In this note, we study the semilinear wave equation with power nonlinearity $|u|^p$ on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any $p>1$, under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.
In this paper we establish existence of a solution to a semilinear equation with free boundary conditions on stratified Lie groups. In the process, a monotonicity condition is proved, which is quintessential in establishing the regularity of the solution.
In this paper, we study the semilinear wave equation with lower order terms (damping and mass) and with power type nonlinearity $|u|^p$ on compact Lie groups. We will prove the global in time existence of small data solutions in the evolution energy space without requiring any lower bounds for $p>1$. In our approach, we employ some results from Fourier analysis on compact Lie groups.
In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.
We study the semilinear elliptic inequality $-Δu\geqφ(δ_K(x))f(u)$ in $R^N\setminus K,$ where $φ, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $δ_K(x)={\rm dist}(x,\partial K)$. We obtain optimal conditions in terms of $φ$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.
We study the Lie point symmetries of semilinear Kohn-Laplace equations on the Heisenberg group H^1 and obtain a complete group classification of these equations.
This paper studies initial value problems for semilinear wave equations with spatial weights in one space dimension. The lifespan estimates of classical solutions for compactly supported data are established in all the cases of polynomial weights. The results are classified into two cases according to the total integral of the initial speed.
We study the Cauchy problem for a semilinear heat equation with initial data non-rarefied at $\infty$. Our interest lies in the discussion of the effect of the non-rarefied factors on the life span of solutions, and some sharp estimates on the life span is established.
For a non-local semilinear eigenvalue problem, we prove simplicity and isolation of the first eigenvalue with homogeneous Dirichlet boundary conditions on open sets supporting a suitable compact Sobolev embedding.
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent concentration-compactness Lemma and a characterization of compact embeddings of $D^{1,2}(\rz^N)$ into weighted Lebesgue spaces.
We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via Bessel capacities.
We discuss the notion of criticality of semilinear differential equations and systems, its relations to scaling transformations and the Noether approach to Pokhozhaev's identities. For this purpose we propose a definition for criticality based on the S. Lie symmetry theory. We show that this definition is compatible with the well-known notion of critical exponent by considering various examples. We also review some related recent papers.
In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.
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 convex potentials $W:\R\to [0,\infty)$ vanishing at $0$ and growing sufficiently fast at $\pm\infty$. Given any open set $Ω\subset\R^n$ with Lipschitz and compact boundary, we prove the existence and uniqueness of a solution of $Δu= W'(u)$ in $Ω$, such that $u=+\infty$ or $u=-\infty$ on $\partial Ω$. Moreover, if $\partial Ω$ is the union of two disjoint compact subsets $A^+$ and $A^-$, there also exists a unique solution satisfying $u=+\infty$ on $A^+$ and $u=-\infty$ on $A^-$.
In the paper we prove the existence of probabilistic solutions to systems of the form $-Au=F(x,u)+μ$, where $F$ satisfies a generalized sign condition and $μ$ is a smooth measure. As for $A$ we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on $L^1$. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhe…
Lie-Trotter method for abstract semilinear evolution equations
Universidad Nacional de General Sarmiento（圣巴拿马国立大学）；IMAS - CONICET（CONICET IMAS研究所）；Instituto de Humanidad, Universidad Nacional de General Sarmiento（圣巴拿马国立大学人文研究所）
We prove a Hopf bifurcation theorem in Hilbert spaces for abstract semilinear equations, which improves a classical result by Crandall and Rabinowitz in the case where basic spaces are Hilbert spaces. Actually, our theorem does not need any compactness conditions, which leads to wider applications. In particular, our theorem can be applied to semilinear equations in unbounded domains of $\mathbb{R}^n$.
In this paper we proceed with the multiscale analysis of semilinear damped stochastic wave motions. The analysis is made by combining the well-known sigma convergence method with its stochastic counterpart, associated to some compactness results such as the Prokhorov and Skorokhod theorems. We derive the equivalent model, which is of the same type as the micro-model.
