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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]

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]

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 study, a global solution is presented for an initial boundary value problem involving a wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. The solution is obtained through the application of the energy method combined with the Faedo-Galarkin procedure. Additionally, a blow-up result for the solution is established, contingent upon the condition of non-positive initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the Lp$$ {L}^p $$ estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem. [ABSTRACT FROM AUTHOR]

In this paper, an initial boundary value problem of the 2D incompressible Bénard system is concerned. The global existence of a unique strong solution is established by the energy method and a key logarithmic interpolation inequality. In particular, the exponential decay rates for the gradients of velocity and temperature field are obtained. [ABSTRACT FROM AUTHOR]

This paper deals with the following chemotaxis system: u t = ∇ · (γ (v) ∇ u - u ξ (v) ∇ v) + μ u (1 - u) , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary. Here, the positive function γ ∈ C 2 ([ 0 , + ∞)) satisfies γ ′ (s) < 0 and γ ′ ′ (s) ≥ 0 for all s ≥ 0 , also ξ (s) = - (1 - α) γ ′ (s) with α ∈ (0 , 1) . For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on μ. The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that (γ ′ (s)) 2 γ ′ ′ (s) ≤ n 2 (n + 1) 3 and some conditions on initial data and μ. We should mention that in the special cases γ (s) = (1 + s) - k (k > 0) and γ (s) = e - χ s (χ > 0) , the result in Li and Lu (2023) is obtained under conditions on k and χ. But, our result is without any restriction on k and χ. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial boundary value problem for a fourth order wave equation with Newtonian potential. We establish firstly the local existence of solutions by Banach fixed point theorem. Using the potential well argument, we show the global existence, finite time blow-up, asymptotic behavior of solutions since the initial energy is not over the depth of the potential well. Finally, when the initial energy is supercritical, we give some explicit criterion for blow-up in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem for a pseudo‐parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite‐time blow‐up for the weak solutions with initial energy J(u0)≤d$$ J\left({u}_0\right)\le d $$. When the initial energy J(u0)>d$$ J\left({u}_0\right)&gt;d $$, we find another criterion for the vanishing solution and blow‐up solution. We also establish the decay rate of the global solution and estimate the life span of the blow‐up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem of the generalized pseudo‐parabolic equation containing viscoelastic terms and associated with Robin conditions. We establish first the local existence of solutions by the standard Galerkin method. Then, we prove blow‐up results for solutions when the initial energy is negative or nonnegative but small enough or positive arbitrary high initial energy, respectively. We also establish the lifespan and the blow‐up rate for the weak solution by finding the upper bound and the lower bound for the blow‐up times and the upper bound and the lower bound for the blow‐up rate. For negative energy, we introduce a new method to prove blow‐up results with a sharper estimate for the upper bound for the blow‐up times. Finally, we prove both the global existence of the solution and the general decay of the energy functions under some restrictions on the initial data. [ABSTRACT FROM AUTHOR]

In this paper, we research initial boundary value problems for time-space fractional Ginzburg–Landau equations with Caputo time fractional derivatives and fractional Laplacian operators. As far as we know, there have been few studies concerning stability and optimal control for the corresponding system. First, by Galërkin method, we establish global existence and a few decay estimates of weak solutions. What's more, we study the dependence of weak solutions on these given data, then we get stability of the system. Further, we obtain an optimal control of the time-space fractional Ginzburg–Landau system. [ABSTRACT FROM AUTHOR]

In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution. [ABSTRACT FROM AUTHOR]

This work is concerned with a Kirchhoff wave equation with strong nonlinear damping. We establish the local existence of the solution and provide some sufficient conditions for the global existence and finite time blow‐up of the solution. Moreover, we derive the exponential decay estimation of the energy functional and the upper and lower bounds of the blow‐up time. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies g ′ (t) ≤ − ξ (t) g r (t) , t ≥ 0 , 1 ≤ r < 3 2 . The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial boundary value problem of m-Laplacian parabolic equation arising in various physical models. We tackle this problem at three different initial energy levels. For the sub-critical initial energy, we obtain the blow-up result and estimate the lower and upper bounds of the blow-up time. For the critical initial energy, we show the global existence, asymptotic behavior, finite time blow-up and the lower bound of the blow-up time. For the sup-critical initial energy, we prove the finite time blow-up and estimate the lower and upper bounds of the blow-up time. [ABSTRACT FROM AUTHOR]

