Unique Global Classical Solution Research Articles

The Cauchy problem for the degenerate convective Cahn-Hilliard equation

In this paper, we study the degenerate convective Cahn-Hilliard equation, which is a special case of the general convective Cahn-Hilliard equation with $M(u,\nabla u)=diag(0,1,\ldots ,1)$. We obtain the uniform a priori decay estimates of a solution by use of the long-short wave method and the frequency decomposition method. We prove the existence of the unique global classical solution with small initial data by establishing the uniform estimates of the solution. Decay estimates are also discussed.

Parabolic–elliptic chemotaxis model with space–time-dependent logistic sources on ℝN. I. Persistence and asymptotic spreading

The current series of three papers is concerned with the asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space- and time-dependent logistic source: [Formula: see text] where [Formula: see text] is a positive integer, [Formula: see text] and [Formula: see text] are positive constants, and the functions [Formula: see text] and [Formula: see text] are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading in [Formula: see text]. To this end, under some explicit condition on the parameters, we first show that [Formula: see text] has a unique nonnegative time global classical solution [Formula: see text] with [Formula: see text] for every [Formula: see text] and every nonnegative bounded and uniformly continuous initial function [Formula: see text]. Next we show the pointwise persistence phenomena of the solutions in the sense that, for any solution [Formula: see text] of [Formula: see text] with strictly positive initial function [Formula: see text], there are [Formula: see text] such that [Formula: see text] and show the uniform persistence phenomena of solutions in the sense that there are [Formula: see text] such that for any strictly positive initial function [Formula: see text], there is [Formula: see text] such that [Formula: see text] We then discuss the spreading properties of solutions to [Formula: see text] with compactly supported initial function and prove that there are positive constants [Formula: see text] such that for every [Formula: see text] and every nonnegative initial function [Formula: see text] with nonempty compact support, we have that [Formula: see text] and [Formula: see text] We also discuss the spreading properties of solutions to [Formula: see text] with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions of [Formula: see text] and the existence of transition fronts of [Formula: see text], respectively.

Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling

We consider a cancer invasion model comprising a strongly coupled PDE–ODE system in two and three space dimensions. The system consists of a parabolic equation describing cancer cell migration arising from a combination of chemotaxis and haptotaxis, a parabolic/elliptic equation describing the dynamics of matrix degrading enzymes (MDEs), and an ODE describing the evolution and re-modeling of the extracellular matrix (ECM). We point out that this strongly coupled PDE–ODE setup presents new mathematical difficulties, which are overcome by developing new integral estimate techniques. We prove that the system admits a unique global classical solution which is uniformly bounded in time in the two-dimensional spatial setting at all cancer cell proliferation rates. We also prove that, in the case of three-dimensional convex spatial domain, when cancer cell proliferation is suitably small, the system also possesses a unique classical solution for appropriately small initial data. These results improve previously known ones.

Asymptotic-stability of the inhomogeneous Boltzmann equation in the Robertson–Walker space–time with Israel particles

ABSTRACTIn this paper, we consider the Cauchy problem for the spatially inhomogeneous relativistic Boltzmann equation where the collision kernel is generated by Israel particle. Unique global (in time) classical solution is obtained in a suitable weighted space by considering small initial data and the Robertson–Walker space–time as background. We also studied the asymptotic behaviour of the solutions as well as the -stability.

Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion

This paper is concerned with two frequency-dependent susceptible–infected–susceptible epidemic reaction–diffusion models in heterogeneous environment, with a cross-diffusion term modelling the effect that susceptible individuals tend to move away from higher concentration of infected individuals. It is first shown that the corresponding Neumann initial-boundary value problem in an n-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly in-time bounded regardless of the strength of the cross-diffusion and the spatial dimension n. It is further shown that, even in the presence of cross-diffusion, the models still admit threshold-type dynamics in terms of the basic reproduction number $\mathcal {R}_0$ – i.e. the unique disease-free equilibrium is globally stable if $\mathcal {R}_0\lt1$, while if $\mathcal {R}_0\gt1$, the disease is uniformly persistent and there is an endemic equilibrium (EE), which is globally stable in some special cases with weak chemotactic sensitivity. Our results on the asymptotic profiles of EE illustrate that restricting the motility of susceptible population may eliminate the infectious disease entirely for the first model with constant total population but fails for the second model with varying total population. In particular, this implies that such cross-diffusion does not contribute to the elimination of the infectious disease modelled by the second one.

