Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

Stationary solutions of reaction-diffusion equations

Given a semilinear reaction-diffusion equation. If the corresponding ordinary differential equation admits a convex compact positively invariant set and the boundary data assume their values in this set then the first and third boundary value problem have stationary solutions. The proofs are based on Weinberger's strong invariance principle, some related tools and the Leray-Schauder degree. The theorem is applied to several equations from theoretical biology, also in the case of distinct diffusion rates.     

Spherically symmetric solutions of a reaction-diffusion equation

Two theorems are proved for the spherically symmetric solutions of the “bistable” reaction-diffusion equation $\text{u}{}_{\mathrm{t}}\text{= Δ}{}_{\mathrm{x}}\text{u + ?(u)}$, where ? is cubic-like and x ∈ Rⁿ. The first theorem says that, for a suitable class of initial data, there are only two types of asymptotic behavior, u(x, t) tends to an equilibrium solution as t → + ∞ or u(x, t) → 1 uniformly on compact sets. The second theorem says that in the latter case, if the solution is followed out along any ray, it approaches, in shape, the one-dimensional travelling wave.     

Integral bounds for solutions of nonlinear reaction-diffusion equations

Differential inequality techniques are used to obtain upper bounds for theL ¹ norm of solutions of nonlinear reaction-diffusion equations. Essential use is made of Sobolev type integral inequalities. An extension to third order pseudo-parabolic equations is included.

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Résumé On utilise des techniques d'inégalités differentielles afin d'obtenir des bornes supérieures pour les normes de typeL ¹ des solutions des équations des réaction et diffusion non linéaires. On utilise de façon essentielle des inégalités intégrales du type de Sobolev. On inclut une extension aux équations pseudo-paraboliques du troisième ordre.

     

Continuously decreasing solutions for polynomial-like iterative equations

Most known results on existence,uniqueness and stability for solutions of the polynomial-like iterative equation Σni=1λifi(x)=F(x) were obtained in the case of λ 1 = 0.In this paper,we construct C 0 decreasing solutions of the iterative equation in the case that λ 1 can vanish to answer the Leading Coefficient Problem.Moreover,we also give the necessary and sufficiently condition for uniqueness of solutions.     

Exact number of solutions of stationary reaction-diffusion equations

We study both existence and the exact number of positive solutions of the problem

     

Positive decreasing solutions of quasilinear dynamic equations

We consider a quasilinear dynamic equation reducing to a half-linear equation, an Emden–Fowler equation or a Sturm–Liouville equation under some conditions. Any nontrivial solution of the quasilinear dynamic equation is eventually monotone. In other words, it can be either positive decreasing (negative increasing) or positive increasing (negative decreasing). In particular, we investigate the asymptotic behavior of all positive decreasing solutions which are classified according to certain integral conditions. The approach is based on the Tychonov fixed point theorem.     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

Positivity and boundedness of solutions of impulsive reaction-diffusion equations

Employing the generalized quasilinearization for nonlinear reaction-diffusion equations, existence of positive bounded solution is proved.     

Generic properties of stationary state solutions of reaction-diffusion equations

The problem of constructing variational principles for a given second-order quasi-linear partial differential equation is considered. In particular, we address the problem of finding a first-order function f whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian. Two sets of equations for such a function f are obtained. Necessary and sufficient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set. In certain special cases, such as the case when the operator is elliptic, the problem is completely solved. The utility of our results is illustrated by a variety of examples.     

Stability of spatially homogeneous periodic solutions of reaction-diffusion equations

When a certain condition is satisfied, a reaction-diffusion equation has a spatially homogeneous periodic solution, i.e. a temporally periodic solution that does not depend on spatial variables. We analyse the orbital stability of this periodic solution. A sufficient condition is given for the homogeneity breaking instability, which is stated in terms of the manner of dependency of its temporal period on a certain parameter of the system.     

Eliminating indeterminacy for localized solutions to reaction-diffusion equations in multidimensional domains

Exponentially ill-conditioned localized solutions are constructed asymptotically in the limit of small diffusion for two classes of singularly perturbed reaction-diffusion equations in a multidimensional domain.     

Global bounds of solutions for a strongly coupled system of reaction-diffusion equations

Boundedness results for a strongly coupled system of reaction-diffusion equations on spatially bounded region are proved.     

Blowup phenomena of solutions to Euler-Poisson equations

In this paper, we consider the Euler-Poisson equations governing the evolution of the gaseous stars with the Poisson equation describing the energy potential for the self-gravitating force. By assuming that the initial density is of compact support in , we first give a family of blowup solutions for non-isentropic polytropic gas when γ=(2N−2)/N which generalizes the known result for the isentropic case. Then we extend the previous result on non-blowup phenomena to the case when (2N−2)/N?γ<2 in N-dimensional space. Here γ is the adiabatic gas constant.     

Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity

Entire solutions for monostable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as t→−∞ in term of appropriate subsolutions and supersolutions. Two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results.     

Spherically symmetric solutions of Euclidean Yang-Mills equations

We consider the Euclidean Yang-Mills equations with structure groupSu(2). The action functional and the topological charge are invariant under the transformations, whereg runs over the set of unit length quaternions, andgx denotes the product ofg by the quaternionx=x ₄+ ix₁+jx_g+kx₃. ThisSu(2)-symmetry permits us to apply Coleman's principle. For the potentials A_µ, we obtain the following spherically symmetric Ansatz: while the Yang-Mills equations and the duality equations reduce to ordinary differential equations for the function. We show that every solution of the form (1) of the Yang-Mills equations with finite action and positive (negative) charge satisfies the duality equationsF=* F (respectively,F=–* F) and has charge 1 (respectively, –1). Moreover, we describe explicitly all solutions of form (1) of the duality equations; among them there are the 1-instanton solutions of Belavin, Polyakov, et al.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 126–132, 1984.     

A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.