The main goal of this work is to study an initial boundary value problem for a Kirchhoff-type equation with nonlinear boundary delay and source terms. This paper is devoted to prove the global existence, decay, and the blow up of solutions. To the best of our knowledge, there are not results on the Kirchhoff type-equation with nonlinear boundary delay and source terms. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial boundary value problem for the one-dimensional thermodiffusion equations of type III. By the semigroup approach and the energy method, we establish the global existence and exponential stability for the solutions in a bounded region. The main work of this paper is to extend the study of thermodiffusion equations to the model of type III, which can be used as a reference for the study of other types of partial differential equations. [ABSTRACT FROM AUTHOR]

We are concerned with the initial boundary value problems of chemotaxis-Navier-Stokes systems involving double chemosignals: one is an attractant consumed by the cells themselves and the other is an attractant or a repellent produced by the cells themselves. This work reveals that for any properly regular initial data, if the initial total cells mass is less than some threshold, then the corresponding attraction-atraction Navier-Stokes system admits a unique global classical solution, which, under extra two smallness assumptions on the initial data, is globally bounded and converges at exponential rate to the spatial equilibrium in the large time limit. As byproducts, the global classical solvability and stabilization of the Keller-Segel Navier-Stokes system and the uniform boundedness and large time behavior of the global classical solution of the associated attraction-repulsion Navier-Stokes system are simultaneously achieved. [ABSTRACT FROM AUTHOR]

We investigate an initial-(periodic-)boundary value problem for a continuum equation, which is a model for motion of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and integrated the effects of a diverse range of thermodynamic driving forces. We first prove the global-in-time existence and uniqueness of weak solution to this initial-boundary value problem in the case with positive equilibrium disconnection density parameter B, and then investigate the asymptotic behavior of the solutions as B goes to zero. The main difficulties in the proof of main theorems are due to the degeneracy of B = 0, a non-local term with singularity, and a non-smooth coefficient of the highest derivative associated with the gradient of the unknown. The key ingredients in the proof are the energy method, an estimate for a singular integral of the Hilbert type, and a compactness lemma. [ABSTRACT FROM AUTHOR]

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with nonlinear diffusion and a complicated nonlinear coupling term. The mathematical model considered herein appears as CNF { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n ∇ c) + ∇ ⋅ (n ∇ ϕ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u − n ∇ ϕ + n ∇ c , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , where Ω ⊆ R 3 is a general bounded domain with smooth boundary and ϕ ∈ W 2 , ∞ (Ω) is a given gravitational potential function. Here, one of the novelties is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. Based on a new energy-type argument combined with maximal Sobolev regularity theory, we conclude that if m > 7 6 , an associated initial-boundary value problem admits at least one globally weak solution which is uniformly bounded. This extends a recent result by Wang and Zhao (J. Differ. Equ. 269:148–179, 2020) which asserts global existence of weak solutions under the constraints m > 4 3 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a fourth-order equation with strong damping and logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. By means of potential well method and Rellich inequality, we obtain the global existence and the decay estimate of global solutions under some appropriate conditions. Furthermore, we prove the finite time blow-up results of weak solutions, and establish the upper and lower bounds for blow-up time. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively. [ABSTRACT FROM AUTHOR]

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation... [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem for a fractional viscoelastic equation of the Kirchhoff type. In suitable functional spaces, we define a potential well. In the framework of the potential well theory, we obtain the global existence of solutions by using the Galerkin approximations. Moreover, we derive the asymptotic behavior of solutions by means of the perturbed energy method. Our main results provide sufficient conditions for the qualitative properties of solutions in time. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of wave equations of Hartree type on a bounded smooth convex domain with Dirichlet boundary condition. By applying the standard semigroup theory, we get the local existence result. Using potential well theory, we derive the condition of global existence of weak solutions. With the help of potential well theory and convexity method, we give blowup results for solutions when the initial energy is nonnegative or negative. [ABSTRACT FROM AUTHOR]

