Unique Classical Solution Research Articles

Solvability of Nonlocal Problems for Systems of Sobolev-Type Differential Equations with a Multipoint Condition

We consider a nonlocal problem for a system of loaded differential equations of the Sobolev type with a multipoint constraint. By introducing additional unknown functions, we reduce the problem under consideration to an equivalent problem consisting of a nonlocal multipoint problem for a system of loaded hyperbolic equations of the second order with functional parameters and integral correlations. We propose algorithms for solving the equivalent problem. Moreover, we establish conditions for the well-posedness of the nonlocal multipoint problem for the system of loaded hyperbolic equations of the second order and conditions for the existence of a unique classical solution to the nonlocal problem for the system of differential equations of the Sobolev type with a multipoint constraint.

Global boundedness of classical solutions to a two species cancer invasion haptotaxis model with tissue remodeling

In this paper, we deal with a haptotaxis cancer invasion model describing the migration and proliferation of two families of cancer cells, the epithelial-mesenchymal transition between two families cancer cells, the dynamics of matrix degrading enzymes, and the evolution and re-modeling of the extracellular matrix. Under appropriate regularity assumptions on initial data, by making some a priori estimates and applying iterative techniques, we establish the global existence and uniform boundedness of the unique classical solution in two-dimensional spatial domain for arbitrary cancer cells proliferation rates and in three-dimensional spatial setting for large cancer cells proliferation rates. These results improve and extend previously known ones.

Global existence and asymptotic behavior of classical solutions to a fractional logistic Keller–Segel system

In this paper we consider a fractional parabolic–elliptic Keller–Segel system with a logistic source on RN. First, we establish the regularity of weak solutions of the fractional parabolic equation, using blow-up arguments combined with Liouville-type theorems. Next, by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions. Moreover, the global existence and boundedness of classical solutions for given initial data are obtained under some conditions. Finally, we show the asymptotic behavior of the global solutions with strictly positive initial data.

Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Λα dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α>0 always possesses a unique global classical solution.

Boundary stabilization of quasilinear Maxwell equations

We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & Müller type in a smooth, bounded, strictly star-shaped domain of R3. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is employed to prove that small initial data admit unique classical solutions that exist globally and decay to zero at an exponential rate. Our approach is based on a recently established local well-posedness theory in a class of H3-valued functions.

Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, [Formula: see text]where [Formula: see text], a, b, [Formula: see text], and [Formula: see text] are positive constants. Assume [Formula: see text] . Then if in addition [Formula: see text] holds, it is proved that [Formula: see text] is the spreading speed of the solutions of (0.1) with nonnegative continuous initial function [Formula: see text] with nonempty compact support, that is, [Formula: see text]and [Formula: see text]where [Formula: see text] is the unique global classical solution of (0.1) with [Formula: see text]. It is also proved that, if [Formula: see text] and [Formula: see text] holds, then [Formula: see text] is the minimal speed of the traveling wave solutions of (0.1) connecting (0,0) and [Formula: see text], that is, for any [Formula: see text], (0.1) has a traveling wave solution connecting (0,0) and [Formula: see text] with speed c, and (0.1) has no such traveling wave solutions with speed less than [Formula: see text]. Note that [Formula: see text] is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, [Formula: see text]Hence, if [Formula: see text] and [Formula: see text], or [Formula: see text] and [Formula: see text], then the chemotaxis neither speeds up nor slows down the spatial spreading in (0.1).

Global existence and large time behavior of solutions to a haptotaxis model with self-remodeling mechanisms

In this paper, we deal with the initial-boundary value problem for the followinghaptotaxis model with self-remodeling mechanisms:beginalign*begincasesu_t=Δ u-ξ∇·(u∇ ω),v_t=Δ v-v+u,ωt=-vω+ηω(1-u-ω)endcasesendalign*in a bounded domain of $\\mathbb~R^2$ with zero-flux boundary conditions.We show that for any $\\eta>~0$, there existsa unique global classical solution. In particular, we show that the solution isuniformly bounded when $\\eta$ is appropriately small. On the basis of this, we also establish theglobal asymptotic stability of the constant steady state $(\\overline~u_0,~\\overline~u_0,~0)$.

The Dynamics of a Predator–Prey Model with Diffusion and Indirect Prey-Taxis

This paper concerns with a reaction–diffusion system modeling the population dynamics of the predator and prey, in which the predator moves toward the gradient of concentration of some chemical released by prey instead of moving directly toward the higher density of prey. The first objective is to investigate the global existence and boundedness of the unique classical solution. Then we study the asymptotic stabilities of nonnegative spatially homogeneous equilibria. Moreover, the convergence rates are also studied.

