We investigate the monotonicity of the minimal period of periodic solutions of quasilinear differential equations involving the $p$-Laplace operator. First, the monotonicity of the period is obtained as a function of a Hamiltonian energy in two cases. We extend to $p\ge2$ classical results due to Chow-Wang and Chicone for $p=2$. Then we consider a differential equation associated with a fundamental interpolation inequality in Sobolev spaces. In that case, we generalize monotonicity results by Miyamoto-Yagasaki and Benguria-Depassier-Loss to $p\ge2$.
We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure.
We consider a quasilinear elliptic equation involving a first order term, under zero Dirichlet boundary condition in half spaces. We prove that any positive solution is monotone increasing w.r.t. the direction orthogonal to the boundary. The main ingredient in the proof is a new comparison principle in unbounded domains. As a consequence of our analysis, we also obtain some new Liouville type theorems.
We prove a weak comparison principle in narrow unbounded domains for solutions to $-Δ_p u=f(u)$ in the case $2<p< 3$ and $f(\cdot)$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot)$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-Δ_p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.
In this paper we prove the monotonicity of positive solutions to $ -Δ_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
In this paper, we investigate a general quasilinear elliptic and singular system. By monotonicity methods, we give some existence and uniqueness results. Next, we give some applications to biological models.
The main goal of this paper is to show how some monotonicity methods related with the subdifferential of suitable convex functions and its extensions as m-accretive operators in Banach spaces lead to new and unexpected results showing, for instance, the continuous and monotone dependence of solutions with the respect to the data (and coefficients) of the problem. In this way, this paper offers `a common roof' to several methods and results concerning monotone and non-monotone frameworks. Besides to present here some new results, this paper offers also a peculiar survey to some topics which att…
The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differe…
We are concerned with a class of second order quasilinear elliptic equations driven by a nonhomogeneous differential operator introduced by C.A. Stuart and whose study is motivated by models in Nonlinear Optics. We establish sufficient conditions for the existence of at least one or two non-negative solutions. Our analysis considers the cases when the reaction has either a sublinear or a linear growth. In the sublinear case, we also prove a nonexistence property. The proofs combine energy estimates and variational methods. In particular, the monotonicity trick is applied in order to overcome t…
We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown. Secondly, we show that this evolution equation has a quasilinear structure, which is at a formal level not obvious, and we also disclose the parabolic character of the equation. Exploiting the…
We provide the rigorous justification of the NLS approximation, in Sobolev regularity, for a class of quasilinear Hamiltonian Klein Gordon equations with quadratic nonlinearities on large one-dimensional tori $\T_L:=\mathbb{R}/(2πL \mathbb{Z})$, $L\gg 1$. We prove the validity of this approximation over a \emph{long-time} scale, meaning that it holds beyond the cubic nonlinear time scale. To achieve this result we need to perform a second-order analysis and deal with higher order resonant wave-interactions. The main difficulties are provided by the quasi-linear nature of the problem and the pr…
We investigate the effect of frequency on the principal eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow and Poláčik [2001 J. Math. Biol.].
In this paper, we consider the quasilinear Schrödinger equation \begin{equation*} -Δu+V(x)u-uΔ(u^2)=g(u),\ \ x\in \mathbb{R}^{3}, \end{equation*} where $V$ and $g$ are continuous functions. Without the coercive condition on $V$ or the monotonicity condition on $g$, we show that the problem above has a least energy sign-changing solution and infinitely many sign-changing solutions. Our results especially solve the problem above in the case where $g(u)=|u|^{p-2}u$ ($2<p<4$) and complete some recent related works on sign-changing solutions, in the sense that, in the literature only the case $g(u)…
In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x, \nabla u))&=& μ\quad \text{in} ~Ω, u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*} where $μ$ is a finite signed Radon measure in $Ω$, $Ω\subset \mathbb{R}^n$ is a bounded domain such that its complement $\mathbb{R}^n\backslashΩ$ is uniformly $p$-thick and $\mathcal{A}$ is a Carathéodory vector valued function satisfying growth and monotonic…
In this paper, we consider the inverse scattering problem to reconstruct the defect in an open periodic waveguide from near field data. Our first aim is to mention that there is a mistake in the factorization method of Lechleiter 2008. By this we can not apply it to solve this inverse problem. Our second aim is to give ways to understand the defect from inside (Theorem 1.1) and outside (Theorem 1.2) based on the monotonicity method.
In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.
We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or la…
Periodic operators over a component domain and homogenization of some class of quasi-linear elliptic problems in two-component domain with interfacial resistance
We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus $\T^3$, with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.
