We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of linear wave operator with periodic-Dirichlet boundary conditions on its range. In the pioneering work of Barbu and Pavel (1997), it gives the existence and regularity of periodic solution for Lipschitz, nonresonant and monotone nonlinearity under the assumption $η_u>0$ (see Sect. 2 for its definition) on the coefficient $u(x)$ and leaves the case $η_u=0$ as an open problem. In this paper, by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory, we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions. Our result not only does not need any requirements on the coefficient except for the natural positivity assumption (i.e., $u(x)>0$), but also does not need the monotonicity assumption on the nonlinearity. In particular, when the nonlinear term is an odd function and satisfies the global nonresonance conditions, there is only one (trivial) solution to this problem on the invariant subspace.
We adapt a technique of nonsmooth critical point theory developed by Degiovanni-Zani for a semilinear problem involving the Laplacian to the the case of the $p$-Laplacian. We suppose only coercivity conditions on the potential and impose no growth condition of the nonlinearity. The coercivity is obtained using similar nonresonance conditions to [Mawhin-Ward-Willem] and to [Landesman-Lazer] in two different results and using some comparison functions and comparison spaces in a third one. It is also shown that neither of the three theorems implies the two others.
Higher Regularity of Time-Periodic Solutions to Nonautonomous Hyperbolic Problems: Away from Resonances
Institute of Mathematics, Humboldt University of Berlin（柏林洪堡大学数学研究所）；Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences（乌克兰国家科学院应用力学与数学问题研究所）
We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.
We prove the existence of infinitely many classical periodic solutions for a class of semilinear wave equations with periodic boundary conditions. Our argument relies on some new estimates for the linear problem with periodic boundary conditions, by combining Littlewood-Paley techniques, the Hausdorff-Young theorem, and a variational formulation due to Rabinowitz. We also develop a new approach to the regularity of the distributional solutions, by employing Gagliardo-Nirenberg estimates.
We prove existence of infinitely many classical periodic solutions with periodic boundary conditions for a class of monotone semilinear wave equations. Our argument relies on some new estimates for the linear problem with periodic boundary conditions by combining Littewood-Paley techniques, the Hausdorff-Young theorem and a variational formulation due to Rabinowitz. We also develop a new approach to the regularity of distributional solutions by differentiating the equations and employing Gagliardo-Nirenberg estimates.
We prove the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations: \[ u_{tt}-u_{xx}+|u|^{s-1}u=f(x,t), \] for all $s>1$. In particular we prove the existence of infinitely many classical solutions for the case $s=3$ posed by Brézis in \cite{BrezisBAMS}. The proof relies on a new upper a priori estimates for minimax values of, a pertubed from symmetry, strongly indefinite functional,depending on a small parameter.
We study topological structure of the $ω$-limit sets of the skew-product semiflow generated by the following scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f(t,u,u_x)$ is $C^2$-admissible with time-recurrent structure including almost-periodicity and almost-automorphy. Contrary to the time-periodic cases (for which any $ω$-limit set can be imbedded into a periodically forced circle flow), it is shown that one cannot expect that any $ω$-limit set can be imbedded into an almost-periodically force…
An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation
Institute of Mathematics, Humboldt University of Berlin（柏林洪堡大学数学研究所）；Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院应用力学与数学问题研究所）；Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University（利沃夫国立理工大学应用数学与基础科学研究所计算数学与编程系）；Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）
There exists an exponentially decreasing function $f$ such that any singly $2π$-periodic positive solution $u$ of $-Δu +u-u^p=0$ in $[0,2π]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'|)$. We prove that with the same period and with the same function $f$, any singly periodic positive solution of $-\ep^2Δu-u+u^p=0$ in $[0,2π]\times \R^{N-1}$ verifies $u(x_1,x')\leq f(|x'| /\ep )$ . We have a similar estimate for the gradient.
We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data.
In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L \end{equation} with periodic boundary condition \begin{equation} \label{bdc1} u(t,0)=u(t,L),\quad u_x(t,0)=u_x(t,L), \end{equation} where $f$ is uniformly almost periodic in $t$. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we…
We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenomenon modelled by the underlying problem. We also provide an appropriate scheme for the approximation of the effective coefficients. To illustrate our approach, we work out some concrete problems such as the periodic homogenization problem, the almost…
Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include various asymptotic approximations that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Preliminary results on continuing a numerical solution beyond the singularity are also presented.
In this work, we present basic results and applications of Stepanov pseudo almost periodic functions with measures. Using only the continuity assumption, we prove a new composition result of $μ$-pseudo almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions in Banach spaces with weak regular forcing terms. We prove the existence and uniqueness of $μ$-pseudo almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations, respectively, provided t…
We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\in H^r_{\per}(\R)$ for $r\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\ast$ such that for $p\in (1,p^\ast)$ both in the "+" and the "-" case we can us…
In this paper we prove the existence of infinitely many small energy solution of a semilinear Schrodinger equation via the dual form of the generalized fountain theorem. This equation is with periodic potential and concave-convex nonlinearities.
We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: $-Δu=λu^p - u^q$ in $\mathbb{R}^{N+1}$, where $ 0< q < p \leq 1$, $λ\in \mathbb{R}$. The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. The limit value of the parameter $λ$, where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the …
We consider the nonlinear Klein-Gordon equation $\partial_t^2u(x,t)-\partial_x^2u(x,t)+αu(x,t)=\pm|u(x,t)|^{p-1}u(x,t)$ on a periodic metric graph (necklace graph) for $p>1$ with Kirchhoff conditions at the vertices. Under suitable assumptions on the frequency we prove the existence and regularity of infinitely many spatially localized time-periodic solutions (breathers) by variational methods. We compare our results with previous results obtained via spatial dynamics and center manifold techniques. Moreover, we deduce regularity properties of the solutions and show that they are weak solution…
We obtain a lower bound for the period of periodic solutions of semilinear evolution equations for the full range of nonlinear terms for which standard local existence theory applies. This lower bound depends on the Lipschitz constant of the nonlinear term as an operator acting on the domain of a fractional power of the linear operator into the base space.
