Existence Of Global Solutions Research Articles

Global existence and stability of the wave equation with boundary variable damping

AbstractIn this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities.

Global existence and blowup of solutions for nonlinear Love equation

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.

Evolution of dispersal and the analysis of a resource flourished population model with harvesting

This study explores a spatially distributed harvesting model that signifies the outcome of the competition of two species in a heterogeneous environment. The model is controlled by reaction-diffusion equations with resource-based diffusion strategies. Two different situations are maintained by the harvesting effects: when the harvesting rates are independent in space and do not exceed the intrinsic growth rate; and when they are proportional to the time-independent intrinsic growth rate. In particular, the competition between both species differs only by their corresponding migration strategy and harvesting intensity. We have computed the main results for the global existence of solutions that represent either coexistence or competitive exclusion of two competing species depending on the harvesting levels and different imposed diffusion strategies. We also established some estimates on harvesting efforts for which coexistence is apparent. Also, some numerical results are exhibited in one and two spatial dimensions, which shed some light on the ecological implementation of the model.

A Minkowski-type inequality for capillary hypersurfaces in a half-space

In this article, we investigate a flow of inverse mean curvature type for capillary hypersurfaces in the half-space. We establish the global existence of solutions for this flow and demonstrate that it converges smoothly to a spherical cap as time tends to infinity. As a result, we derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces for the whole range of contact angle θ∈(0,π).

Longitudinal vibrations of a bar in memoriam to Professor Luiz Adauto Medeiros

This paper is concerned with the deduction of a mathematical model for the small longitudinal vibrations of a bar where one end is clamped and in the other end is glued a concentrated mass. The existence of global solutions of this model perturbed with a general nonlinear internal dissipation and with a general nonlinear boundary dissipation is derived. For a particular case, the exponential decay rate of the energy is obtained.

Dynamical analysis of a diffusive population-toxicant model with toxicant-taxis in polluted aquatic environments

This paper deals with a diffusive population-toxicant model in polluted aquatic environments, with a toxicant-taxis term describing a toxicant-induced behavior change, that is, the population tends to move away from locations with high-level toxicants. The global existence of solutions is established by the techniques of the semigroup estimation and Moser iteration. Based on a detailed study on the properties of the principal eigenvalue for non-self-adjoint eigenvalue problems, we investigated the local and global stability of the toxin-only steady-state solution and the existence of positive steady state, which yields sufficient conditions that lead to population persistence or extinction. Finally, by numerical simulations, we studied the effects of some key parameters, such as toxicant-taxis coefficient, advection rate, and effect coefficient of the toxicant on population growth, on population persistence. Both numerical and analytical results show that a weak chemotaxis effect, a small advection rate of the population, and a weak effect of the toxicant on population growth are favorable for population persistence.

Small-data global existence of solutions for the Pitaevskii model of superfluidity

We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959 Sov. Phys. JETP 8 282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in T2 —strong in wavefunction and velocity, and weak in density.

Global existence of solutions for a parabolic systems with logarithmic nonlinearity

Global existence of solutions for a parabolic systems with logarithmic nonlinearity

Global existence of solutions for a free‐boundary tumor model with angiogenesis and a necrotic core

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 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.

Asymptotic behavior in a two-species chemotaxis system with Lotka-Volterra type competition and two signals

