In the paper, we study the inverse problem with the resonant data of fast decaying potential $V$. We review Froese' construction of the Born's approximation and Neumann series to analyze the growth of scattering determinant. Assuming all the the resonances are given, we deduce the certain inverse uniqueness on $V$ from the Nevanlinna type of representation theorem.
The initial value problem for an evolution equation of type $v' + Av + BKv = f$ is studied, where $A:V_A \to V_A'$ is a monotone, coercive operator and where $B:V_B \to V_B'$ induces an inner product. The Banach space $V_A$ is not required to be embedded in $V_B$ or vice versa. The operator $K$ incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-…
This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison principle for Hamilton-Jacobi equations, we are able to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined.…
In this article, we continue our investigation into the unique continuation properties of real-valued solutions to elliptic equations in the plane. More precisely, we make another step towards proving a quantitative version of Landis' conjecture by establishing unique continuation at infinity estimates for solutions to equations of the form $- Δu + V u = 0$ in $\mathbb{R}^2$, where $V = V_+ - V_-$, $V_+ \in L^\infty$, and $V_-$ is a non-trivial function that exhibits exponential decay at infinity. The main tool in the proof of this theorem is an order of vanishing estimate in combination with …
We consider self-adjoint operators of black-box type which are exponentially close to the free Laplacian near infinity, and prove an exponential bound for the resolvent in a strip away from resonances. Here the resonances are defined as poles of the meromorphic continuation of the resolvent between appropriate exponentially weighted spaces. We then use a local version of the maximum principle to prove that any cluster of real quasimodes generates at least as many resonances, with multiplicity, rapidly converging to the quasimodes.
We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Δ+ V - iεx^2$, $|V(x)|\leq C e^{-2γ|x|}$ converge, as $ ε\to 0+ $, to the poles of the meromorphic continuation of $ ( -Δ+ V -λ^2 )^{-1} $ uniformly on compact subsets of $\textrm{Re}\,λ>0$, $\textrm{Im}\,λ>-γ$, $\argλ> -π/8$.
We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime, uniqueness and global asymptotic stability. These solutions are non-decaying as $|x| \to +\infty$, which leads to ambiguity in the velocity $\vec{R}^\perp θ$. This ambiguity is corrected by imposing $m$-fold rotational symmetry. The self-similar solutions exhibited here lie just beyond the known well-posedness theory and are expected to shed light on potential no…
Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the…
This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time…
We prove that radial, monotonic, superexponentially decaying potentials in R^n, n greater than or equal to 1 odd, are determined by the resonances of the associated semiclassical Schrodinger operator among all superexponentially decaying potentials in R^n.
We study Schrödinger operators on $L^2 (\RR^d)$ and $\ell^2(\ZZ^d)$ with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation a Wegner estimate which is polynomial in the volume of the box and linear in the size of the energy interval holds. We apply the established Wegner est…
We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $Ω$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold $a^{*}>0$ such that minimizers exist for $0<a<a^{*}$ and the minimizer does not exist for any $a>a^{*}$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^*$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, ba…
We study solutions of the focusing energy-critical nonlinear heat equation $u_t = Δu - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating from initial data with energy and $\dot{H}^1-$norm below those of the stationary solution $W$ are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argu…
We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves…
The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. Building on the results of Hamel and Roques '10, in this paper we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the…
We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results …
This paper is concerned with some spreading properties of monostable Lotka--Volterra two-species competition--diffusion systems when the initial values are null or exponentially decaying in a right half-line. Thanks to a careful construction of super-solutions and sub-solutions, we improve previously known results and settle open questions. In particular, we show that if the weaker competitor is also the faster one, then it is able to evade the stronger and slower competitor by invading first into unoccupied territories. The pair of speeds depends on the initial values. If these are null in a …
We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate involving the projections P+ and P- onto the positive and negative frequencies.
We consider the KdV equation $$ \partial_t u +\partial^3_x u +u\partial_x u=0 $$ with quasi-periodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with exponentially decaying Fourier coefficients, of a solution on a small interval of time, the length of which depends on the given data and the frequency vector involved. For a Diophantine frequency vector and for small quasi-periodic data (i.e., when the Fourier coefficients obey $|c(m)| \le \varepsilon \exp(-κ_0 |m|)$ with $\varepsilon > 0$ suffi…
We consider the Dirichlet-to-Neumann map associated to the Schrödinger equation with a potential in a bounded Lipschitz domain in three or more dimensions. We show that the integral of the potential over a two-plane is determined by the Cauchy data of certain exponentially growing solutions on any neighborhood of the intersection of the two-plane with the boundary.
