 1 = m 2 . [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the non-isentropic compressible Euler equations in ducts with varying cross-sectional area and friction. We analyze the combined effect of cross-sectional area and friction subtly and investigate the motion trend of the corresponding steady flow along ducts. Moreover, for the time-dependent system, when the boundary perturbations are temporal-periodical, we can utilize an energy function and Gronwall's inequality to demonstrate that the smooth supersonic solution is a temporal periodic solution, after a certain start-up time, possessing the same period as the boundary conditions. The semi-global existence of a $ C^{1} $ solution to the mixed initial-boundary value problem is used in the proof. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for a singular parabolic $ p $-biharmonic equation with gradient-type logarithmic nonlinearity. By the cut-off technique, combined with the methods of Faedo-Galerkin approximation and multiplier, we establish the local solvability. Further, by virtue of the family of potential wells and the modified differential inequality technique, we derive the threshold results of global existence and nonexistence with low initial energy conditions ($ J(u_{0})\leq d $), the decay property, blow-up with negative initial energy, the estimates of blow-up rate, and the bounds of life span of blow-up solutions with subcritical initial energy. Furthermore, we adopt the Hardy-Sobolev inequality to show the finite time blow-up result with arbitrary initial energy. [ABSTRACT FROM AUTHOR]

In this paper, a class of mixed pseudo-parabolic-Kirchhoff equation is considered. We focus on the interaction between the nonlocal diffusion and the nonstandard growth condition. More specifically, we establish a threshold result for the global solutions and finite time blow-up of the weak solution for low and critical initial energy. Meanwhile, we give the upper and lower bound estimates of blow-up time for the blow-up solution. And then we investigate the convergent relation between the global solutions and the ground-state solutions. Finally, we give some conditions to ensure the global existence or blow-up of solutions for the high initial energy. [ABSTRACT FROM AUTHOR]

We consider the global existence and regularity of the solution to the stage-structured SIR system for rumor spreading. To obtain the local existence result of a weak solution to the system, we reduce the system to a single equation using the SIR structure and employ a fixed point argument with a Banach space and Volterra's formula. By defining an appropriate weak solution of the system, we show that the total mass is conserved. Using the local solution to the single equation and the mass conservation property of the system, we verify the global existence of a weak solution to the stage-structured SIR system. If the initial data are regular and satisfy some compatible conditions, then we prove that the solution is piecewise continuously differentiable. [ABSTRACT FROM AUTHOR]

We consider a dissipative Baer-Nunziato type system for a mixture of two compressible heat conducting gases in the whole space. The existence theory of the global solution to the system is established in H N (R 3) -framework (N ≥ 2) when only the H 2 -norm of the initial perturbation is small. The weighted energy method combined with the low-frequency and high-frequency decomposition is used to derive the desired a priori estimates and hence the global existence. As a by-product, we obtain the decay rates of spatial derivatives of the solution. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate an initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation. Under the different initial energy, we study the existence, uniqueness and asymptotic behavior of the global solution and the blow up phenomenon of the weak solution. Moreover, we obtain lower and upper bound estimates for the blow-up time and rate in different initial energy cases. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the nonlinear Love equation, which describes some sort of "propagation process" in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow‐up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow‐up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well‐posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567‐2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation. [ABSTRACT FROM AUTHOR]

The chemotaxis-consumption model { u t = ∇ ⋅ ((u + 1) α ∇ u) − ∇ ⋅ (u (u + 1) β − 1 ∇ v) , x ∈ Ω , t > 0 0 = Δ v − u v , x ∈ Ω , t > 0 is considered in a smooth bounded domain Ω ⊂ R n under the boundary conditions (u + 1) α ∂ ν u = u (u + 1) β − 1 ∂ ν v and ∂ ν v = (γ − v) g on ∂Ω, with parameters α , β ∈ R , γ > 0 and the nonnegative function g ∈ C 1 + ω (Ω ¯) for some ω ∈ (0 , 1). If β − α ≤ 1 , it is proved that there exists a global bounded classical solution (u , v) for suitably regular initial data u 0. Furthermore, for the given mass m = ∫ Ω u ∞ > 0 , the corresponding stationary system admits a unique classical solution (u ∞ , v ∞) , appearing as a large-time limit in the sense that if γ > 0 is small enough, then ‖ u (⋅ , t) − u ∞ ‖ L ∞ (Ω) + ‖ v (⋅ , t) − v ∞ ‖ L ∞ (Ω) → 0 as t → ∞ , with ∫ Ω u ∞ = ∫ Ω u 0 = m > 0. [ABSTRACT FROM AUTHOR]