This paper deals with the study of asymptotic behavior of the following two-species chemotaxis system with Lotka-Volterra type competition and two signals{∂tu1=Δu1−∇⋅(χ11(υ1)u1∇υ1)−∇⋅(χ12(υ2)u1∇υ2)+μ1u1(1−u1−a1u2),x∈Ω,t>0,∂tu2=Δu2−∇⋅(χ21(υ1)u2∇υ1)−∇⋅(χ22(υ2)u2∇υ2)+μ2u2(1−u2−a2u1),x∈Ω,t>0,∂tυ1=Δυ1−λ1υ1+α11u1+α12u2,x∈Ω,t>0,∂tυ2=Δυ2−λ2υ2+α21u1+α22u2,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, where parameters μi≥0,ai≥0,λi>0, αij>0(i,j=1,2) and χij(i,j=1,2) are the chemotactic sensitivity functions. The boundedness and global existence of solutions to this system have been studied [22]. By constructing appropriate energy function, the purpose of this paper is to prove the asymptotic behavior for the cases•If a1,a2∈(0,1) and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is exponentially converges to (1−a11−a1a2,1−a21−a1a2,α11(1−a1)+α12(1−a2)λ1(1−a1a2),α21(1−a1)+α22(1−a2)λ2(1−a1a2)) as t→∞.•If a1>1>a2>0 and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is exponentially converges to (0,1,α12λ1,α22λ2) as t→∞.•If a1=1>a2>0 and μ1,μ2 are suitably large, then the solution (u1,u2,υ1,υ2) is algebraically converges to (0,1,α12λ1,α22λ2) as t→∞.

On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work.

On the existence of global solutions for the 3D chemorepulsion system

On the existence of global solutions for the 3D chemorepulsion system

Existence, Decay, and Blow-up of Solutions for a Weighted m-Biharmonic Equation with Nonlinear Damping and Source Terms

In this paper, we consider the weighted m-biharmonic equation with nonlinear damping and source terms. We proved the global existence of solutions. Later, the decay of the energy is established by using Nakao’s inequality. Finally, we proved the blow-up of solutions in finite time.

A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

A class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs are studied. Under suitable assumptions on the exponents of nonlinear source terms and the memory kernel, the existence of global solutions in an appropriate functional space is established by a combination of the theory of potential wells and the Galerkin approximations. Furthermore, the asymptotic behavior of global solutions is obtained by a combination of the theory of potential wells and the perturbed energy method.

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.

On the Existence of Global Solutions for the Vlasov–Poisson System in a Half-space and Plasma Confinement

We consider the Vlasov–Poisson system with external magnetic field in a half-space with the Neumann boundary condition for the electric potential and specular reflection on a boundary. For arbitrary compactly supported initial density distribution functions, we obtain sufficient conditions for external magnetic field, which provide global existence of density distribution functions with compact supports lying at some distance from a boundary.

A necessary and sufficient condition for the existence of global solutions to reaction-diffusion equations on bounded domains

AbstractThe purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations $$ u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), $$ u t = Δ u + ψ ( t ) f ( u ) , in Ω × ( 0 , ∞ ) , under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case $f(u)=u^{p}$ f ( u ) = u p . As a matter of fact, we prove: $$ \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} $$ there is no global solution for any initial data if and only if ∫ 0 ∞ ψ ( t ) f ( ∥ S ( t ) u 0 ∥ ∞ ) ∥ S ( t ) u 0 ∥ ∞ d t = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 ( Ω ) . Here, $(S(t))_{t\geq 0}$ ( S ( t ) ) t ≥ 0 is the heat semigroup with the mixed boundary condition.

The fully parabolic multi-species chemotaxis system in

Abstract This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$ . The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.

Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences

This research paper revolves around investigating the phenomenon of quenching in reaction-diffusion systems and highlighting its significance. The primary focus is on analyzing a specific type of parabolic singular reaction-diffusion model that incorporates positive Dirichlet boundary conditions. The objective is to establish the sufficiency of certain conditions for quenching to occur within a finite time frame and to demonstrate the global existence of solutions. The novelty of this paper lies in the simplicity of the conditions imposed on the nonlinearity. This simplicity allows us to choose it from a wide range of possibilities, thus facilitating the application of the model to numerous singular reaction-diffusion phenomena. To bolster our findings, we will present various real-world applications in the fields of bioengineering and life sciences, showcasing the practical relevance of quenching phenomena. Finally, the paper ends with a conclusion and some potential future perspectives for further research in this area.

A study for a higher order Riemann-Liouville fractional differential equation with weakly singularity

<abstract><p>In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems.</p></abstract>