Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
In this paper we explore the weak solution of a time-dependent inverse source problem and inverse initial problem for $q$-analogue of the heat equation. As an over-determination condition we have used integral type condition on space-variable (in the case of inverse source problem) and final time condition (in the case of inverse initial problem). Series form of considered problems have been obtained via the method of spectral expansions.
We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.
In this paper, an inverse initial-boundary value problem for the heat equation in three dimensions is studied. Assume that a three-dimensional heat conductive body contains several cavities of strictly convex. In the outside boundary of this body, a single pair of the temperature and heat flux is given as an observation datum for the inverse problem. It is found the minimum length of broken paths connecting arbitrary fixed point in the outside, a point on the boundary of the cavities and a point on the outside boundary in this order, if the minimum path is not line segment.
Let $H:=-Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of $e^{-tH}$ and $\nabla e^{-tH}$ in Lorentz spaces.
The paper considers an inverse source problem for a one-dimensional time-fractional heat equation with the generalized impedance boundary condition. The inverse problem is the time dependent source parameter identification together with the temperature distrubution from the energy measurement. The well-posedness of the inverse problem is shown by applying the Fourier expansion in terms of eigenfunctions of a spectral problem which has the spectral parameter also in the boundary condition and by using the results on Volterra type integral equation with the kernel may have a diagonal singularity…
A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed.
In this paper we explore the weak solutions of the Cauchy problem and an inverse source problem for the heat equation in the quantum calculus, formulated in abstract Hilbert spaces. For this we use the Fourier series expansions. Moreover, we prove the existence, uniqueness and stability of the weak solution of the inverse problem with a final determination condition. We give some examples such as the q-Sturm-Liouville problem, the q-Bessel operator, the q-deformed Hamiltonian, the fractional Sturm-Liouville operator, and the restricted fractional Laplacian, covered by our analysis.
We consider the unitary group for the Schrödinger operator with inverse-square potential. We adapt Combes-Thomas estimates to show that, when restricted to non-radial functions, the operator enjoys much better estimates that mirror those of the Laplacian.
In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrödinger equation with an inverse-square potential.
In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.
We analyze controllability properties for the one-dimensional heat equation with singular inverse-square potential $$ u_t-u_{xx}-\fracμ{x^2}u=0,\;\;\; (x,t)\in(0,1)\times(0,T).$$ For any $μ<1/4$, we prove that the equation is null controllable through a boundary control $f\in H^1(0,T)$ acting at the singularity point $x=0$. This result is obtained employing the moment method by Fattorini and Russell.
The initial-to-final inverse problem for the heat operator
Basque Center for Applied Mathematics（巴斯克应用数学中心）；MaLGa Center, Department of Mathematics, University of Genoa（热那亚大学数学系MaLGa中心）；Ikerbasque (Basque Foundation for Science)（伊克尔巴斯科（巴斯克科学基金会））
In this paper, we consider the energy critical nonlinear Schrödinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the corresponding nonlinear Schrödinger equation without a potential. We investigate time behavior of the radial solutions with such initial data.
We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.
We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocusing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.
We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-Δu+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper, based on the arguments in [K. Ishige and A. Mukai, preprint (arXiv:1709.00809)], we classify the large time behavior of the hot spots of $u$ and reveal the relationship between the behavior of the hot spots and the harmonic functions for $-Δ+V$.
In this paper, the Choquard equation with an inverse-square potential and both focusing and defocusing nonlinearities in the energy-subcritical regime is investigated. For all the cases, the local well-posedness result in $H^1(\mathbb{R}^N)$ is established. Moreover, the global existence result for arbitrary initial values is proved in the defocusing case while a global existence/blowup dichotomy below the ground state is established in the focusing case.
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold.
We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^αu,\] where $\mathcal{L}_a=-Δ+ a |x|^{-2}.$ We establish some fixed-time decay estimate for $e^{-t\mathcal{L}_a}$ associated with inhomogeneous nonlinearity $|\cdot|^{-b}$ in Lebesgue spaces. We then develop local theory in $L^q-$ scaling critical and super-critical regime and small data global well-posedness in critical Lebegue spaces. We further study asymptotic behaviour of global solutions by using self-similar solutions, provided the initial data satisfies cert…
In this paper, we consider radial standing waves to a nonlinear Klein-Gordon equation with a repulsive inverse-square potential. It is known that existence of a "radial" ground state to the stationary problem of the nonlinear Klein-Gordon equation. Here, the "radial" ground state is a solution with the least energy among radial solutions to the stationary problem. We deal with stability and instability of the standing wave for the "radial" ground state.