Global existence of the classical solutions for the inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time

In this paper, we consider the Cauchy problem for the spatially inhomogeneous relativistic Boltzmann equation with near vacuum initial data. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson–Walker space-time. Unique global (with respect to the direction of time corresponding to the expansion of the universe) classical solution is obtained. This is done in a suitable weighted space.

Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals

We study a nonlinear system of differential equations describing the evolution of a competitive two-species chemotaxis system with two chemicals in a bounded domain. The system consists of four PDEs, two equations of parabolic type describing the evolution of the competitive species and two elliptic equations modeling the distribution of the chemicals. By introducing global competitive/cooperative factors, we obtain, for different ranges of parameters, that any positive and bounded solution converges to a spatially homogeneous state. The proofs rely on the comparison principle for spatially homogeneous sub- and super-solutions. The existence and uniqueness of global classical solution are proved under assumptions on the initial data and appropriate conditions on the parameters of the system. Such solution stabilizes to spatially homogeneous equilibria in the large time limit, given coexistence or extinction of solutions for a range of parameters.

Global and almost global existence for general quasilinear wave equations in two space dimensions

For 2-D quasilinear wave equations of the form □u=F(u,Du,D2u), we show that the corresponding Cauchy problem admits a unique global classical solution with small initial data, if both the quadratic nonlinearity and the cubic nonlinearity satisfy the corresponding null conditions. We also prove that the classical solution is almost global, i.e., the lifespan of solution Tε≥exp⁡(cε−2), where ε>0 denotes the amplitude of the initial data and c is a positive constant, provided that the quadratic nonlinearity satisfies the null condition and ∂uβF(0,0,0)=0, β=3,4. These results extend Alinhac's seminal work Alinhac (2001) [9] to the general case in a unified way.

Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity

We prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations vtt=c(|v|s−1v)xx with the nonnegative initial data v0(x) and s>1. These results are extensions of the results in the second author’s paper (Sugiyama, 2016), where the existence and the nonexistence of the unique global classical solution were studied with a threshold on ∫−∞∞v1(x)dx and the non-degeneracy condition v0(x)≥c0>0 on the initial data.

Global wellposedness of magnetohydrodynamics system with temperature-dependent viscosity

The initial boundary value problem of the one-dimensional magneto-hydrodynamics system, when the viscosity, thermal conductivity, and magnetic diffusion coefficients are general smooth functions of temperature, is considered in this article. A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1.

Global well-posedness of the 3D non-isothermal compressible fluid model of Korteweg type

The aim of this work is to study the global existence and large-time behavior of solutions to the non-isothermal model of capillary compressible fluids derived by Dunn and Serrin (1985). It is proved that the three-dimensional non-isothermal compressible Navier–Stokes–Korteweg system admits a unique global classical solution, provided that the initial energy is suitably small. This result improves previous results obtained by Hattori and Li (1996), where the existence of global classical solutions is established for the initial data close to an equilibrium in some Sobolev space Hs(s≥3). Furthermore, the large time behavior of the solution is also investigated.

Global classical solutions of compressible isentropic Navier–Stokes equations with small density

This paper concerns the Cauchy problem of compressible isentropic Navier–Stokes equations in the whole space R3. First, we show that if ρ0∈Lγ∩H3, then the problem has a unique global classical solution on R3×[0,T] with any T∈(0,∞), provided the upper bound of the initial density is suitably small and the adiabatic exponent γ∈(1,6). If, in addition, the conservation law of the total mass is satisfied (i.e., ρ0∈L1), then the global existence theorem with small density holds for any γ>1. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the results obtained particularly extend the one due to Huang–Li–Xin (Huang et al., 2012), where the global well-posedness of classical solutions with small energy was proved.

Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model

We consider a haptotaxis cancer invasion model that includes two families of cancer cells. Both families, migrate on the extracellular matrix and proliferate. Moreover the model describes an epithelial-to-mesenchymal-like transition between the two families, as well as a degradation and a self-reconstruction process of the extracellular matrix. We prove positivity and conditional global existence and uniqueness of the classical solutions of the problem for large initial data.

Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions

In this paper, we study the generalized chemotaxis system with fractional Laplacian. The existence and the uniqueness of global classical solution are proved under the assumption that the initial data are small enough. During the proof, with the help of the fixed point theorem, the asymptotic decay behaviors of \begin{document}$ u $\end{document} and \begin{document}$ \nabla{v} $\end{document} are also shown.

On the dynamics of finite temperature trapped Bose gases

The system that describes the dynamics of a Bose–Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C12 and C22, which describe collisions of the condensate and the non-condensate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collision operator similar to C22, which possesses a blow-up positive radial solution with respect to the L∞ norm (cf. [29]). On the other hand, at the very low temperature regime (only a portion of the transition temperature TBEC), the system can be simplified into an equation of C12, with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L1 norm (cf. [3]). In our model, we first decouple the QB, which contains C12+C22, and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [29], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [29]. A new theory is then supplied.

