We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem:
\begin{align*} \left\{\begin{array}{l l} { d(-Δ)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $Ω,$ }
{ \mathcal{N}_{s}u=0 } \text{ in $\mathbb{R}^{n}\setminus \overlineΩ,$}
{u>0} \text{ in $Ω,$} \end{array} \right.\end{align*} where $Ω\subset \mathbb{R}^{n}$ is a bounded domain of class $C^{1,1}$, $1<p<\frac{n+s}{n-s},\,n> \max \left\{1, 2s \right\}, 0<s<1,\,d>0$ and $\mathcal{N}_{s}u$ is the nonlocal Neumann derivative. We show that for small $d,$ the least energy solutions $u_d$ of the above problem achieves $L^{\infty}$ bound independent of $d.$ Using this together with suitable $L^{r}$-estimates on $u_d,$ we show that least energy solution $u_d$ achieve maximum on the boundary of $Ω$ for $d$ sufficiently small.
The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a\neq 0$, on a Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and the coefficient $a$ up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.
In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time $T$ if and only if its energy is negative at some time before $T$. The proof of this result is based on a Gamma-convergence technique.
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $Δ^{\fracα{2}} u=ρ(x)\,φ(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
In this paper, we establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
In this paper, we consider an inverse problem to determine a semilinear term of a parabolic equation from a single boundary measurement of Neumann type. For this problem, a reconstruction algorithm is established by the spectral representation for the fundamental solution of heat equation with homogeneous Neumann boundary condition and the Whitney extension theorem.
Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning the non-existence of positive solutions in the half space. The regularity of solutions are also investigated.
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
The semilinear space-time fractional Schrödinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.
An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.
In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation with Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional. To prove our result, we first derive Lr-Lq linear estimates for the solution to the inhomogeneous linear Cauchy problem and then we apply a contraction argument.
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 study the Cauchy problem for the semilinear fractional heat equation $u_{t}=\triangle^{α/2}u+f(u)$ with non-negative initial value $u_{0}\in L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For $f$ satisfying the Osgood-type condition $\int_{1}^{\infty}\frac{ds}{f(s)}=\infty$, we show that there exist initial conditions such that the equation has no local solution in $L^{1}_{loc}(\mathbb{R}^{n})$.
We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and the derivative of the Dirichlet-to-Neumann map hold. Based on the strong monotonicity relations, we study a constructive global uniqueness for coefficients and inclusion detection for the fractional Calderón type inverse problem. Meanwhile, we can also derive the Lipschitz stability with finitely many measurements. The results hold for any $n\geq 1$.
We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.
We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-Δ)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.
Let $0<α<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{α/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.
For the following semilinear equation with Hilfer- Hadamard fractional derivative \begin{equation*} \mathcal{D}^{α_1,β}_{a^+} u-Δ\mathcal{D}^{α_2,β}_{a^+} u-Δu =\vert u\vert^p, \qquad t>a>0, \qquad x\inΩ, \end{equation*} where $Ω\subset \mathbb{R}^N$ $(N\geqslant 1)$, $p>1$, $0<α_{2}<α_{1}<1$ and $0<β<1$. $\mathcal{D}^{α_i,β}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $α_i$ and of type $β$, we establish the necessary conditions for the existence of global solutions.
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, with $N\geq 5$, $a>0$, $α\geq 0$ and $2^*=\frac{2N}{N-2}$. We show that the the exponent $q=\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem $$ \left\{\begin{array}{ll} -Δu+au=u^{2^*-1}-αu^{q-1}&\mbox{in}\ Ω,\\ u>0&\mbox{in}\ Ω,\\ \frac{\partial u}{\partialν}=0&\mbox{on}\ \partialΩ. \end{array}\right. $$ Namely, we prove that when $q=\frac{2(N-1)}{N-2}$ there exists an $α_{0}>0$ such that the problem has a least energy solution if $α<α_{0}$ and has no leas…
We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.
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.).
In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces en…
We consider non-local in time semilinear subdiffusion equations on a bounded domain, where the kernel in the integro-differential operator belongs to a large class, which covers many relevant cases from physics applications, in particular the important case of fractional dynamics. The elliptic operator in the equation is given in divergence form with bounded measurable coefficients. We prove a well-posedness result in the setting of bounded weak solutions and study the stability and instability of the zero function in the special case where the nonlinearity vanishes at 0. We also establish a b…
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-Δ)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient $a(x,u)$ can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient ca…
For $γ>0$, we are interested in blow up solutions $u\in C^+(B)$ of the fractional problem in the unit ball $B$ \begin{equation}\label{2nov} \left\{\begin{array} {rcll} Δ^{\fracα{2}} u &=& u^γ&\ \text{in }B\\ u &=& 0&\ \text{in }B^c.\end{array}\right. \end{equation} We distinguish particularly two orders of singularity at the boundary: solutions exploding at the same rate than $δ^{1-\fracα{2}}$ ($δ$ denotes the Euclidean distance) and those higher singular than $δ^{1-\fracα{2}}.$ As a consequence, it will be shown that the classical Keller-Osserman condition can not be readopted in the fraction…
We study the existence of solutions to the fractional elliptic equation (E1) $(-Δ)^αu+εg(|\nabla u|)=ν$ in a bounded regular domain $Ω$ of $\R^N (N\ge2)$, subject to the condition (E2) $u=0$ in $Ω^c$, where $ε=1$ or $-1$, $(-Δ)^α$ denotes the fractional Laplacian with $α\in(1/2,1)$, $ν$ is a Radon measure and $g:\R_+\mapsto\R_+$ is a continuous function. We prove the existence of weak solutions for problem (E1)-(E2) when $g$ is subcritical. Furthermore, the asymptotic behavior and uniqueness of solutions are described when $ν$ is Dirac mass, $g(s)=s^p$, $p\geq 1$ and $ε=1$.
Let $Ω$ be a $\mathcal C^2$-bounded domain of $\mathbb R^d$, $d=2,3$, and fix $Q=(0,T)\timesΩ$ with $T\in(0,+\infty]$. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation $\partial_t^αu+\mathcal A u=f_b(u)$ in $Q$ where $1<α<2$, $\partial_t^α$ corresponds to the Caputo fractional derivative of order $α$, $\mathcal A$ is an elliptic operator and the nonlinearity $f_b\in \mathcal C^1( \mathbb R)$ satisfies $f_b(0)=0$ and $|f_b'(u)|\leq C|u|^{b-1}$ for some $b>1$. We first provide a definition of local weak solutions of…
We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form \[ \begin{cases} (-Δ)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases} \] where $s \in (0,1)$, $(-Δ)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $ \varepsilon \to 0^+$, i.e. the positive and…