Explicit lower bound of blow–up time for an attraction–repulsion chemotaxis system

In this paper we study classical solutions to the zero–flux attraction–repulsion chemotaxis–system(◇){ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)inΩ×(0,t⁎),0=Δv+αu−βvinΩ×(0,t⁎),0=Δw+γu−δwinΩ×(0,t⁎), where Ω is a smooth and bounded domain of R2, t⁎ is the blow–up time and α,β,γ,δ,χ,ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u(x,0)=u0∈C0(Ω¯), system (◇) has a unique classical solution which becomes unbounded as t↗t⁎. The main result of this investigation is to provide an explicit lower bound for t⁎ estimated in terms of ∫Ωu02dx and attained by means of well–established techniques based on ordinary differential inequalities.

The logistic chemotaxis system with singular sensitivity and signal absorption in dimension two

In this paper we study the non-flux chemotaxis system: ut=Δu−χ∇⋅(uv∇v)+bu−μu2, vt=Δv−uv, in a bounded and smooth domain Ω⊂R2, with b∈R and χ,μ>0. The existing literature, on the premise of relatively weak cross-diffusion with 0<χ<1, has established the global existence of classical solutions for arbitrary μ>0. In the present paper, we extend to prove that for any χ≥1, there exists μ∗(χ)>0 such that whenever μ>μ∗(χ), any reasonably smooth initial data emanates a unique global classical solution. Moreover, it is shown that there exist α(Ω)>0 and β(Ω)>0 such that when b>0, if μ>α(Ω)b+β(Ω) for 0<χ<1, or μ>max{μ∗(χ),α(Ω)b+β(Ω)} for χ≥1, then any classical solution must be globally bounded with the convergence that (u(⋅,t),|∇v(⋅,t)|v(⋅,t),v(⋅,t))→(bμ,0,0) in [L∞(Ω)]3 as t→∞. Instead, when b≤0, all classical solutions remain uniformly bounded in time as long as μ>0 for 0<χ<1, or μ>μ∗(χ) for χ≥1, and it also holds that (u(⋅,t),|∇v(⋅,t)|v(⋅,t),v(⋅,t))→(0,0,λ) in [L∞(Ω)]3 as t→∞ with 0≤λ<1|Ω|∫Ωv(⋅,0).

Global existence and boundedness of classical solutions to a forager–exploiter model with volume-filling effects

This work is concerned with a chemotactic model for the dynamics of social interactions between two species — foragers u and exploiters v, as well as the dynamics of food resources w consumed by these two species. The foragers search for food directly, while the exploiters head for food by following the foragers. Specifically, the parabolic system in a smoothly bounded convex n-dimensional domain Ω, ut=Δu−∇⋅(S1(u)∇w),x∈Ω,t>0,vt=Δv−∇⋅(S2(v)∇u),x∈Ω,t>0,wt=dΔw−λ(u+v)w−μw+r(x,t),x∈Ω,t>0,is considered for the constants d,λ,μ>0 and r∈C0(Ω×[0,∞)) with a uniform bound. Volume-filling effects account for a simple version by taking S1(u)=u(1−u),S2(v)=v(1−v).We prove the global existence and boundedness of the unique classical solution to this forager–exploiter model associated with no-flux boundary conditions under the mild assumption that the initial data u0, v0, w0 satisfy 0≤u0,v0≤1 and w0≥0.

Steady-State Heat Distribution in Bimaterial with an Interface Crack: Part 1

The transmission problem describing a steady-state temperature distribution in a plane consisting of two half-planes occupied by different materials with exponential internal thermal conductivities with a single finite crack along the interface is considered. The compatibility conditions for the boundary functions are formulated under which the problem has a unique classical solution. Closed-form representations of the classical solution are found. The weak solution to the problem is studied without making additional assumptions, and asymptotic expansions are constructed for the weak solution and its first derivatives near the ends of the crack.

Long time behavior of the solution to a parabolic–elliptic system

In this paper, we are concerned with a parabolic–elliptic model for biological fluid transportation networks with the large diffusivity and activation. Our investigations reveal that local existence of the unique classical solution to the correspondinginitial–boundary value problem can always be achieved in the n-dimensional setting with n≤3. Moreover, we show that this local solution can be extended to a global one as long as the source term and the initial data are made small enough in a suitable sense, and that the classical solution converges exponentially to a trivial steady state; meanwhile, we present that the smallness condition on the initial data can be removed in the one-dimensional setting. Our result implies that we cannot expect network formation when the diffusivity is sufficiently large, which is a first step toward a qualitative analysis for the role of diffusion in biological transportation networks.

Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system with flux limitation

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation,{ut=∇⋅(up∇uu2+|∇u|2)−χ∇⋅(uq∇v1+|∇v|2),0=Δv−μ+u, under no-flux boundary conditions in balls Ω⊂Rn, and the initial condition u|t=0=u0 for a radially symmetric and positive initial data u0∈C3(Ω‾), where χ>0 and μ:=1|Ω|∫Ωu0. Bellomo–Winkler [3] proved local existence of unique classical solutions and extensibility criterion ruling out gradient blow-up as well as global existence and boundedness of solutions when p=q=1 under some conditions for χ and ∫Ωu0. This paper derives local existence and extensibility criterion ruling out gradient blow-up when p,q≥1, and moreover shows global existence and boundedness of solutions when p>q+1−1n.

On classical solutions to the first mixed problem for the Vlasov-Poisson system in an infinite cylinder

The first mixed problem for the Vlasov-Poisson system in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a high-temperature two-component plasma under an external magnetic field. For an arbitrary electric field potential and a sufficiently strong external magnetic field, it is shown that the characteristics of the Vlasov equations do not reach the boundary of the cylinder. It is proved that the Vlasov-Poisson system with ion and electron distribution density functions supported at some distance from the cylinder boundary has a unique classical solution.

Analysis of a chemotaxis model with indirect signal absorption

We consider the chemotaxis model{ut=Δu−∇⋅(u∇v),vt=Δv−vw,wt=−δw+u in smooth, bounded domains Ω⊂Rn, n∈N, where δ>0 is a given parameter.If either n≤2 or ‖v0‖L∞(Ω)≤13n we show the existence of a unique global classical solution (u,v,w) and convergence of (u(⋅,t),v(⋅,t),w(⋅,t)) towards a spatially constant equilibrium, as t→∞.The proof of global existence for the case n≤2 relies on a bootstrap procedure. As a starting point we derive a functional inequality for a functional being sublinear in u, which appears to be novel in this context.

Boundedness and stabilization in a two-species chemotaxis system with signal absorption

This paper is concerned with the Neumann initial–boundary value problem for the two-species chemotaxis system with consumption of chemoattractant ut=Δu−χ1∇⋅(u∇w),x∈Ω,t>0,vt=Δv−χ2∇⋅(v∇w),x∈Ω,t>0,wt=Δw−(αu+βv)w,x∈Ω,t>0in a smooth bounded domain Ω⊂Rn (n≥2), where the parameters χ1, χ2, α and β are positive. It is proved that if max{χ1,χ2}‖w(x,0)‖L∞(Ω)<2nπthe problem possesses a unique global classical solution that is uniformly bounded. Moreover, we prove that u(x,t)→1|Ω|∫Ωu(x,0),v(x,t)→1|Ω|∫Ωv(x,0)andw(x,t)→0ast→∞uniformly with respect x∈Ω.

Exact Solutions of the Cauchy Problem for the Friedman Equation

We reduce the cosmological Friedmann equation for a universe filled with a scalar field to a system of two first-order equations, one of which is an equation with separable variables. For this equation, we write exact solutions for a quadratic potential as a series in the helical and attractor domains and also for quite arbitrary potentials both in the neighborhood of a finite point and in a neighborhood of infinity. We prove the existence and uniqueness of classical solutions of the Cauchy problem of the Friedmann equation in certain cases and the presence of exactly two solutions in other cases.

Solution of initial-boundary value problem for a system of partial differential equations of the third order

The author consider initial-boundary value problem for a system of differential equations in partial derivatives of third order. We study questions of existence and uniqueness of classical solution of the problem, and ways for its evaluation. The introduction of a new desired function enables us to reduce the problem to equivalent nonlocal problem with integral condition for a system of integro-differential euqtions of hyperbolic type with a functional relation. We establish conditions for unique solvability of the nonlocal problem by means of the method of functional parameters, propose algorithms for solving of the equivalent problem and prove their convergence. As a result, there are obtained conditions for existence of unique classical solution of initial-boundary value problem for the considered system of differential equations of third order in terms of initial data.

Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations

We consider a linear differential-algebraic system of PDEs with special matrix coefficients. Two cases are investigated. In the first one, the system has a small index, and the matrix at unknown vector-function, while the system written in the canonical form, is arbitrary. In the second case, the system has an arbitrary index, while a matrix at the small term is triangular. In both the cases, using the methods of characteristics and successive approximations, we prove the existence of a unique classical solution of mixed problems for the considered differential-algebraic systems of PDEs.