One-dimensional viscous diffusion equation of higher order with gradient dependent potentials and sources

In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ(s) and sources A(s). We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ(s) and A(s) are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T > 0, the solution will never blow up during the period 0 < t < T, so long as the viscous coefficient is large enough.

Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production

We study the Neumann initial-boundary problem for the chemotaxis system \left\{ \begin{array}{ll} u_t= \Delta u - \nabla \cdot (u\nabla v), &amp; x\in \Omega, \, t&gt;0, \\ 0=\Delta v - \mu(t)+w, &amp; x\in \Omega, \, t&gt;0, \\ \tau w_t + \delta w = u, &amp; x\in \Omega, \, t&gt;0, \end{array} \right. \qquad \qquad (\star) in the unit disk \Omega:=B_1(0)\subset R^2 , where \delta\ge 0 and \tau&gt;0 are given parameters and \mu(t):=\frac{1}{|\Omega|} \int_\Omega w(x,t)dx , t&gt;0 . It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking \tau\to 0 , in that it refers to blow-up in infinite time rather than in finite time. Specifically, it is first proved that for any sufficiently regular nonnegative initial data u_0 and w_0 , ( \star ) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in ( \star ) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that \|u(\cdot,t)\|_{L^\infty(\Omega)} \to \infty \qquad \text{as } t\to\infty.

The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system

This paper deals with an initial-boundary value problem for the chemotaxis-(Navier–)Stokes system $$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot (n\nabla c), \qquad &{}\quad x\in \Omega , \ t>0, \\ c_t + u\cdot \nabla c &{}=&{}\Delta c - nc, \qquad &{}\quad x\in \Omega , \ t>0, \\ u_t + \kappa (u\cdot \nabla )u &{}=&{} \Delta u - \nabla P + n\nabla \phi , \qquad &{}\quad x\in \Omega , \ t>0, \\ \nabla \cdot u =0, &{} &{} \qquad &{}\quad x\in \Omega , t>0, \end{array} \right. \end{aligned}$$ in a bounded convex domain \(\Omega \subset \mathbb {R}^2\) with smooth boundary, with \(\kappa \in \mathbb {R}\) and a given smooth potential \(\phi :\Omega \rightarrow \mathbb {R}\). It is known that for each \(\kappa \in \mathbb {R}\) and all sufficiently smooth initial data this problem possesses a unique global classical solution \((n^{(\kappa )},c^{(\kappa )},u^{(\kappa }))\). The present work asserts that these solutions stabilize to \((n^{(0)},c^{(0)},u^{(0)})\) uniformly with respect to the time variable. More precisely, it is shown that there exist \(\mu >0\) and \(C>0\) such that whenever \(\kappa \in (-1,1)\), $$\begin{aligned}&\Big \Vert n^{(\kappa )}(\cdot ,t)-n^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} + \Big \Vert c^{(\kappa )}(\cdot ,t)-c^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \\&\quad +\, \Big \Vert u^{(\kappa )}(\cdot ,t)-u^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \le C |\kappa | e^{-\mu t} \end{aligned}$$ for all \(t>0\). This result thereby provides an example for a rigorous quantification of stability properties in the Stokes limit process, as frequently considered in the literature on chemotaxis-fluid systems in application contexts involving low Reynolds numbers.

Boundedness in a two-species chemotaxis parabolic system with two chemicals

This paper is devoted to the chemotaxis system \begin{document}$\left\{\begin{aligned}u'>which models the interaction between two species in presence of two chemicals, where \begin{document}$χ, \, ξ∈\mathbb{R}$\end{document} , \begin{document}$Ω\subset\mathbb{R}^2$\end{document} are bounded domains with smooth boundary. It is shown that under the homogeneous Neumann boundary conditions the system possesses a unique global classical solution which is bounded whenever both \begin{document}$\int_Ω u_0dx$\end{document} and \begin{document}$\int_Ω w_0dx$\end{document} are appropriately small. In particular, we extend the recent results obtained by Tao and Winkler (2015, Disc. Cont. Dyn. Syst. B) to the fully parabolic case, i.e., the case of \begin{document}$τ=1$\end{document} .

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)ut=+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0, is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂R3. Here χ>0, ξ>0 and μ>0, D(u)≥cDum−1 for all u>0 with some cD>0 and D(u)>0 for all u≥0. It is shown that if the ratio χμ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.