In this paper, we study the long time behavior of the solution of nonlinear Schrödinger equation with a singular potential. We prove scattering below the ground state for the radial NLS with inverse-square potential in dimension two $$iu_t+Δu-\frac{a u}{|x|^2}= -|u|^pu$$ when $2<p<\infty$ and $a>0$. This work extends the result in [13, 14, 16] to dimension 2D. The key point is a modified version of Arora-Dodson-Murphy's approach [2].
We investigate the properties of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground states, we establish the orbital stability of standing waves for mass sub-critical nonlinearity. Owing to the non-compactness and the absence of translational invariance of the problem, we apply a profile decomposition argument. We obtain convergent minimizing sequences by comparing the problem to the problem at "infinity" (i.e., the equation without inverse s…
Let $\mathcal{L}_a$ be a Schrödinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.
We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\leq d \leq 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.
This article is devoted to the analysis of control properties for a heat equation with singular potential $μ/δ^2$, defined on a bounded $C^2$ domain $Ω\subset\mathbb{R}^N$, where $δ$ is the distance to the boundary function. More precisely, we show that for any $μ\leq 1/4$ the system is exactly null controllable using a distributed control located in any open subset of $Ω$, while for $μ>1/4$ there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.
We consider the final state problem for the inhomogeneous nonlinear Schrödinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which converges to the profile. As a consequence, the existence of modified wave operators for localized small scattering data is obtained. We also study the same problem for the case with the critical inverse-square potential under the radial symmetry. In particular, we construct the modified wave operators for the …
The purpose of this note is to prove global-in-time smoothing effects for the Schrödinger equation with potentials exhibiting critical singularity. A typical example of admissible potentials is the inverse-square potential $a|x|^{-2}$ with $a>-(n-2)^2/4$. This particularly gives an affirmative answer to a question raised by Bui-D'Ancona-Duong-Li-Ly [4]. The proof employs a uniform resolvent estimate proved by Barceló-Vega-Zubeldia [1] and an abstract perturbation method by Bouclet-Mizutani [3].
In this paper, we utilize the method in Dodson-Murphy [4] to establish the radial scattering result for the focusing nonlinear Schrödinger equation with inverse square potential $i\pa_tu-\la u=-|u|^{p-1}u$ in the energy space $H^1_a(\R^d)$ in dimensions $d\geq3$, which extends the result of [10,11] to higher dimensions cases but with radial initial data. The new ingredient is to establish the dispersive estimate for radial function and overcome the weak dispersive estimate when $a<0$.
We study the $L^p$-theory for the Schrödinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with those defined via $(-Δ)^{s/2}$. We consider all regularities $0<s<2$. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator $\mathcal L_a$.
In this paper, we consider the scattering theory of the radial solution to focusing energy-subcritical Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$ using the method from \cite{Dodson2016}. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when $a<0$, using the dispersive estimate established by \cite{zheng…
We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schrödinger equation with inverse-square potential \[ i\partial_t u + Δu + c|x|^{-2} u = - |u|^α u, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $u: \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{C}$, $c\ne 0$ satisfies $c<λ(d):=\left(\frac{d-2}{2}\right)^2$ and $\frac{4}{d} <α<\frac{4}{d-2}$. This result extends a recent result of Bensouilah-Dinh-Zhu [{\it On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse…
In this paper, we establish the soliton resolution for the energy critical wave equation with inverse square potential in the radial case and in all dimensions $N\geq3$. The structure of the radial linear operator $\mathcal{L}_a :=-Δ+\frac{a}{|x|^2}=A^*A$, is essential for the channel of energy, where $A$ is a first order differential operator and $A^*$ is its adjoint operator. Modulation and analysis of the multi-solitons are performed in the function spaces $\dot{H}^1_a(\Bbb R^N)\times L^2(\Bbb R^N)$ associated with $\mathcal{L}_a$.