Reaction–diffusion systems with mass dissipation are known to possess blow‐up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension 1, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, that is, nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this paper, we remove this extra entropy assumption completely and obtain global boundedness for reaction–diffusion systems with cubic intermediate sum condition. The novel idea is to show a nonconcentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space M1,δ(Ω)$\mathsf {M}^{1,\delta }(\Omega)$ for some δ>0$\delta >0$. As far as we are concerned, it is the first time such a bound is derived for mass dissipating reaction–diffusion systems. The results are then applied to obtain global existence and boundedness of solutions to an oscillatory Belousov–Zhabotinsky system, which satisfies cubic intermediate sum condition but does not fulfill the entropy assumption. Extensions include global existence mass controlled systems with slightly super cubic intermediate sum condition. [ABSTRACT FROM AUTHOR]

In this paper, we study a chemotaxis system with nonlinear indirect signal production { u t = Δ (γ (v) u) + r u − μ u l , x ∈ Ω , t > 0 , v t = Δ v − v + w β , x ∈ Ω , t > 0 , w t = − δ w + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 2) , where the parameters r , μ , β , δ > 0 , and l > 1 , the motility function γ ∈ C 3 ([ 0 , ∞)) , γ (v) > 0 is bounded, γ ′ (v) < 0 , and γ ′ (v) γ (v) is bounded. We show that if l β > n 2 , the system has a unique global classical solution. Moreover, the solution exponentially converges to ((r μ) 1 l − 1 , (1 δ) β (r μ) β l − 1 , 1 δ (r μ) 1 l − 1 )) in the large time limit under some extra hypotheses. [ABSTRACT FROM AUTHOR]

This work considers the doubly degenerate nutrient model u t = ∇ · u m - 1 v ∇ u - ∇ · f (u) v ∇ v + ℓ u v , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under no-flux boundary conditions in a smoothly bounded convex domain Ω ⊂ R n ( n ≤ 2 ), where the nonnegative function f ∈ C 1 ([ 0 , ∞)) is assumed to satisfy f (s) ≤ C f s α with α > 0 and C f > 0 for all s ≥ 1 . When m = 2 , it was shown that a global weak solution exists, either in one-dimensional setting with α = 2 , or in two-dimensional version with α ∈ 1 , 3 2 . The main results in this paper assert the global existence of weak solutions for 1 ≤ m < 3 and classical solutions for 3 ≤ m < 4 to the above system under the assumption α ∈ m - 1 , min m , m 2 + 1 if n = 1 , and m - 1 , min m , m 2 + 1 if n = 2 , which extend the range α ∈ (1 , 3 2) to α ∈ (1 , 2) in two dimensions for the case m = 2 . Our proof will be based on a new observation on the coupled energy-type functional and on an inequality with general form. [ABSTRACT FROM AUTHOR]

Keywords: Regularity criterion; bootstrap argument; Green's function; global existence; decay rate of solution.GraphBy Jie Qi and Weike WangReported by Author; Author [Extracted from the article]

In this paper, we consider 3D compressible magneto-micropolar fluids without resistivity and spin viscosity in a strip domain. The prominent character of the governing model is the presence of the microstructure, a linear coupling structure involving derivatives of the velocity fields, which along with the lack of spin viscosity brings several challenges to the analysis. By exploiting the two-tier energy method developed in Guo and Tice (Arch Ration Mech Anal, 2013), we prove the global existence of classical solutions to the governing model around a uniform magnetic field that is non-parallel to the horizontal boundary. Moreover, we show that the solution converges to the steady state at an almost exponential rate as time goes to infinity. One of the main ingredients in our analysis, compared with previous works on micropolar fluids, is that we deal with the microstructure by establishing some delicate estimates based on the analysis of the div-curl decomposition, and the coupling between the fluid velocity and the vorticity of angular velocity. [ABSTRACT FROM AUTHOR]