In this paper, we consider initial boundary value problems for abstract fractional diffusion equations ∂βt u + (−∆)su = g(t, x, u) with the Caputo time fractional derivatives and fractional Laplacian operators. When g(t, x, u) satisfies condition (G), problems can be applied by a strong maximum principle involving time-space fractional derivatives. Hence, we establish the global existence and uniqueness of mild solution by upper and lower solutions method. Moreover, the mild solution mentioned above turns out to be a classical solution. When condition (G) does not hold, then we study nonexistence of global solutions under certain conditions, and we obtain the local existence and blow-up of mild solutions. Further, we conclude that the first eigenvalue λ1 seems to be a critical value for nonlinear problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate initial boundary value problems for the following time–space fractional Rosenau equation ∂tβu+(−Δ)su+(−Δ)1+su+(−Δ)1+s∂tβu=−(−Δ)sf(u),with the Caputo time fractional derivatives and the spectral fractional Laplacian operators. To the best of our knowledge, there are few results concerning the time–space fractional Rosenau equations. The main difficulties are the nonlocal effects generated by the operators ∂tβ and (− Δ)γ. First, we establish some rough and rigorous decay estimates of weak solutions to the corresponding linear equations, respectively. Based on the decay estimates, under small initial value condition, we prove the global existence and asymptotic behavior of weak solutions in the time‐weighted Sobolev spaces by the contraction mapping principle. Furthermore, we discuss the regularity of weak solutions when initial value data are strengthened from ℍs(Ω) to ℍs+1(Ω). [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial–Neumann boundary value problem for a parabolic–elliptic Keller–Segel system with signal-dependent motility and a source term. Previous research has rigorously shown that the source-free version of this system exhibits an infinite-time blowup phenomenon when dimension N ≥ 2. In the current work, when N ≤ 3 , we establish uniform boundedness of global classical solutions with an additional source term that involves slightly super-linear degradation effect on the density, of a maximum growth order s log s , unveiling a sufficient blowup suppression mechanism. The motility function considered in our work takes a rather general form compared with recent works (Fujie and Jiang, 2020; Lyu and Wang, 2023) which were restricted to the monotone non-increasing case. The cornerstone of our proof lies in deriving an upper bound for the second component of the system and an entropy-like estimate, which are achieved through tricky comparison skills and energy methods, respectively. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for a higher-order parabolic equation with logarithmic source term ... . By employing the potential wells technique we show the global existence of the weak solution. Also, we obtain the exponential decay for the weak solutions. [ABSTRACT FROM AUTHOR]

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype ⎧ ⎩ ⎨ ⎪ ⎪ u t = Δ u − ∇ ⋅ (u ∇ v) + μ u (1 − u) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , τ w t + δ w = u , x ∈ Ω , t > 0 (∗) { u t = Δ u − ∇ ⋅ (u ∇ v) + μ u (1 − u) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , τ w t + δ w = u , x ∈ Ω , t > 0 (∗) in a smoothly bounded domain Ω ⊂ R N (N ≥ 1) Ω ⊂ R N (N ≥ 1) under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters μ μ as well as δ δ and τ τ are positive. Based on an new energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever μ > { 0 , i f N ≤ 4 , (N − 4) + N − 2 max { 1 , λ 0 } , i f N ≥ 5 μ > { 0 , i f N ≤ 4 , (N − 4) + N − 2 max { 1 , λ 0 } , i f N ≥ 5 and the initial data (u 0 , v 0 , w 0) (u 0 , v 0 , w 0) are sufficiently regular. Here λ 0 λ 0 is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem for the p-Laplacian equation with weak and p-Laplacian damping terms, nonlinear boundary, delay and source terms acting on the boundary. By introducing suitable energy and perturbed Lyapunov functionals, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases p > 2 and p = 2. To our best knowledge, there is no results of the p-Laplacian equation with a nonlinear boundary delay term. [ABSTRACT FROM AUTHOR]