We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $\mathbb{R}\times\mathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $g\in L^{\infty}(\mathbb{R})$ is even with $g\not\equiv 0$ and $h(x)=γ\,δ_0(x)$ with $γ\in{\mathbb{R}}\backslash\{0\}$ and $δ_0$ the delta-distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k=-\frac{d^2}{dx^2}-k^2ω^2g$ on $L^2(\mathbb{R})$ for all $…
We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} where $σ$ is a nonnegative locally finite measure in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and ${\rm div}\, \mathcal{A}(x,\nabla u)$ is the $\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ($1<p<\infty$) on $\mat…
We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2] by Coulombel, Gues, and Williams studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulti…
This paper is concerned with the wave length $λ$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map $a\longmapsto λ(a)$, i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of $λ(a)$, namely monotonicity and unimodality. The key point is to r…
We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of $\varepsilon^{-5/3+}$, for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of…
In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein-Gordon type equations, which include the well known $ϕ^4$ and $ϕ^6$ models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of exp…
We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions $V$ and $Γ$. We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coeffcient $Γ$ we distinguish between two cases: (a) $Γ\in L^\infty$ being regular and (b) $Γ=γδ_0$ being a multiple of the delta potential at zero. For both cases w…
In this paper, we study the good-$λ$ type bounds for renormalized solutions to nonlinear elliptic problem: \begin{align*} \begin{cases} -÷(A(x,\nabla u)) &= μ\quad \text{in} \ \ Ω, \\ u &=0 \quad \text{on} \ \ \partial Ω. \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$, $μ$ is a finite Radon measure and $A$ is a monotone Carathédory vector valued function defined on $W^{1,p}_0(Ω)$. The operator $A$ satisfies growth and monotonicity conditions, and the $p$-capacity uniform thickness condition is imposed on $\mathbb{R}^n \setminus Ω$, for the singular case $\frac{3n-2}{2n-1} < p \le 2- \f…
We study the well-posedness of the initial value problem on periodic intervals for linear and quasilinear evolution equations for which the leading-order terms have three spatial derivatives. In such equations, there is a competition between the dispersive effects which stem from the leading-order term, and anti-diffusion which stems from the lower-order terms with two spatial derivatives. We show that the dispersive effects can dominate the backwards diffusion: we find a condition which guarantees well-posedness of the initial value problem for linear, variable coefficient equations of this k…
In this paper we are concerned with the fractional Schrödinger equation $(-Δ)^α u+V(x)u =f(x, u)$, $x\in \rn$, where $f$ is superlinear, subcritical growth and $u\mapsto\frac{f(x, u)}{\vert u\vert}$ is nondecreasing. When $V$ and $f$ are periodic in $x_{1},\ldots, x_{N}$, we show the existence of ground states and the infinitely many solutions if $f$ is odd in $u$. When $V$ is coercive or $V$ has a bounded potential well and $f(x, u)=f(u)$, the ground states are obtained. When $V$ and $f$ are asymptotically periodic in $x$, we also obtain the ground states solutions. In the previous research, …
We consider the linear wave equation $V(x) u_{tt}(x, t) - u_{xx}(x, t) = 0$ on $[0, \infty)\times[0, \infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x(0, t) = (f(u_t(0,t)))_t$ at $x=0$. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where $f\colon\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there …
We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.
We study the following quasilinear elliptic system for all $i=1,\cdots,m$ \begin{equation*} \label{} -div(Φ'(|\nabla u_i|^2) \nabla u_i) = H_i(u) \quad \text{in} \ \ \mathbb{R}^n \end{equation*} where $u=(u_i)_{i=1}^m: \mathbb R^n\to \mathbb R^m$ and the nonlinearity $ H_i(u) \in C^1(\mathbb R^m)\to \mathbb R$ is a general nonlinearity. Several celebrated operators such as the prescribed mean curvature, the Laplacian and the $p$-Laplacian operators fit in the above form, for appropriate $Φ$. We establish a Hamiltonian identity of the following form for all $x_n\in\mathbb R$ \begin{equation*}\l…
In this paper we mainly investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem with some unknown functions in the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for a almost periodic function of one variable. The bifurcation theory ensures us to obtain a existence resu…
We study the Schrödinger equations $-Δu + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-Δu - λu = f(x,u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and $u\mapsto f(x,u)/|u|$ is nondecreasing. In $\mathbb{R}^N$ we also assume that $V$ and $f$ are periodic in $x_1,\ldots,x_N$. We show that these equations have a ground state and that there exist infinitely many solutions if $f$ is odd in $u$. Our results generalize those in \cite{sw1} where $u\mapsto f(x,u)/|u|$ was assumed to be strictly increasing. This seemingly small change forces us to…
We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous result in \cite{Mietka}, generalising the considered class of equations and improving the regularity assumption on the initial data.