This paper is concerned with existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, precise lower bound estimates of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem.
Symmetry plays a basic role in variational problems (settled e.g. in $\mathbb R^{n}$ or in a more general manifold), for example to deal with the lack of compactness which naturally appear when the problem is invariant under the action of a noncompact group. In $\mathbb R^n$, a compactness result for invariant functions with respect to a subgroup $G$ of $\mathrm{O}(n)$ has been proved under the condition that the $G$ action on $\mathbb R^n$ is compatible, see \cite{willem}. As a first result we generalize this and show here that the compactness is recovered for particular subgroups of the isom…
In this paper, we study the semilinear elliptic equation of the form \begin{eqnarray*} -Δu+a(x)|u|^{p-2}u-b(x)|u|^{q-2}u=0 \end{eqnarray*} on lattice graphs $\mathbb{Z}^{N}$, where $N\geq 2$ and $2\leq p<q<+\infty$. By the Brézis-Lieb lemma and concentration compactness principle, we prove the existence of positive solutions to the above equation with constant coefficients $\bar{a},\bar{b}$ and the decomposition of bounded Palais-Smale sequences for the functional with variable coefficients, which tend to some constants $\bar{a},\bar{b}$ at infinity, respectively.
Starting with approximate solutions of the equation $-Δu=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9], where the linear case was treated. In addition, we deduce some compactness properties of concentration sets (e.g. moving interfaces) when dealing with singular limits of certain nonlinear wave equations.
We consider a functional semilinear Rayleigh-Stokes equation involving fractional derivative. Our aim is to analyze some circumstances, in those the global solvability and some results on asymptotic behavior of solutions take place. By establishing a new Halanay type inequality, we show the dissipativity and asymptotic stability of solutions to our problem. In addition, we prove the existence of a compact set of decay solutions by using local estimates and fixed point arguments.
We consider the Cauchy problem for systems of semilinear wave equations in two space dimensions. We present a structural condition on the nonlinearity under which the energy decreases to zero as time tends to infinity if the Cauchy data are sufficiently small, smooth and compactly-supported.
We will show that in $\RR^{2+1}$ semilinear wave equations of the form $-\Box u = u Q(\del u; \del u)$ possess global-in-time solutions if the null condition on $Q(\del u; \del u)$ is assumed. As a consequence, we also provide a new proof, after \cite{Wong}, on the small data global solutions to the wave map equation in $\RR^{2+1}$ and no compactness assumptions on the initial data are needed.
In this paper, using the subvariant functional method due to Favard \cite{Favard}, we prove the existence of aunique compact almost automorphic solution for a class of semilinear evolution equations in Banach spaces. More specifically, we improve the assumptions in \cite{CieuEzz}, we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order $1\leq p <\infty$) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). We distinguish two cases, $ p=1 $ and $ p>1$. Moreover, we propose to study a class …
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 prove the existence of solutions $u$ in $H^1(\mathbb{R}^N,\mathbb{R}^M)$ of the following strongly coupled semilinear system of second order elliptic PDEs on $\mathbb{R}^N$ \[ \mathcal{P}[u] = f(x,u,\nabla u), \quad x\in \mathbb{R}^N, \] whith pointwise constraints. We present the construction of the suitable topoligical degree which allows us to solve the above system on bounded domains. The key step in the proof consists of showing that the sequence of solutions of the truncated system is compact in $H^1$ by the use of the so-called tail estimates.
{\bf Abstract} \,\,We prove exponential decay of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation with locally distributed damping on bounded domain. One novelty compared to previous results, is to give a checkable condition of the inhomogeneous and anisotropic medias. Another novelty is to establish a framework to study the stability of the damped semilinear inhomogeneous and anisotropic elastic wave equation, which is hard to apply Carleman estimates to deal with. We develop the Morawetz estimates and the compactness-uniqueness arguments for the sem…
In this paper we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\frlap u=u^{\frac{Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $\frlap$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.