This work is concerned with the initial boundary value problem for a nonlinear viscoelastic Petrovsky wave equation utt + Δ²u - ... g(t - τ) Δ²u(τ)dτ - Δut - Δutt + ut |ut|m-1 = u|u|p-1. Under suitable conditions on the relaxation function g, the global existence of solutions is obtained without any relation between m and p. The uniform decay of solutions is proved by adapting the perturbed energy method. For p > m and sufficient conditions on g, an unboundedness result of solutions is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations ∂ β t u + M (∥ u ∥ 2 H s 0 (Ω)) (− Δ) s u = γ | u | ρ u + g (t , x) ∂ t β u + M (‖ u ‖ H 0 s (Ω) 2) (− Δ) s u = γ | u | ρ u + g (t , x) with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let 0 < ρ < 4 s N − 2 s 0 < ρ < 4 s N − 2 s when γ < 0 γ < 0 , and 0 < ρ < min { 4 s N , 2 s N − 2 s } 0 < ρ < min { 4 s N , 2 s N − 2 s } when γ > 0 γ > 0 , then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in L 2 (Ω) L 2 (Ω) and L ρ + 2 (Ω) L ρ + 2 (Ω). Remarkably, the decay rate differs from that in the case β = 1 β = 1. [ABSTRACT FROM AUTHOR]

This paper discusses initial boundary value problem for a semilinear edge degenerate parabolic equation and corresponding stationary problem. We first find some initial conditions with different energy levels such that the solution exists globally and blows up in finite time, respectively. We also study the asymptotic behaviors like exponential decay and exponential growth for solution and energy function. Especially, we show the solution of evolution problem will converge to the steady state solution. Additionally, we find that there are two explicit vacuum regions which are ball and annulus respectively, that is to say, there is no solution belongs to them and all solutions are isolated by them. Finally, we discuss the existence of ground state solution to the stationary problem. The instability of the ground state solution is considered and we prove that there exists initial value such that the instability occurs as a blow-up in finite time. [ABSTRACT FROM AUTHOR]

This paper deals with the doubly degenerate nutrient taxis system in an open bounded interval , with , which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of Bacillus subtilis growing on the surface of thin agar plates. It is shown that under the mere assumption that an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution , where and in , and that moreover there exists such that the solution approaches the pair in the large time limit with respect to the topology. This extends comparable results recently obtained in [ 17 ], the latter crucially relying on the additional requirement that , to situations involving nontrivially supported initial data , which seems to be of particular relevance in the addressed application context of sparsely distributed populations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy J (u 0) ≤ d. When the initial energy J (u 0) > d , we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy. [ABSTRACT FROM AUTHOR]

We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using potential well method and energy method, we obtain the existence of global solutions and finite-time blow-up solutions. Moreover, the upper bound of blow-up time is discussed under suitable assumptions. [ABSTRACT FROM AUTHOR]

In this paper we study the initial boundary value problem for the system u t − Δ u m = − div (u q ∇ v) , v t − Δ v + v = u. This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as m > q > 0 , thereby substantially generalizing the known results in this area. Furthermore, our result seems to imply that the Keller-Segel model can have bounded solutions and blow-up ones simultaneously. [ABSTRACT FROM AUTHOR]

In this paper, we consider the zero-flux chemotaxis system u t = Δ u − ∇ ⋅ (u χ (v) ∇ v) in Ω × (0 , ∞) , 0 = Δ v − v + g (u) in Ω × (0 , ∞) , in a smooth and bounded domain Ω of R 2 . The chemotactic sensitivity χ is a general nonnegative function from C 1 ((0 , ∞)) while g, the production of the chemical signal v, belongs to C 1 ([ 0 , ∞)) and satisfies λ 1 ≤ g (s) ≤ λ 2 (1 + s) β , for all s ≥ 0 , 0 ≤ β < 1 and 0 < λ 1 ≤ λ 2. It is established that no chemotactic collapse for the cell distribution u occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data u (x , 0) emanates a unique pair of global and uniformly bounded functions (u , v) which classically solve the corresponding initial boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations. [ABSTRACT FROM AUTHOR]

