Related papers: Periodic solutions of forced Kirchhoff equations
We point out a rather effective approach for solving the time-dependent harmonic oscillator $\ddot q=-\omega^2 q$ under various regularity assumptions. Where $\omega(t )$ is $C^1$ this is reduced to Hamilton equation for the angle variable…
We prove the $W^{1,2}_p$-estimate and solvability for the Dirichlet problem of second-order parabolic equations in simple convex polytopes with time irregular coefficients, when $p\in (1,2]$. We also consider the corresponding Neumann…
We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ \kappa_j \dot{z}_j = J \nabla_{z_j} H_\Omega(z_1,z_2), \quad j=1,2, \] for $z_1,z_2$ in a domain $\Omega\subset\mathbb{R}^2$. In the classical point vortex…
This article concerns time-periodic solutions to a two-dimensional Sellers-type energy balance model coupled to the three-dimensional primitive equations via a dynamic boundary condition. It is shown that the underlying equations admit at…
In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal…
We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function.…
In this paper, we are devoted to consider the periodic problem for the impulsive evolution equations with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of…
The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these…
Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet…
The goal of the paper is to develop a method that will combine the use of variational techniques with regularization methods in order to study existence and multiplicity results for the periodic and the Dirichlet problem associated to the…
In this paper, we study the Cauchy-Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \mbox{$\partial_t u - \operatorname{div} \left( D_\xi f(t, Du)\right) = 0$ } & \mbox{in $\Omega_T$}, \\[5pt] \mbox{$u = u_o$} & \mbox{on…
We consider the conformal wave equation on the Einstein cylinder with a defocusing cubic non-linearity. Motivated by a method developed by Rostworowski-Maliborski on the existence of time periodic solutions to the spherically symmetric…
The periodic solutions of a type of nonlinear hyperbolic partial differential equations with a localized nonlinearity are investigated. For instance, these equations are known to describe several acoustical systems with fluid-structure…
Consider the equation of the linear oscillator $u"+u=h(\theta)$, where the forcing term $h:\mathbb R\to\mathbb R$ is $2\pi$-periodic and positive. We show that the existence of a periodic solution implies the existence of a positive…
The time-periodic problem on the Boltzmann equation with a given time-periodic external force in the three-dimensional whole space has remained open since it was first studied in [15] for only spatial dimensions not less than five. The goal…
In this paper we consider a generalized fourth order nonlinear Kirchhoff equation in a bounded domain in $\mathbb R^{N}, N\geq2$ under Navier boundary conditions and with sublinear nonlinearity. We employ a change of variable which reduces…
In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions…
This revision includes clarified exposition and simplified analysis. Solutions of the Einstein equations which are periodic and have standing gravitational waves are valuable approximations to more physically realistic solutions with…
As a first step in exploring time-periodic solutions of the Einstein equations with a negative cosmological constant, we study the cubic conformal wave equation on the Einstein cylinder. Using a combination of numerical and perturbative…
In this note we present some recent results for Kirchhoff equations in generalized Gevrey spaces. We show that these spaces are the natural framework where classical results can be unified and extended. In particular we focus on existence…