Sub-and Supersolutions Research Articles

Periodic traveling waves for a diffusive influenza model with treatment and seasonality

This paper is concerned with the existence and nonexistence of time-periodic traveling waves for a diffusive influenza model with treatment and seasonality. By using the next generation operator theory, we first get basic reproduction number R0 for the corresponding periodic ODEs. Then, by constructing sub-and super-solutions and using Schauder’s fixed point theorem, we obtain the existence of time-periodic traveling wave solutions for the system with wave speed c>c∗ and R0>1. We further prove the existence of time-periodic traveling waves with wave speed c=c∗ by a delicate limitation argument. For du=dh, the nonexistence of traveling waves is proved by a contradiction argument for two cases involved with c∗ and R0, while the exponential decay of traveling waves with the critical speed is obtained by a dynamical system approach combined with Laplace transform.

On the theory of entropy suband supersolutions of nonlinear degenerate parabolic equations

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem, understood in the sense of Chen-Perthame, is both an entropy sub- and supersolution. It is established that the maximum of entropy subsolutions of the Cauchy problem is also an entropy subsolution of this problem. This result is used to prove the existence of the largest entropy subsolution (and the smallest entropy supersolution). It is also shown that the largest entropy subsolution and the smallest entropy supersolution are also entropy solutions.

Front Profile in Time Backward for the Bistable Reaction-Diffusion Equation on Metric Graphs

In the bistable reaction-diffusion equation on a half-line there exists an entire solution exhibiting the front propagation from the infinity of the line. We glue an arbitrary number of half-lines at a single point (junction) to build a metric graph and examine the front propagation on given half-lines (upstream branches). The aim of the article is to show the existence of an entire solution which converges to the profile of the traveling wave with an arbitrarily assigned phase-shift in each upstream branch as $$t\rightarrow -\infty $$ . The proof is carried out by constructing an appropriate pair of sub-and super-solutions.

On the Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be non-unique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy suband super-solutions, in the case when at least one of the initial functions is periodic. The obtained results are generalization of the results known for conservation laws to the parabolic case.

Thin Front Limit of an Integro-differential Fisher-KPP Equation with Fat-Tailed Kernels

We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton-Jacobi limit.

Regularity properties of viscosity solutions for fully nonlinear equations on the model of the anisotropic p →-Laplacian

This paper is devoted to some Lipschitz estimates between sub-and super-solutions of Fully Nonlinear equations on the model of the anisotropic p-Laplacian. In particular we derive from the results enclosed that the continuous viscosity solutions for the equation N 1 ∂i(∂iu| p i −2 ∂iu) = f are Lipschitz continuous when sup i pi < infi pi + 1, where p = i piei.

On Neumann problems for nonlocal Hamilton\u2013Jacobi equations with dominating gradient terms

We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton–Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order strictly less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron’s method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data.

Existence of weak positive solution for a singular elliptic problem with supercritical nonlinearity

In this paper we consider the existence of weak positive solutions for an elliptic problems with the nonlinearity containing both singular and supercritical terms. By means of a priori estimate and sub-and supersolutions method, a positive weak solution is obtained.

Existence results for a class of Kirchhoff type systems with Caffarelli-Kohn-Nirenberg exponents

Abstract This paper is concerned with the existence of positive solutions for a class of infinite semipositone kirchhoff type systems with singular weights. Our aim is to establish the existence of positive solution for λ large enough. The arguments rely on the method of sub-and super-solutions.

Representation formulas for solutions of Isaacs integro-PDE

We prove suband super-optimality inequalities of dynamic programming for viscosity solutions of Isaacs integro-PDE associated with two-player, zero-sum stochastic differential game driven by a Levy type noise. This implies that the lower and upper value functions of the game satisfy the dynamic programming principle and they are the unique viscosity solutions of the lower and upper Isaacs integro-PDE. We show how to regularize viscosity suband super-solutions of Isaacs equations to smooth suband super-solutions of slightly perturbed equations.

On a sub-supersolution method for the prescribed mean curvature problem

The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub-and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub-and supersolutions are established.