In this paper we prove some symmetry results for entire solutions to the semilinear equation $-Δu=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give some sufficient conditions for them to be radially symmetric with respect to those variables, such as periodicity or the pointwise decay of some derivatives.
We analyze nonlinear Schrödinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.
In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.
We establish the existence of weak solutions $u$ of the semilinear wave equation $\partial_t^2 u-\textrm{div}_x(a(t,x)\nabla_xu)=f_k(u)$ where $a(t,x)$ is equal to $1$ outside a compact set with respect to $x$ and a non-linear term $f_k$ which satisfies $\vert f_k(u)\vert\leq C\vert u\vert^k$. For some non-trapping time-periodic perturbations $a(t,x)$, we obtain the long time existence of solution for small initial data.
We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity $g(t,x,u,u_x)$ either not depending on $t$, or periodic in $t$. While the topological and geometric structure of their attractors has been investigated in depth, we focus here on ergodic-theoretical properties. The main result is that the union of supports of all the invariant measures projects one-to-one to $\mathbb{R}^2$. We rely on a novel application of the zero-number techniques with respect to evolution of measures …
We consider positive solutions $u$ of the semilinear biharmonic equation $Δ^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a periodic function of $\ln |x|$ and we classify all such solutions.
We study bounded solutions to the fractional equation $(-Δ)^s u + u - |u|^{q-2}u = 0$ in $\mathbb R^n$ for $n\ge2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov (2020) we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.).
This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find…
A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.
In this paper, we study the following semilinear Schrödinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term $f(x,t)$ satisfies some superlinear growth conditions and need not be odd or increasing strictly in $t$. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are …
We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0$ for $(x,t)\in \mathbb{R}^3\times\mathbb{R}$ and arbitrary $p>1$. Depending on the coefficients $s, q, V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our soluti…
This paper shows that the nonlinear periodic eigenvalue problem $${cases} -Δu + V(x) u - f(x,u) = λu, u \in H^1(\IR^N), {cases}$$ has a nontrivial branch of solutions emanating from the upper bound of every spectral gap of $-Δ+ V$. No convexity condition is assumed. The following result of independent interest is also proven: the direct sum $Y \oplus Z$ in $H^1(\IR^N)$ associated to a decomposition of the spectrum of $-Δ+V$ remains "topologically direct" in the $L^p$'s (in the sense that the projections from $Y+Z$ onto $Y$ and $Z$ are $L^p$-continuous).
We study the existence of nontrivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in $\mathbb{R}^N$. By combining variational methods and the concentration-compactness principle we obtain a nontrivial solution for asymptotically periodic problem and a ground state solution for the periodic problem. In the proofs we apply the Mountain Pass Theorem and its local version.
We obtain some existence theorems for periodic solutions to several linear equations involving fractional Laplacian. We also prove that the lower bound of all periods for semilinear elliptic equations involving fractional Laplacian is not larger than some exact positive constant. Hamiltonian identity, Modica-type inequalities and an estimate of the energy for periodic solutions are also established.
In this paper we study second order non-linear periodic systems driven by the ordinary vector $p$-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by the $p$-Laplacian. In the last section of the paper we examine the s…
We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions. Such problems are posed on compact submanifolds evolving periodically in time. The discussion is based on the principal eigenvalue of periodic parabolic operators. The study is motivated by biological models on the effect of growth and curvature on patterns formation. The Ricci curvature plays an important role.
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 consider the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0 \mbox{ for } (x,t)\in \mathbb{R}^3\times\mathbb{R}$. For any $p>1$ we prove the existence of time-periodic spatially localized real-valued solutions (breathers) both for the $+$ and the $-$ case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to $0$ as $|x|\to \infty$. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the …
We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-Δu + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$ we obtain that the set of positive solutions is loca…
In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 \le p<1+\frac{2}{N}$, \begin{equation*} \begin{cases} u_t+v \cdot \nabla u-Δu=|u|^p-\int_{\mathbb T^N} |u|^p \quad & \textrm{on} \quad \mathbb T^N, \\ \\ u \ \textrm{periodic} \quad & \textrm{on} \quad \partial \mathbb T^N \end{cases} \end{equation*} with initial data $u_0$ defined on $\mathbb T^N$. Here $v$ is an incompressible flow, and $\mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbit…
We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant manifolds attached to fixed points or periodic orbits. The core of the study is writing down the invariance condition as a zero of an operator. These operators are in general not continuous, so one needs to smooth them by means of preconditioners before classical fixed point theorems can be applied. We develop in detail all the aspects of how to work with th…
This paper is dedicated to studying the semilinear Schrödinger equation $\left\{\begin{array}{ll}-\Nabla u+V(x)u=f(x, u), \ \ \ \ x\in {\R}^{N},u\in H^{1}({\R}^{N}),\end{array}\right.$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\R^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[σ(-\triangle +V_0)\cap (-\infty, 0)]<0<\inf[σ(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\R^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results.
We prove the existence of strong time-periodic solutions and their asymptotic stability with the total energy of the perturbations decaying to zero at an exponential decay rate as $t \rightarrow \infty$ for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domain. The mathematical model includes a mechanical dissipation and a periodic forcing function of period $T$. In the second part of the paper, we consider a magnetoelastic system in the form of a semilinear initial boundary value problem in a bounded, simply-connected two-dimensional d…
We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.
The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. We use the ``$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}({\Bbb T^{d}})$ threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions…
