Radial Stationary Solution Research Articles

Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation

Pressure jump and radial stationary solutions of the degenerate Cahn–Hilliard equation

Analytic results of a double-layered radial tumor model with different consumption rates

This paper is concerned with the evolution of a double-layered spherically symmetric tumor model containing two kinds of living cells: quiescent cells and proliferating cells. The main feature of our work is to take different nutrient consumption rates λ1 and λ2 of quiescent cells and proliferating cells into rigorous analysis. Under the condition λ1⩽λ2 and some reasonable biological assumptions, we prove the transient solutions of the model exist globally in time and investigate the asymptotic behavior of these solutions towards different radial stationary solutions such as (σs,Rs) and (σs,ηs,Rs), where Rs is the steady radius of the dormant tumor and ηs is the steady width of the quiescent-cell layer. We also point out the potential of controlling the internal structure and development trend of tumors by adjusting the external nutrient concentration. This might have important implications for tumor research and cancer treatment.

Soliton resolution for the radial critical wave equation in all odd space dimensions

Consider the energy-critical focusing wave equation in odd space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. The proof essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in our previous work, Cambridge Journal of Mathematics 2013, arXiv:1204.0031) by reducing the study of a multisoliton solution to a finite dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multisolitons.

Scattering to a stationary solution for the superquintic radial wave equation outside an obstacle

We consider the focusing wave equation outside a ball of ℝ 3 , with Dirichlet boundary condition and a superquintic power nonlinearity. We classify all radial stationary solutions, and prove that all radial global solutions are asymptotically the sum of a stationary solution and a radiation term.

Asymptotic stability of contraction-driven cell motion.

We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two- dimensional free-boundary model that generalizes a previous one-dimensional model [P. Recho, T. Putelat, and L. Truskinovsky, Phys. Rev. Lett. 111, 108102 (2013)10.1103/PhysRevLett.111.108102] by combining a Keller-Segel model, a Hele-Shaw boundary condition, and the Young-Laplace law with a regularizing term which precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We find a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is nonlinear asymptotic stability of traveling solutions that model observable steady cell motion. We derive an explicit asymptotic formula for the stability-determining eigenvalue via asymptotic expansions in small speed. This formula greatly simplifies computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability.

Radial spiky steady states of a flux‐limited Keller–Segel model: Existence, asymptotics, and stability

AbstractThis paper is concerned with the radial stationary problem of a flux‐limited Keller–Segel model derived in a multidimensional bounded domain with Neumann boundary conditions. With the global bifurcation theory and Helly compactness theorem by treating the chemotactic coefficient as a bifurcation parameter, we establish the existence of nonconstant monotone radial stationary solutions and further show that the radial stationary solution will tend to a Dirac delta mass as the chemotactic coefficient tends to infinity. By using the stability criterion of Crandall and Rabinnowitz, we prove the linearized stability of bifurcating stationary solutions near the bifurcation points.

Decay Estimates for Nonradiative Solutions of the Energy-Critical Focusing Wave Equation

We consider the energy-critical focusing wave equation in space dimension \(N\ge 3\). The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation, behaves asymptotically as a sum of modulated Ws, decoupled by the scaling, and a radiation term. A nonradiative solution of the equation is by definition a solution of which energy in the exterior \(\{|x|>|t|\}\) of the wave cone vanishes asymptotically as \(t\rightarrow +\infty \) and \(t\rightarrow -\infty \). In our previous work [9], we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, 0 and \(\pm W\). This was crucial in the proof of soliton resolution in [9]. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behavior as \(r\rightarrow \infty \). We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag, and the second author in [15]. We also study the propagation of the support of nonzero radial solutions with compactly supported initial data and prove that these solutions cannot be nonradiative.

The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution θ=|x|. We use a modified energy method to prove the existence time of classical solutions from 1ϵ to a time scale of 1ϵ4. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.

Radial stationary solutions to a class of wave system as well as their asymptotical behavior

For a stationary version to a class of wave system $ \left\{ \begin{aligned} & -\left(a_1 + b_1\int_{\mathbb{R}^3} {|\nabla u|^2} dx + c\int_{\mathbb{R}^3} {|\nabla v|^2} dx\right) \Delta u + u = \frac{p}{Q} |u|^{p-2}u|v|^q, \\ & -\left(a_2 + b_2\int_{\mathbb{R}^3} {|\nabla v|^2} dx +c\int_{\mathbb{R}^3} {|\nabla u|^2} dx\right) \Delta v + v = \frac{q}{Q} |u|^p |v|^{q-2}v, \end{aligned} \right. $ $u, v \in H_r^1 (\mathbb{R}^3)$, by establishing a variant variational identity and constraint set, we prove that for $a_s \gt 0$, $b_s \gt 0$, $(s = 1, 2)$, $c\geq 0$ and $p \gt 1$, $q \gt 1$ with $Q: = p+q \in (2, 6)$, the system admits a positive radially symmetric ground state solution in $H_r^1 (\mathbb{R}^3)\times H_r^1 (\mathbb{R}^3)$. Moreover, for any fixed $a_1 \gt 0$ and $a_2 \gt 0$, as $b_1^2 + b_2^2 + c^2 \to 0$, this solution converges to a positive radially symmetric solution to $ - a_1 \Delta u + u = \frac{p}{Q} |u|^{p-2}u|v|^q, \quad - a_2 \Delta v + v = \frac{q}{Q} |u|^p |v|^{q-2}v,\quad u, v\in H_r^1 (\mathbb{R}^3). $

The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

Abstract We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

Analysis of a tumor-model free boundary problem with a nonlinear boundary condition

In this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with angiogenesis. This model differs from the other tumor models studied in existing literatures at the point that it has a nonlinear boundary value condition for the nutrient concentration. We first study spherically symmetric version of this model. We prove that there exists a unique spherically symmetric stationary solution which is asymptotically stable under spherically symmetric perturbation. Next we make rigorous analysis to the spherically asymmetric version of this model. By using some abstract theory of parabolic differential equations in Banach manifold, we prove that this free boundary problem is locally well-posed in little Hölder spaces and the radial stationary solution is asymptotically stable in case the surface tension coefficient γ is larger than a threshold value, whereas unstable in case γ is less than this threshold value.