This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption. [ABSTRACT FROM AUTHOR]

Abstract This paper presents the investigation of a boundedness solution for a Keller–Segel chemotaxis (KSC) system with nonlocal diffusion. In order to achieve boundedness, we derive the necessary and sufficient condition based on the assumption of ensuring a unique classical global bounded solution for the proposed KSC system. In this work, the proposed system’s boundedness condition is determined based on three particular situations: when χ < χ ∗ $\chi

We study a system of inhomogeneous nonlinear Schrödinger equations that emerge in optical media with a $ \chi^{(2)} $ nonlinearity. This nonlinearity, whose local strength is subject to a cusp-shaped spatial modulation, $ \chi^{(2)}\sim |x|^{-\alpha} $ where $ \alpha > 0 $, can be induced by spatially non-uniform poling. Our first step is to establish a vectorial Gagliardo–Nirenberg type inequality related to the system. This allows us to identify the necessary conditions on the initial data that lead to the existence of global in time solutions. By exploiting the spatial decay at infinity of the nonlinearity, we demonstrate the non-radial energy scattering in the mass-supercritical regime for global solutions. These solutions have initial data that lie below a mass-energy threshold, regardless of whether the system is mass-resonant or non-mass resonant. Lastly, we provide the criteria for the existence of non-radial blow-up solutions with mass-critical and mass-supercritical nonlinearities in both mass and non-mass resonance cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behaviors of solutions to the inhomogeneous Navier-Stokes-Vlasov system in $ \mathbb{R}^{3}\times\mathbb{R}^{3} $, where the initial fluid density is allowed to vanish. We establish the uniform bound of the macroscopic density associated with the distribution function and prove the global existence and uniqueness of strong solutions to the Cauchy problem with vacuum for either small initial energy or large viscosity coefficient. The uniform boundedness and the presence of vacuum enable us to show that as the time evolves, the fluid velocity decays, while the distribution function concentrates towards a Dirac measure in velocity centred at $ 0 $, with an exponential rate. In order to overcome the degeneracy in the momentum equations, we develop an energy argument based on higher order functional inequalities designed for fluid-particle coupled structures. [ABSTRACT FROM AUTHOR]

This paper is concerned with weak solutions to an initial-boundary value problem for a fourth order nonlinear degenerate parabolic equation, which is used to simulate the convection phenomenon dominated by the surface-tension-driven mechanism. First, we construct a modified problem of the initial-boundary value problem. Then, constructing an energy functional with special form and using the energy method, we obtain the uniform a-priori estimates of the solutions to the modified problem. Next, the Aubin-lions lemma is used to take the limit of the solutions to get the weak solutions to an approximate problem of the initial-boundary value problem. Finally, introducing the special integral functions, we prove the non-negativity of the solutions and obtain the global existence of weak solutions to the initial-boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift–Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta–Kawasaki equation, providing a framework for handling derivatives in nonlinear terms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the following parabolic–elliptic–ODE system with indirect signal production mechanism u t ∗ = Δ u ∗ - ∇ · (u ∗ ∇ v ∗) , x ∈ R 2 , t > 0 , 0 = Δ v ∗ + w ∗ , x ∈ R 2 , t > 0 , τ ∗ w t ∗ = - δ w ∗ + u ∗ , x ∈ R 2 , t > 0 , u ∗ (x , 0) = u 0 ∗ (x) , w ∗ (x , 0) = w 0 ∗ (x) , x ∈ R 2 with τ ∗ > 0 and δ > 0 . Our first result asserts that for all reasonably regular initial data, this system admits a unique global solution, which drastically differs from the classical Keller–Segel system with direct signal production mechanism obtained by taking τ ∗ = 0 formally. Moreover, the solution remains uniformly bounded whenever the mass ∫ R 2 u 0 ∗ (x) d x < 4 π δ . Our second result further reveals that in the radially symmetric setting, there is a threshold value ∫ R 2 u 0 ∗ (x) d x = 8 π δ separating two different behaviors: all global solutions are uniformly bounded when the mass is below 8 π δ , while there are unbounded solutions starting from initial conditions having a mass exceeding 8 π δ . [ABSTRACT FROM AUTHOR]