Existence and asymptotic behavior of entire blow-up solutions for quasilinear elliptic equation

In this article, we discuss the blow-up problem of entire solutions of a class of second-order quasilinear elliptic equation Δ p u ≡ div(|∇u| p−2∇u) = ρ(x)f(u), x ∈ R N . No monotonicity condition is assumed upon f(u). Our method used to get the existence of the solution is based on sub-and supersolutions techniques.

Sub-supersolutions in a variational inequality related to a sandpile problem

AbstractIn this paper we study a variational inequality in which the principal operator is a generalised Laplacian with fast growth at infinity and slow growth at 0. By defining appropriate sub-and super-solutions, we show the existence of solutions and extremal solutions of this inequality above the subsolutions or between the sub- and super-solutions.

Existence and multiplicity results for quasilinear elliptic equations

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f(x, s), we show the follwing problem: , where Ω is a bounded open subset of RN, N ≥ 2, with smooth boundary, λ is a positive parameter and ∆p is the p-Laplacian operator with p &gt; 1, possesses at least two positive solutions for large λ.

Existence of positive solutions for some problems with nonlinear diffusion

In this paper we study the existence of positive solutions for problems of the type − Δ p u ( x ) a m p ; = u ( x ) q − 1 h ( x , u ( x ) ) , a m p ; a m p ; x ∈ Ω , u ( x ) a m p ; = 0 , a m p ; a m p ; x ∈ ∂ Ω , \begin{equation*} \begin {aligned} -\Delta _pu(x) &amp;=u(x)^{q-1}h(x,u(x)), &amp;&amp; x\in \Omega , \\ u(x)&amp;=0, &amp;&amp; x\in \partial \Omega , \end{aligned} \end{equation*} where Δ p \Delta _p is the p p -Laplace operator and p , q &gt; 1 p,q&gt;1 . If p = 2 p=2 , such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases p = q p=q , p &gt; q p&gt;q and p &gt; q p&gt;q , respectively. Also, some systems of equations are considered.

Uniqueness of global positive solution branches of nonlinear elliptic problems

has been studied in many publications (the first global description was given in [19]). Nonetheless there were open questions some of which are answered in this paper. We motivate some of these questions now: For a suitable class of (smooth) fimctions f : IR+ ~ IR problem (1.1) has infinitely many solutions for 2 = 1. This was shown in [15] via the method of suband supersolutions which also gives an order of these solutions: 0 0 (which is not excluded in [15]) there is also a global branch ( = continuum) C + of solutions (2 ,u) bifurcating from the trivial solution line {(2, 0)} at 2 = 21 where #1 = 21 f ' ( 0 ) is the principal eigenvalue of the linearization

Nonlinear Elliptic Variational Inequalities

AbstractWe consider the problem of finding a solution to a class of nonlinear elliptic variational inequalities. These inequalities may be defined on bounded or unbounded domains Ω, and the nonlinearity can depend on gradient terms. Appropriate definitions of sub‐and supersolutions relative to the constraint sets are given. By using a mixture of maximal monotone operator theory and compactness arguments we prove the existence of a H2(Ω) solution lying between a given subsolution φ1 and a given supersolution φ2≧φ1, when Ω is bounded, and a H1(Ω) solution when Ω is unbounded.

Cauchy problems for certain Isaacs-Bellman equations and games of survival

Two person zero sum differential games of survival are considered; these terminate as soon as the trajectory enters a given closed set F, at which time a cost or payoff is computed. One controller, or player, chooses his control values to make the payoff as large as possible, the other player chooses his controls to make the payoff as small as possible. A strategy is a function telling a player how to choose his control variable and values of the game are introduced in connection with there being a delay before a player adopts a strategy. It is shown that various values of the differential game satisfy dynamic programming identities or inequalities and these results enable one to show that if the value functions are continuous on the boundary of F then they are continuous everywhere. To discuss continuity of the values on the boundary of F certain comparison theorems for the values of the game are established. In particular if there are suband super-solutions of a related Isaacs-Bellman equation then these provide upper and lower bounds for the appropriate value function. Thus in discussingyalue functions of a game of survival one is studying solutions of a Cauchy problem for the Isaacs-Bellman equation and there are interesting analogies with certain techniques of classical potential theory.