In this paper we consider the critical exponent problem for the semilinear wave equation with space-time dependent damping. When the damping is effective, it is expected that the critical exponent agrees with that of only space dependent coefficient case. We shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent. Moreover, we do not assume that the data are compactly supported. However, it is still open whether there exists a blow-up solution if the power of nonlinearity is smaller than the expected exponent.
In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity …
We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter λ. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter λ …
In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold $M$: \begin{equation*} \begin{cases} u_{t}-\mathfrak{L}_{M} u=f(u), \;x\in M, \;t>0, \\u(0,x)=u_{0}(x), \;x\in M, \end{cases} \end{equation*} for $u_{0}\geq 0$, where $\mathfrak{L}_{M}$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\mathbb G$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_{0}\not\equiv 0$, blow up in finite time if and only if $1<p\leq p_…
We study the existence of a maximal solution of $-\Gd u+g(u)=f(x)$ in a domain $\Gw\subset \BBR^N$ with compact boundary, assuming that $f\in (L^1_{loc}(\Gw))_+$ and that $g$ is nondecreasing, $g(0)\geq 0$ and $g$ satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classical $C_{1,2}$ Wiener criterion then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition we discuss the question of uniqueness of large solutions.
In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$ u_t-Δu=|u|^{p-1}u, \ \ in \ \ Ω\times (0,T) $$ and $u=0$ on the boundary $\partialΩ\times [0,T)$ and $u=ϕ$ at $t=0$, where $Ω\subset R^n$ is a compact $C^1$ domain, $1<p\leq p_S$ is a fixed constant, and $ϕ\in C^2_0(Ω)$ is a given smooth function. Introducing new idea, we show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $ϕ\in W$, there is a global positive solution $u(t)\in \tilde{W}$ with $h^1$ omega limit $\{0\}$ and for $ϕ\in \tilde{Z}$, the solution blows up at finite…
We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-Δu+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of wavelength $\sim h$ and amplitude $\sim 1$. They propagate in a regime where the non-linearity affects the subprincipal but not the principal term, except for the zeroth harmonics. We measure the transmitted wave when it exits $\text{supp}_x f$. We show that one can recover $f(x,u)$ when it is an odd function of $u$, and we can recover $α(x)$ when $f(x,u)=α(x)u^{2m}…
We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, $0<α<β<1$, of the involved nonlinearites. Suitable assumptions are made on the spatial domain $Ω$ where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain $Ω$. Our results also clarify some previous works in the literature. The main techniques of proof are a …
We establish new bounds of the Sobolev norms of solutions of semilinear wave equations for data lying in the Hs, s<1, closure of compactly supported data inside a ball of radius R, with R a fixed and positive number. In order to do that we perform an analysis in the neighborhood of the cone, using an almost Shatah-Struwe estimate, an almost conservation law and some estimates for localized functions: this allows to prove a decay estimate and establish a low frequency estimate of the position of the solution. Then, in order to establish a high frequency estimate of the position and an estimate …
We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.
In this paper, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and diffusion terms (resp. in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameter…
In the past few decades, much attention has been paid to the bubbling problem for semilinear Neumann elliptic equation with the critical and subcritical polynomial nonlinearity, much less is known if the polynomial nonlinearity is replaced by the exponential nonlinearity. In this paper, we consider the following semilinear Neumann elliptic problem with the Trudinger-Moser exponential growth: \begin{equation*}\begin{cases} -dΔu_d+u_d=u_d(e^{u^2_{d}}-1)\ \ \mbox{in}\ Ω,\\ \frac{\partial u_d}{\partialν}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial Ω,\\ \end{cases}\end{equa…
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry, results for solutions to overdetermined problems on Riemannian manifolds of nonconstant curvature.