Asymptotic behavior of solutions of a free-boundary tumor model with angiogenesis

This paper is concerned with a free boundary problem modeling the growth of solid tumor spheroid with angiogenesis. The model comprises a coupled system of two elliptic equations describing the distribution of nutrient concentration σ and inner pressure p within the tumor tissue. Angiogenesis results in a new boundary condition ∂nσ+βσ−σ¯=0 instead of the widely studied condition σ=σ¯ over the moving boundary, where β is a positive constant. We first prove that this problem admits a unique radial stationary solution, and this solution is globally asymptotically stable under radial perturbations. Then we establish local well-posedness of the problem and study asymptotic stability of the radial stationary solution under non-radial perturbations. A positive threshold value γ∗ is obtained such that the radial stationary solution is asymptotically stable for γ>γ∗ and unstable for 0<γ<γ∗.

Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis

In this paper we study a free boundary problem modeling the growth of vascularized tumors. The model is a modification to the Byrne–Chaplain tumor model that has been intensively studied during the past two decades. The modification is made by replacing the Dirichlet boundary value condition with the Robin condition, which causes some new difficulties in making rigorous analysis of the model, particularly on existence and uniqueness of a radial stationary solution. In this paper we successfully solve this problem. We prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient, whereas unstable for small surface tension coefficient. Tools used in this analysis are the geometric theory of abstract parabolic differential equations in Banach spaces and spectral analysis of the linearized operator.

Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relation

In this paper we study a free boundary problem modeling the growth of solid tumor spheroid. It consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumor, respectively. The new feature is that nutrient concentration on the boundary is less than external supply due to a Gibbs–Thomson relation and the problem has two radial stationary solutions, which differs from widely studied tumor spheroid model with surface tension effect. We first establish local well-posedness by using a functional approach based on Fourier multiplier method and analytic semigroup theory. Then we investigate stability of each radial stationary solution. By employing a generalized principle of linearized stability, we prove that the radial stationary solution with a smaller radius is always unstable, and there exists a positive threshold value γ⁎ of cell-to-cell adhesiveness γ, such that the radial stationary solution with a larger radius is asymptotically stable for γ>γ⁎, and unstable for 0<γ<γ⁎.

Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition

In this paper we study existence of nonradial stationary solutions of a free boundary problem modeling the growth of nonnecrotic tumors. Unlike the models studied in existing literatures on this topic where boundary value condition for the nutrient concentration $ \sigma $ is linear, in this model this is a nonlinear boundary condition. By using the bifurcation method, we prove that nonradial stationary solutions do exist when the surface tension coefficient $ \gamma $ takes values in small neighborhoods of certain eigenvalues of the linearized problem at the radial stationary solution.

Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs–Thomson relation

In this paper we study a free boundary problem modeling tumor growth. The model consists of two elliptic equations describing nutrient diffusion and pressure distribution within tumors, respectively, and a first-order partial differential equation governing the free boundary, on which a Gibbs–Thomson relation is taken into account. We first show that the problem may have none, one or two radial stationary solutions depending on model parameters. Then by bifurcation analysis we show that there exist infinite many branches of non-radial stationary solutions bifurcating from given radial stationary solution. The result implies that cell-to-cell adhesiveness is the key parameter which plays a crucial role on tumor invasion.

Stationary solutions of Keller–Segel-type crowd motion and herding models : Multiplicity and dynamical stability

In this paper we study two models for crowd motion and herding. Each of the models is of Keller-Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated to such solutions. The dynamical stability in a neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter and all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss qualitative properties of the solutions using theoretical methods and numerical computations.

Linearized stability for a multi-dimensional free boundary problem modelling two-phase tumour growth

This paper is concerned with a multi-dimensional free boundary problem modelling the growth of a tumour with two species of cells: proliferating cells and quiescent cells. This free boundary problem has a unique radial stationary solution. By using the Fourier expansion of functions on unit sphere via spherical harmonics, we establish some decay estimates for the solution of the linearized system of this tumour model at the radial stationary solution, so proving that the radial stationary solution is linearly asymptotically stable when neglecting translations.

On radial stationary solutions to a model of non-equilibrium growth

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.

Bifurcation analysis of a free boundary problem modelling tumour growth under the action of inhibitors

In this paper we investigate non-radial stationary solutions of a free boundary problem modelling tumour growth under the action of inhibitors. The model consists of two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the velocity of tumour cells and internal pressure. The ratio μ/γ of the proliferation rate μ and the cell-to-cell adhesiveness γ plays the role of the bifurcation parameter. We prove that in certain situations there exists a positive sequence such that for each (μ/γ)n(n even ⩾n*) there exist non-radial stationary solutions bifurcating from the radial stationary solution, while in the other situations there exists at most a finite number of bifurcation points. This is a remarkable difference from the corresponding inhibitor-free model where there always exist infinitely many branches of non-radial stationary bifurcation solutions. Our analysis also indicates that inhibitor supply may lower the ability of tumour invasion, and even make the tumour unaggressive and stable.