We consider the initial boundary value problem of pseudo-parabolic equation with singular potential. We obtain global existence, asymptotic behavior and blowup of solutions with initial energy J (u 0) ≤ d. Moreover, we estimate the upper bound of the blowup time for J (u 0) < 0 and 0 < J (u 0) < d respectively. Finally, we prove the finite time blowup and estimate the upper bound of the blowup time for the high energy level, i.e., J (u 0) > 0. [ABSTRACT FROM AUTHOR]

This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diffusion systems, with an additional chemotactic influence and constant coefficients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diffusion and are attracted/ repulsed by chemical stimulus produced by the other. The biological species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diffusion, along with two elliptic equations describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for different ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state. [ABSTRACT FROM AUTHOR]

This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p‐Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent. [ABSTRACT FROM AUTHOR]

In this paper we analyze the porous medium equation ut = Δum + a ∫Ω up − buq − c|∇√u|2 in Ω × I, (◇) where Ω is a bounded and smooth domain of RN, with N ≥ 1, and I = [0,t∗) is the maximal interval of existence for u. The constants a,b,c are positive, m,p,q proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of u. Under some hypotheses on the data, including intrinsic relations between m,p and q, and assuming that for some positive and sufficiently regular function u0(x) the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution u = u(x,t) on Ω × I:▹ when p>q and in 2- and 3-dimensional domains, we determine a lower bound oft∗ for those u becoming unbounded in Lm(p−1)(Ω) at such t∗;▹ when p < q and in N-dimensional settings, we establish a global existence criterion for u. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time. [ABSTRACT FROM AUTHOR]

We study the existence of global in time and uniform decay of weak solutions to the initial-boundary value problem related to the dynamic behavior of evolution equation accounting for rotational inertial forces along with a linear nonlocal frictional damping arises in viscoelastic materials. By constructing appropriate Lyapunov functional, we show the solution converges to the equilibrium state polynomially in the energy space. [ABSTRACT FROM AUTHOR]

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow‐up result for solutions with positive and negative initial energy respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Keller–Segel–Navier–Stokes system n t + u · ∇ n = Δ n + ∇ · (n ∇ c) , x ∈ Ω , t > 0 , u · ∇ c = Δ c - c + n , x ∈ Ω , t > 0 , u t + (u · ∇) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , x ∈ Ω , t > 0 , (⋆) with a given smooth gravitational potential Φ . It is shown that for all suitably regular initial data, a corresponding no-flux/no-flux/Dirichlet initial boundary value problem posed in a smoothly bounded planar domain admits a uniquely determined global classical solution (n, c, u, P) which has the additional property that n remains uniformly bounded. This partially goes beyond a recent result asserting global classical solvability, but without any boundedness information, in a related slightly more complex variant of (⋆ ) accounting for parabolic evolution of the quantity c. In particular, the obtained outcome therefore provides further evidence indicating that the considered fluid interaction does not substantially reduce a certain explosion-avoiding character of the Keller–Segel-type chemorepulsion mechanisms, as known to form an essential feature of corresponding fluid-free analogues. The reasoning at its core relies on the use of a quasi-Lyapunov inequality which operates at regularity levels that seem rather unusual in this and related contexts, but which in the considered two-dimensional setting can be seen to serve as a starting point sufficient for a bootstrap-type series of arguments, finally providing global boundedness. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized predator–prey system with prey-taxis under Neumann boundary condition, that is, the predators can survive even in the absence of the prey species. It is proved that for an arbitrary spatial dimension, the corresponding initial boundary value problem possesses a unique global bounded classical solution when the prey-taxis is restricted to a small range. Moreover, the local stabilities of constant steady states (including trivial, semi-trivial and positive constant steady states) are investigated. A further study on the coexistence steady state implies that the prey-taxis term suppresses the global asymptotical stability and influences the steady-state/Hopf bifurcations (if they exist). Analyses of steady-state bifurcation, Hopf bifurcation, and even Hopf/steady-state mode interaction are carried out in detail by means of the Lyapunov–Schmidt procedure. In particular, we obtain stable or unstable steady states, time-periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found in numerical simulations. [ABSTRACT FROM AUTHOR]